[2010]57R58; 57R55; 57M27

We consider the set of connected surfaces in the 4-ball that bound a fixed knot in the 3-sphere.
We define the stabilization distance between two surfaces as the minimal $g$ such that
we can get from one to the other using stabilizations and destabilizations through surfaces of genus at most $g$.
Similarly, we obtain the double point distance between two surfaces of the same genus
by minimizing the maximal number of  double points appearing in a regular homotopy connecting them.

To many of the concordance invariants defined using Heegaard Floer homology,
we construct an analogous invariant for a pair of surfaces. We show
that these give lower bounds on the stabilization distance and the double point distance.
We compute our invariants for some pairs of deform-spun slice disks
by proving a trace formula on the full infinity knot Floer complex, and by
determining the action on knot Floer homology of an automorphism of the connected sum of
a knot with itself that swaps the two summands.

§ INTRODUCTION

It is a classical result of Wall <cit.> that if two smooth, simply-connected 4-manifolds
are homeomorphic, then they become diffeomorphic after taking connected sums with
some number of copies of $\Sphere^2 \times \Sphere^2$. It is an open conjecture
whether a single copy of $\Sphere^2 \times \Sphere^2$ always suffices.
Since $(\Sphere^2 \times \Sphere^2) \# \CP^2$ and $\CP^2 \# \CP^2 \# \overline{\CP}^2$ are diffeomorphic,
one could alternatively take connected sums with copies of $\CP^2$ and $\overline{\CP}^2$.

Analogously, Baykur and Sunukjian <cit.>
considered the stabilization operation on smooth, oriented surfaces in 4-manifolds
where one attaches an embedded 1-handle to the surface.
When the 1-handle is unknotted, this does not change the fundamental group of the surface complement.
They asked the following question: If two surfaces are topologically isotopic,
then do they become smoothly isotopic after a single unknotted 1-handle stabilization?
They verified this
for most known constructions of pairs of exotic surfaces, such as rim surgery.
Stabilizations of surfaces and 4-manifolds are related:
An unknotted 1-handle stabilization of a surface corresponds to a stabilization
of its branched double cover with $\Sphere^2 \times \Sphere^2$.

It is apparent from the above discussion that it is of great interest
to construct invariants that provide lower bounds on the number of
1-handle stabilizations required to make two surfaces isotopic.
Using a generalization of 1-handle stabilization, we endow the set of smooth, connected, oriented,
and properly embedded surfaces in $B^4$ that bound a knot $K$ with a type of metric
that we call the stabilization distance. The stabilization distance between two surfaces
bounds from below the number of 1-handle stabilizations required to make them isotopic.
Using the link Floer TQFT, we define several invariants of pairs of surfaces bounding $K$ that
give lower bounds on the stabilization distance. We compute these invariants for certain pairs
of slice disks arising from deform-spinning, and observe they often give non-trivial lower bounds.

Throughout this paper, we work in the smooth category, and all manifolds
are assumed to be oriented, unless otherwise stated.

§.§ Metric filtrations on the set of surfaces bounding a knot

In Definition <ref>, we introduce a very general type of stabilization operation for surfaces in 4-manifolds
that extends the 1-handle stabilization considered by Baykur and Sunukjian.
Let $S$ be a properly embedded surface in a 4-manifold $W$.
To obtain the stabilization of $S$, we choose a 4-ball $B \subset \Int(W)$ such that $B \cap S$
is a collection of disks that can be isotoped into $\d B^4$ relative to their boundaries.
In particular, $\d B \cap S$ is an unlink. We then replace $S \cap B$
with a properly embedded surface $S_0 \subset B$ such that $\d S_0 = \d B \cap S$.
Any two surfaces in the same relative homology class can be related by a finite sequence of
such stabilizations and destabilizations (in fact, 1-handle stabilizations suffice).

Let $K$ be a knot in $\Sphere^3$. We denote by $\Surf(K)$ the set of isotopy classes of connected, properly embedded surfaces
in $B^4$ bounding $K$. In Definition <ref>, we introduce the stabilization distance
$\mu_{\st}(S, S')$ of a pair of surfaces $S$, $S' \in \Surf(K)$
to be the minimum of

\[
\max\{g(S_1), \dots, g(S_k) \}
\]

over sequences of connected, properly embedded surfaces $S_1$, …, $S_k$ in $\Surf(K)$
connecting $S$ and $S'$ such that consecutive surfaces are related by a stabilization or a destabilization.
Furthermore, if $K$ is slice and $S \in \Surf(K)$, then
we define the destabilizing genus $g_{\dest}(S)$ to be the stabilization distance of $S$ from
the subspace of slice disks.

We also define the M-distance function $M_{(S, S')} \colon [0,2] \to \R^{\ge 0}$
of a pair of surfaces $S$, $S' \in \Surf(K)$, which is similar to the stabilization distance,
but where the stabilization operation is allowed to change
the ambient 4-manifold. Instead of changing the surface in a 4-ball,
one can glue in a pair $(X_0, S_0)$, where $\d X_0 = \Sphere^3$ and $\d S_0$ is an unlink,
and $b_2^+(X_0) = b_1(X_0) = 0$.
The $M$-degree of a pair $(W, S)$, where $W$ is a compact 4-manifold and $S$ is a
properly embedded surface, is defined in Section <ref>. It is a function
on $[0,2]$ that measures not only the genus of $S$,
but also a homological quantity depending on $[S] \in H_2(W)$
and the intersection form $Q_W$ of $W$.
The $M$-distance of a pair of surfaces $S$, $S' \in \Surf(K)$
minimizes the maximal $M$-degree along sequences $(W_1, S_1) \dots, (W_n, S_n)$
connecting $(B^4, S)$ and $(B^4, S')$ such that $(W_i, S_i)$
and $(W_{i+1}, S_{i+1})$ are related by the above stabilization operation.

For $g \in \N$, let $\Surf_g(K)$ denote the subset of $\Surf(K)$ consisting
of genus $g$ surfaces. If $S$, $S' \in \Surf_g(K)$, then they are regularly homotopic relative to $K$.
We define the double point distance $\mu_{\Sing}(S, S')$ as $\tilde{\mu}_{\Sing}(S,S') + g$, where
$\tilde{\mu}_{\Sing}(S,S')$ is obtained by minimizing half the maximal number of double points
that appear during a regular homotopy from $S$ to $S'$; see Definition <ref>.
When $g(S) \neq g(S')$, we set $\mu_{\Sing}(S, S') = \infty$.

Both $\mu_{\st}$ and $\mu_{\Sing}$ are metric filtrations on $\Surf(K)$;
i.e., they are nonnegative, symmetric, and satisfy the ultrametric inequality

\[
\mu(S, S'') \le \max\{\, \mu(S, S'), \mu(S', S'') \,\}
\]

for any $S$, $S'$, $S'' \in \Surf(K)$; see Section <ref>.
Furthermore, $M_{(S,S')}(t)$ is also a metric filtration for every $t \in [0,2]$.
If $S$, $S' \in \Surf_0(K)$, then $\mu_{\st}(S, S') = 0$ if and only if $S$ and $S'$ are related
by genus zero stabilizations, hence $\Surf_0(K)/\{\text{2-knots}\}$ is an ultrametric space.
However, in general, $\mu_{\st}(S, S) = g(S)$, and if $\mu_{\st}(S, S') = g(S) = g(S')$,
it is possible that $S$ and $S'$ are not related by genus zero stabilizations.

§.§ Lower bounds from Heegaard Floer homology

Our aim is to provide computable lower bounds on the stabilization distance, the double point distance,
and the destabilizing genus using the link Floer TQFT of the second author <cit.>, which extends the TQFT of the
first author <cit.> from $\widehat{\HFL}$ to the full infinity complex $\cCFL^\infty$.
If $\bK = (K, w, z)$ is a doubly-based knot in $\Sphere^3$, then $\cCFL^\infty(\bK)$ is a $\Z\oplus \Z$
filtered chain complex over the two-variable Laurent polynomial ring

\[
\cR^\infty:=\bF_2[U,V,U^{-1},V^{-1}].
\]

We give an overview of  $\cCFL^\infty(\bK)$ in Section <ref>.
There is a variation, denoted $\cCFL^-(\bK)$, corresponding to the subspace
of $\cCFL^\infty(\bK)$ in nonnegative bi-filtration, which is a module over the ring

\[
\cR^-:=\bF_2[U,V].
\]

Knot Floer homology has been used by many authors to provide numerical concordance invariants which provide deep geometric information. Important examples are the knot invariants $\tau$, $\nu$, $V_k$ for $k \in \N$,
and $\Upsilon$, introduced by Ozsváth–Szabó <cit.>,
Rasmussen <cit.>, Hom–Wu <cit.>
and Ozsváth–Stipsicz–Szabó <cit.>, respectively.
Since  $\tau$, $\nu$, $V_k$, and $\Upsilon$ are concordance invariants, they
all vanish when the knot is slice.

It is well known that the knot invariants $\tau$, $\nu$, and $V_k$ give lower bounds on the 4-ball genus $g_4(K)$:

\[
\tau(K) \le \nu(K) \le g_4(K) \text{ and } V_k(K) \le \left\lceil \frac{g_4(K)-k}{2} \right\rceil
\]

whenever $k \le g_4(K)$; see <cit.>*Theorem 1.1,
<cit.>*Section 2, and <cit.>*Theorem 2.3.
The invariant $\Upsilon$ also gives bounds on the 4-ball genus; see
<cit.>*Theorem 1.11. The invariants $\tau$ and $\Upsilon$
satisfy more general versions of the genus bound involving surfaces
in negative definite 4-manifolds $W$ with $b_1(W) = b_2^+(W) = 0$; see
<cit.>*Theorem 1.1 and
<cit.>*Theorem 1.1.

In Section <ref>, we show that, by mirroring their constructions,
we can define secondary versions of all of the above knot invariants for a pair of surfaces
$S$, $S' \in \Surf(K)$ in $B^4$ bounding a knot $K$ in $\Sphere^3$.
We show that these give formally analogous lower bounds on the metric filtrations
$\mu_{\st}$, $\mu_{\Sing}$, and $M$:

Let $S$, $S' \in \Surf(K)$. Then

\[
\tau(S, S') \le \nu(S, S') \le \min\{\, \mu_{\st}(S, S') \text{, } \mu_{\Sing}(S, S') \,\}.
\]

\[
\Upsilon_{(S,S')}(t) \le M_{(S,S')}(t)
\]

for every $t \in [0,2]$.
Finally, for every $k \in \N$,

\[
V_k(S, S') \le \left\lceil \frac{\mu_{\Sing}(S, S') - k}{2} \right \rceil.
\]

If $S$ and $S'$ are disks, then the above inequality is also holds with $\mu_{\st}$ in place of $\mu_{\Sing}$.

The bounds stated in Theorem <ref> are proven separately throughout the paper;
see Theorems <ref>, <ref>, <ref>,
<ref>, and <ref>,
as well as Proposition <ref>. In Section <ref>, we introduce a novel
invariant $\kappa(S, S')$ that does not have an analogue for knots,
and which only gives a lower bound on $\mu_{\Sing}(S, S')$; see Theorem <ref>.

Using the link Floer TQFT, we also introduce another integer invariant
$\cI(S)$ of a surface $S$ which has $g(S) > 0$ and bounds a slice knot $K$;
see Definition <ref>. In Theorem <ref>, we prove that $\cI(S)$ bounds the stabilization distance
between $S$ and the subspace of slice disks:

\[
\cI(S) \le g_{\dest}(S).
\]

However, $\cI(S)$ is in general difficult to compute,
as it involves determining the link cobordism maps for infinitely many
decorations on the surface $S$. Giving a lower bound on $\cI(S)$ is theoretically feasible,
since it only involves finding two decorations on $S$ satisfying a
simple condition, whose induced cobordism maps disagree.
We define an analogue of $\kappa(S, S')$ for a single surface, $\kappa_0(S)$, which gives a computable
lower bound on $g_{\dest}(S)$; see Theorem <ref>.

The invariants $\tau$, $\kappa$, and $\kappa_0$ can all be derived from $\Upsilon$.
As opposed to the case of knots, $\Upsilon_{(S, S')}(t)$ is not a symmetric function.
We will see in Theorem <ref> that, for all $t\in [0,2]$ sufficiently close to 0, we have

\[
\Upsilon_{(S,S')}(t)= \tau(S,S') \cdot t.
\]

However, for $t$ sufficiently close to $2$, we have

\[
\Upsilon_{(S,S')}(t)=\begin{cases} (\kappa_0(S)-g(S))\cdot (2-t)+ g(S)\cdot t& \text{ if } g(S)>g(S'),\\
(\kappa(S,S')-g(S))\cdot (2-t)+g(S)\cdot t& \text{ if } g(S)=g(S').
\end{cases}
\]

We now review the construction of the invariants.
If $S \in \Surf(K)$, then it can be viewed as a
link cobordisms from $\emptyset$ to $(\Sphere^3, K)$.
If we decorate it such that the type-$\ws$ region is a bigon, then it induces a filtered chain map

\[
\ve{t}_{S, \zs}^\infty \colon \cR^\infty \to \cCFL^\infty(\bK),
\]

well-defined up to filtered chain homotopy; see Section <ref>.
If instead the type-$\zs$ region is a bigon, we obtain a map $\ve{t}_{S, \ws}^\infty$.
We call $\ve{t}_{S, \zs}^\infty$ and $\ve{t}_{S, \ws}^\infty$ the extremal principal invariants
of the surface $S$.

Given surfaces $S$, $S' \in \Surf(K)$, the invariants

\[
\tau(S, S') \text{, } \nu(S, S') \text{, } V_k(S, S') \text{, and } \Upsilon_{S, S'}(t)
\]

are all extracted from the pair of maps $(\ve{t}^\infty_{S, \zs}, \ve{t}^\infty_{S', \zs})$
by algebraically mirroring the construction of their knot invariant counterparts.
Hence, we think of our invariants as secondary versions of the knot invariants.
The invariant $\kappa_0(S)$ is derived from $\ve{t}_{S, \ws}^\infty$, and we obtain
$\kappa(S, S')$ from the pair $(\ve{t}^\infty_{S, \ws}, \ve{t}^\infty_{S', \ws})$.

For example, to obtain $\tau$, we set $U = 0$ and $V = 1$ in the $\cR$-module
$\cCFL^-(\bK)$, and obtain the $\Z$-filtered complex $\hCFKfz(\bK)$ whose homology is $\HFh(\Sphere^3) \cong \bF_2$.
Given a pair of surfaces $S$, $S' \in \Surf(K)$,
the elements $\ve{t}_{S, \zs}^-(1)$ and $\ve{t}_{S', \zs}^-(1)$ of $\cCFL^-(\bK)$ give rise to
elements $\hat{t}_{S, \zs}(1)$ and $\hat{t}_{S', \zs}(1)$ of $\hCFKfz(\bK)$. We define the invariant

\[
\tau(S, S') := \min\{ n \ge \max\{g, g'\} : \left[\hat{t}_{S, \zs}(1) \right] =
\left[\hat{t}_{S', \zs}(1)\right] \in H_*(\hCFKfz_n(\bK))\},
\]

where $\hCFKfz_n(\bK)$ is the part of $\hCFKfz(\bK)$ in filtration at most $n$.

§.§ Computing the invariants for deform-spun disks using the trace formula

In Section <ref>, we compute the secondary invariants for
several pairs of deform-spun slice disks, using the computer algebra software
SageMath <cit.>. We exhibit several
examples where the lower bound on the stabilization distance is 2 or 3. We
note that, in our examples, the pairs of slice disks are not
topologically isotopic relative to their boundaries.

Let $K$ be a knot in $\Sphere^3$, and suppose that the 3-ball $B$ intersects $K$ in an unknotted arc.
Then $(\Sphere^3 \setminus \Int(B), K \setminus \Int(B))$ is a ball-arc pair $(B^3, a)$.
Suppose that we are given an isotopy $\phi_t$ of $\Sphere^3$ that is the identity on $B$, and such
that $\phi_1(K) = K$. Then the deform-spun slice disk $D_{K, \phi} \subset B^4$ is defined by taking

\[
\bigcup_{t \in I} \{t\} \times \phi_t(a) \subset I \times B^3,
\]

and rounding the corners along $\{0, 1\} \times \d B^3$.
This is a slice disk of $-K \# K$. The isotopy class of $D_{K, \phi}$, relative to $-K \# K$,
only depends on the diffeomorphism $\phi_1$, so we will write $D_{K, \phi_1}$ instead of $D_{K, \phi}$.

In this paper, we consider three automorphisms of a ball-arc pair.
The first is the roll-spinning automorphism $r$, which
corresponds to a positive Dehn twist in the longitudinal direction of the knot $K$.
The automorphism $r$ is supported in a neighborhood of $K$.
The second automorphism is twist-spinning, which is similar to roll-spinning,
but instead twists in the meridional direction.
The third automorphism $\ve{R}^\pi$ that we consider is specific to
knots of the form $K \# K$. The summand-swapping automorphism $\ve{R}^\pi$ of $K \# K$ is the
composition of an isometry of $\R^3$ that swaps the two copies of $K$, and a
half Dehn twist that ensures $\ve{R}^\pi$ fixes $K \# K$ pointwise.

We show that, if $K$ is a knot in $\Sphere^3$, then the roll-spun and twist-spun slice disks $D_{K, r}$,
$D_{K, t} \in \Surf_0(-K \# K)$ satisfy

\[
\mu_{st}(D_{K, r}, D_{K, t}) \le 2;
\]

see Proposition <ref>. We conjecture that the upper bound can be improved to $1$.

We obtain lower bounds on the stabilization distance and the double point distance using our secondary invariants.
For this end, we first compute the extremal principal invariant
$\ve{t}_{D_{K,d}}^\infty(1) \in \cCFL^\infty( -K \# K)$ of a deform-spun slice disk:

Let $D_{K, d}$ be a slice disk of the doubly-based knot $-\bK \# \bK = (-K \# K, w, z)$, obtained by deform-spinning
a doubly-based knot $\bK = (K, w, z)$ in $\Sphere^3$ using an automorphism $d$ of $(\Sphere^3, K)$.
If we write $C := \cCFL^\infty(\bK)$, then

\[
E \circ \ve{t}_{D_{K, d}}^\infty \simeq (\id \otimes d_*) \circ \cotr \in
\Hom_{\cR^\infty}(\cR^\infty, C^\vee \otimes C),
\]

where  $E \colon \cCFL^\infty(-\bK \# \bK) \to C^\vee \otimes C$ is the chain homotopy equivalence
induced by a pair-of-pants link cobordism.

Our proof of Theorem <ref> uses a much more general
trace formula for the full link Floer TQFT, Theorem <ref>,
extending a result from our earlier work <cit.> for sutured Floer
homology, as well as a result of the second author for the graph TQFT
<cit.>*Theorem 1.6.

Hence, to compute the secondary invariants of deform-spun slice disks, it
remains to determine the induced diffeomorphism map $d_*$ in specific examples.
The map $r_*$ induced by the basepoint moving diffeomorphism is well-known,
and is given by the formula

\[
r_*\simeq \id+\Phi\circ \Psi,
\]

which was proven on $\HFK^-(\bK)$ by Sarkar <cit.>, and
extended to $\cCFL^\infty(\bK)$ by the second author <cit.>. The maps
$\Phi$ and $\Psi$ are two special endomorphisms of knot Floer homology, which
we describe in Section <ref>. It turns out that $\tau(D_{K,\id}, D_{K,r}) \le 1$
for the canonical deform-spun slice disks $D_{K,\id}$ and the roll-spun slice disks $D_{K,r}$;
see Proposition <ref> .

To provide more interesting examples beyond roll-spinning, we compute a
formula for the map induced by the summand-swapping automorphism $\ve{R}^\pi$ on
$\cCFL^\infty(\bK \# \bK)$ in Theorem <ref>. We prove
that there is a chain homotopy equivalence identifying
$\cCFL^\infty(\bK \# \bK)$ and $\cCFL^\infty(\bK) \otimes \cCFL^\infty(\bK)$ under which

\[
\ve{R}^\pi_* \simeq  \Sw \circ \left(\id \otimes \id + (\Phi \circ \Psi) \otimes \id +
\Phi \otimes \Psi \right),
\]

where $\Sw$ is the map that swaps the two tensor factors.
To our knowledge, after Sarkar's formula, this is the only other known formula
for a mapping class group action on the knot Floer complexes of a family of knots in $\Sphere^3$.

§.§ Acknowledgements

We would like to thank David Gabai and Maggie Miller for helpful conversations.
The first author was supported by a Royal Society Research Fellowship,
and the second author by an NSF Postdoctoral Research Fellowship (DMS-1703685).
This project has received funding from the European Research Council (ERC)
under the European Union's Horizon 2020 research and innovation programme
(grant agreement No 674978).

§ THE STABILIZATION DISTANCE OF A PAIR OF SURFACES

In this section, we first review deform-spun slice disks.
We then introduce the notion of a metric filtration,
which is a generalization of an ultrametric,
but where the distance of a point from itself can be nonzero.
We proceed to define the stabilization distance of a pair of surfaces bounding a knot, which
is an instance of a metric filtration. Finally, we show that the stabilization distance
of a 1-roll-spun and a 1-twist-spun slice disk is always at most two.

§.§ Deform-spun slice disks

We review the definitions of deform-spun slice disks <cit.>.
Originally, Litherland <cit.> introduced deform-spinning to construct
2-knots in $\R^4$, generalizing twist-spinning, due to Zeeman <cit.>,
and roll-spinning, due to Fox <cit.>. An analogous construction can be used to obtain
slice disks in $B^4$. The following is <cit.>:

Let $K$ be a knot in $\Sphere^3 = -B^3 \cup B^3$ such that $K$ intersects $-B^3$ in an unknotted arc,
and write $a = K \cap B^3$.
Furthermore, let $\phi \colon I \times \Sphere^3 \to \Sphere^3$ be an isotopy of $\Sphere^3$ such that
$\phi_0 = \Id_{\Sphere^3}$, $\phi_t|_{-B^3} = \Id_{-B^3}$ for every $t \in I$, and $\phi_1(K) = K$.
The deform-spun slice disk $D_{K, \phi} \subset B^4$ is defined by taking

\[
\bigcup_{t \in I} \{t\} \times \phi_t(a) \subset I \times B^3,
\]

and rounding the corners along $\{0, 1\} \times \d B^3$.
The surface $D_{K, \phi}$ is a slice disk for $-K \# K$, where $-K$ stands for $(-\Sphere^3, -K)$.

Note that $\d \R^4_+ = \R^3 = \R^3_+ \cup \R^3_-$.
Intuitively, we consider the arc $a$ in $\R^3_-$, which we
rotate about the plane $\R^2 = \R^3_+ \cap \R^3_-$ in $\R^4_+$,
while applying the isotopy $\phi$, until we reach $\R^3_+$.
The following result is <cit.>, which immediately follows from
the work of Hatcher <cit.>.

Let $d$ be an automorphism of $(\Sphere^3, K)$ such that $d|_{-B^3} = \Id_{-B^3}$.
Then there is an isotopy $\phi \colon I \times \Sphere^3 \to \Sphere^3$
as in Definition <ref>, such that $\phi_1 = d$. Furthermore,
the isotopy class of the deform-spun disk $D_{K, \phi}$ only depends on $d$,
which we denote by $D_{K, d}$.

We now recall the definition of roll-spinning, based on the description of
Litherland <cit.>*Example 2.2. The following is <cit.>:

Let $K$ be a knot in $\Sphere^3$. Choose a tubular neighborhood $N(K)$ of $K$ as well as an identification
$N(K) \approx K \times B^2$ which induces the Seifert framing of $K$.
Let $X = \Sphere^3 \setminus \Int(N(K))$ be the knot exterior.
Furthermore, let $\d X \times I$ be a collar of $\d X$ in $X$.
We identify $K$ with $\R/\Z$. Choose a smooth monotonic function $\varphi \colon \R \to I$
such that $\varphi(s) = 0$ for $s \le 0$ and $\varphi(s) = 1$ for $s \ge 1$.
We define the rolling diffeomorphism $r \colon (\Sphere^3, K) \to (\Sphere^3, K)$ by the formula

\[
r(\bar{x}, \bar{\theta}, s) = \left(\overline{x + \varphi(s)}, \bar{\theta}, s \right)
\text{ for } (\bar{x}, \bar{\theta}, s) \in K \times \d B^2 \times I \approx \d X \times I,
\]

and let $r(p) = p$ for $p \in \Sphere^3 \setminus (\d X \times I)$.

Similarly, we define the twisting diffeomorphism $t\colon (\Sphere^3,K)\to (\Sphere^3,K)$ by the formula

\[
t(\bar{x}, \bar{\theta}, s) = \left(\overline{x}, \overline{\theta + \varphi(s)}, s \right)
\text{ for } (\bar{x}, \bar{\theta}, s) \in K \times \d B^2 \times I \approx \d X \times I,
\]

and let $t(p) = p$ for $p \in \Sphere^3 \setminus (\d X \times I)$.

Let $B \subset N(K)$ be an open ball that intersects $K$ in an arc.
Then $(\Sphere^3 \setminus B, K \setminus B)$ is diffeomorphic to a ball-arc pair
$(B^3, a)$, and $r|_B = \id_B$. We define the $(k,l)$-twist-roll-spin of $K$ to be $D_{K, t^kr^l}$.

Note that $D_{K, \id}$ is simply the spun slice disk of $K$, obtained using the identity deformation.
We will call this the canonical slice disk of $-K \# K$.

§.§ Metric filtrations

Let $X$ be a set. We say that a function $\mu \colon X \times X \to \R^{\ge 0}$
is a metric filtration on $X$ if it is symmetric, and satisfies the ultrametric inequality

\[
\mu(x,x') \le \max\{\mu(x,x'), \mu(x',x'')\}
\]

for every $x$, $x'$, $x'' \in X$.

The metric filtrations appearing in the paper will all be instances of the following construction.

Let $X$ be a path-connected topological space, and $f \colon X \to \R^{\ge 0}$ a continuous function.
Given points $x$, $x' \in X$, we define

\[
\mu(x,x') := \min_{\substack{\gamma \colon I \to X \\ \gamma(0) = x \text{, } \gamma(1) = x'}} \max_{t \in I} (f\circ\g)(t),
\]

where the minimum is taken over continuous paths $\g$. This is clearly a metric filtration.
Note that $\mu(x,x) = f(x)$.

Similarly, let $G = (V, E)$ be a connected graph, and $f \colon V \to \R^{\ge 0}$ a function.
For $x$, $x' \in V$, we define $\mu(x,x')$ to be the minimum of

\[
\max\{f(x_1), \dots, f(x_n)\}
\]

over paths $x_1, \dots, x_n \in V$ such that $x_1 = x$ and $x_n = x'$.
This is a special case of the above construction: Set $X$ to be the 1-complex
associated to $G$, and extend $f$ over the 1-cells of $X$ linearly.

Let $\mu$ be a metric filtration on the set $X$. Then we define its normalization as

\[
\tilde{\mu}(x,x') := \mu(x,x') - \min\{\mu(x,x), \mu(x',x')\}.
\]

Recall that $g \colon X \times X \to \R^{\ge 0}$ is a pseudometric on $X$ if it is symmetric, satisfies the triangle
inequality, and $g(x,x) = 0$ for every $x \in X$ (but $g(x,y) = 0$ does not necessarily
imply that $x = y$).

Let $\mu$ be a metric filtration on the set $X$. Then its normalization $\tilde{\mu}$
is a pseudometric.

Choose points $x$, $x'$, $x'' \in X$, and
write $a = \mu(x,x)$, $a' = \mu(x',x')$, and $a'' = \mu(x'',x'')$.
It is clear that $\tilde{\mu}(x,x) = 0$.
If we apply the ultrametric inequality to the triple $x$, $x$, $x'$
we obtain that $\mu(x,x') \ge a$. Similarly, by considering the
triple $x$, $x'$, $x'$, we get that $\mu(x,x') \ge a'$, and hence
$\mu(x,x') \ge \max\{a,a'\}$. In particular, $\tilde{\mu}(x,x') \ge 0$.

It remains to prove the triangle inequality

\[
\tilde{\mu}(x, x'') \le \tilde{\mu}(x, x') + \tilde{\mu}(x', x'').
\]

By definition,

\[
\tilde{\mu}(x, x') = \mu(x, x') - \min\{a, a'\} \quad {and} \quad
\tilde{\mu}(x', x'') = \mu(x', x'') - \min\{a', a''\}.
\]

Without loss of generality, we can assume that

\[
\max\{\mu(x,x'), \mu(x',x'')\} = \mu(x',x'').
\]

Hence, by the ultrametric inequality, $\mu(x,x'') \le \mu(x',x'')$.
So it suffices to show that

\[
-\min\{a, a''\} \le \mu(x,x') - \min\{a, a'\} - \min\{a', a''\}.
\]

We saw that $\max\{a, a'\} \le \mu(x,x')$, hence we only need to prove that

\[
\min\{a, a'\} + \min\{a', a''\} \le \max\{a, a'\} + \min\{a, a''\}.
\]

This holds because $\max\{a, a'\}$ bounds both terms on the left-hand side from above,
while $\min\{a, a''\}$ bounds at least one of them from above.

§.§ The stabilization distance

In this section, we describe a collection of topological numerical invariants associated
to pairs of surfaces bounding a knot.
Given a properly embedded surface $S$ in a 4-manifold $W$, we describe
several ways of increasing the genus of $S$ within $W$.  Our description
is inspired by Baykur and Sunukjian <cit.>.
The most general notion, and the one we will focus on in this paper, is the following:

Let $S$ be a properly embedded surface in a 4-manifold $W$.
Suppose that $B^4 \subset \Int(W)$ is an embedded 4-ball such that $\d B^4 \cap S$
is a collection $U_1, \dots, U_m$ of unknots. Furthermore, suppose
that $S \cap B^4$ is a collection of $m$ properly embedded disks
$D_1, \dots, D_m$ that can be smoothly isotoped into $\d B^4$ relative to their boundaries.
Let $S_0$ be an oriented, connected, properly embedded genus $n$ surface in $B^4$ such
that $\d S_0 = U_1 \cup \dots \cup U_m$. The surface

\[
S' := (S \setminus \Int(B^4)) \cup S_0
\]

is the $(m,n)$-stabilization of $S$ along $(B^4, S_0)$.
We say the genus of an $(m,n)$-stabilization is $m + n - 1$,
which we note is $g(S') - g(S)$.

A schematic of an $(m,n)$-stabilization is shown in Figure <ref>.
We define the following special cases of stabilization:

A $(3,3)$-stabilization.

Let $S$ be a properly embedded surface in the 4-manifold $W$,
and suppose that $S'$ is obtained from $S$ by an $(m,n)$-stabilization along $(B^4, S_0)$.

* We say $S'$ is an unknotted surface stabilization of $S$
if $m = 1$ and $S_0$ is smoothly isotopic into $\d B^4$ relative to $\d S_0$.

* We say $S'$ is a 1-handle stabilization of $S$ if $(m,n) = (2,0)$
and $S_0 \cup D_1 \cup D_2$ is the boundary of a 3-dimensional 1-handle that is embedded in $W$.

Unknotted surface stabilization and 1-handle stabilization
have been introduced by Boyle <cit.>,
and studied further by Baykur and Sunukjian <cit.>.
They refer to an unknotted surface stabilization with $n = 1$
as a trivial stabilization, and to a 1-handle stabilization simply as a stabilization.

The surface $S'$ can be obtained from $S$ by a genus $g$ unknotted surface stabilization
if and only if it can be obtained by $g$ disjoint trivial stabilizations.

Suppose that $S'$ is obtained from $S$ by an unknotted surface stabilization along $(B^4, S_0)$.
After isotoping $S_0$ into $\Sphere^3$, it becomes a Seifert surface of the unknot $\d S_0$.
If $g > 0$, the map

\[
\pi_1(S_0 \setminus N(\d S_0)) \to \pi_1(\Sphere^3 \setminus N(\d S_0)) \cong \Z
\]

is not injective, hence $S_0$ is compressible in $\Sphere^3$ by the loop theorem.
Compressing corresponds to reversing a 1-handle stabilization in $\Sphere^3$.
If we push the interior of the 1-handle into $B^4$, it becomes unknotted.
By induction on the genus of $S_0$, we see that $S'$ can be obtained from $S$ by $g$ consecutive
trivial stabilizations. However, in dimension $4$, we can always isotope consecutive trivial stabilizations
to be disjoint from each other. The opposite implication is straightforward.

As in <cit.>*Definition 3.8, we can define the peripheral map

\[
h_S \colon \pi_1(\d W \setminus \d S) \to \pi_1(W \setminus S)
\]

for a properly embedded surface $S$ in $W$. Given surfaces $S$ and $S'$ in $W$
with $\d S = \d S'$,
we say that their peripheral maps are equivalent if there is an

\[
g \colon \pi_1(W \setminus S) \to \pi_1(W \setminus S')
\]

such that $h_{S'} = g \circ h_{S}$. The peripheral map is clearly
an invariant of $S$ up to ambient isotopy in $W$ fixing $\d W$ pointwise.

Suppose that $S'$ is obtained from $S$ by a trivial stabilization.
Then their peripheral maps are equivalent.

Boyle <cit.> showed that $\pi_1(W \setminus S) \cong \pi_1(W \setminus S')$.
Indeed, a trivial stabilization corresponds to taking the connected sum of $(W, S)$ and $(\Sphere^4, T^2)$,
where $T^2$ is an unknotted torus. Since $\Sphere^4 \setminus T^2$ is homotopy equivalent to
the suspension of $\Sphere^3 \setminus T^2$,
we have $\pi_1(\Sphere^4 \setminus T^2) \cong \Z$, generated by the meridian of $T^2$.
Hence, the claim follows from the Seifert–van Kampen theorem.
As the connected sum is taken in the interior of $W$, it follows that the peripheral maps are equivalent.

As shown by Boyle <cit.>, a nontrivial stabilization might change
the fundamental group of the surface complement, though it is always a quotient of
the original group. Based on his work,
Baykur and Sunukjian <cit.> determined when two
1-handle stabilizations give isotopic surfaces. If the 1-handle $h$ is attached along the points $a$, $b \in S$,
then we can act on the homotopy class of the core of $h$ by either adding the class of the
meridian of $S$, or pre- or post-composing with the push-off of a loop in $\pi_1(S, a)$
or $\pi_1(S, b)$. The equivalence class of the homotopy class of the core of $h$
determines the resulting surface up to isotopy, also in the case when $\S$ has boundary.

Let $D_d$ and $D_{d'}$ be deform-spun slice disks of a knot $-K \# K$,
where $d$ and $d'$ are non-isotopic automorphisms of $(\Sphere^3, K)$ that are
fixed in a neighborhood of a point of $K$. Then one cannot obtain $D_{d'}$
from $D_d$ by a sequence of trivial stabilizations (or, equivalently, by
an unknotted surface stabilization).

According to the proof of <cit.>, the peripheral maps
of $D_d$ and $D_{d'}$ are inequivalent. Hence, the result follows from Lemma <ref>.

Let $S$ and $S'$ be compact, properly embedded surfaces in the compact 4-manifold $W$
such that $\d S = \d S'$ and $[S] = [S'] \in H_2(W, \d S)$. Then $\S$ and $\S'$ become ambient isotopic
relative to $\d W$ after finitely many 1-handle stabilizations.

This is a relative version of <cit.>,
and the proof is analogous. The idea is that one removes a neighborhood
of $S \cap S'$, and chooses a relative Seifert manifold $M$ for

\[
(S \cup {-}S') \setminus N(S \cap S').
\]

Then a self-indexing Morse function on $M$ with only index 1 and 2 critical points
that is minimal along $S$ and maximal along $S'$ gives the required handles.

Suppose that $S$ and $S'$ are connected, properly embedded surfaces in the
compact 4-manifold $W$ such that $\d S = \d S'$ is a knot $K$ in $\d W$.
We define the stabilization distance of the pair $(S, S')$, for which we write
$\mu_{\st}(S, S')$, to be the minimum of

\[
\max\{ g(S_1), \dots, g(S_k) \}
\]

over sequences of connected, properly embedded surfaces $S_1, \dots, S_k$ in $W$
such that

* $S_1 = S$ and $S_k = S'$,

* $\d S_i = K$ for $i \in \{1, \dots, k\}$, and

*  $S_i$ and $S_{i+1}$ are related by an $(m,n)$-stabilization or destabilization
for $i \in \{1, \dots, k-1\}$.

Analogously, we define $\overline{\mu}_{\st}(S, S')$ by minimizing

\[
\max\{ g(S_1), \dots, g(S_k) \} - \min\{ g(S_1), \dots, g(S_k) \}
\]

over the same set of sequences. Finally, we let

\[
\tilde{\mu}_{\st}(S, S') = \mu_{\st}(S, S') - \min\{g(S), g(S')\},
\]

which we call the normalized stabilization distance.

The (trivial) 1-handle distance of $S$ and $S'$ is defined similarly to $\mu_{\st}$,
except that $S_i$ and $S_{i+1}$ are related by adding or removing a (trivial) 1-handle.
We denote the 1-handle distance by $\mu_1(S, S')$, and the trivial 1-handle distance by $\mu_1^0(S, S')$.

\[
\tilde{\mu}_{\st}(S, S') \le \overline{\mu}_{\st}(S, S') \le \mu_{\st}(S, S') \le \mu_1(S, S') \le \mu_1^0(S, S').
\]

Furthermore, if $[S] = [S']$ in $H_2(W, K)$,
then $\mu_1(S, S')$ is finite by Proposition <ref>,
and hence so are $\mu_{\st}(S, S')$ and $\overline{\mu}_{\st}(S, S')$.
On the other hand, $\mu_1^0(S, S')$ might be infinite by Corollary <ref>.
Note that $\overline{\mu}_{\st}(S, S') = 0$ if and only if $S$ and $S'$ become
isotopic after taking connected sums with 2-knots. Since $\mu_{\st}(S, S) = g(S)$,
the normalized distance satisfies $\tilde{\mu}_{\st}(S, S) = 0$.

Consider the graph whose vertices are connected surfaces in $W$ bounding the knot $K$
in a fixed relative homology class in $H_2(W, K)$, and whose edges correspond to $(m,n)$-stabilization
for some $m$ and $n$. If we apply the procedure outlined in Example <ref>
to the genus function, we obtain $\mu_{\st}$, which is hence a metric filtration in the sense
of Definition <ref>.
Its normalization in the sense of Definition <ref> is $\tilde{\mu}_{\st}$.
So, as a special case of Lemma <ref>, we obtain the following:

Let $W$ be a compact 4-manifold, and $K$ a knot in $\d W$.
The normalized stabilization distance $\tilde{\mu}_{\st}$ is a pseudometric
when restricted to surfaces in a given class in $H_2(W, K)$.

For a knot $K$ in $\Sphere^3$, let us write $\Surf(K)$ for the set of isotopy classes of
connected, oriented, properly embedded surfaces in $B^4$ bounding $K$,
and $\Surf(K)/\{\text{2-knots}\}$ for $\Surf(K)$ modulo genus 0 stabilizations.
We denote by $\Surf_g(K)$ the subset of genus $g$ surfaces in $\Surf(K)$.
If $K$ is slice, we will write

\[
\cD(K) := \Surf_0(K)
\]

for the set of isotopy classes of slice disks of $K$ in $B^4$,
and $\cD(K)/\{\text{2-knots}\}$ for $\cD(K)$ modulo genus 0 stabilizations.

Note that $(\Surf(K)/\{\text{2-knots}\}, \tilde{\mu}_{\st})$ is a pseudometric space
(i.e., $\tilde{\mu}_{\st}(S, S') = 0$ can hold for $S \neq S'$ in $\Surf(K)/\{\text{2-knots}\}$),
while $(\Surf(K)/\{\text{2-knots}\}, \overline{\mu}_{\st})$ is a metric space.
Furthermore, $\mu_{\st}$ is a metric filtration, so it satisfies the ultrametric inequality

\[
\mu_{\st}(S, S'') \le \max\{ \mu_{\st}(S, S'), \mu_{\st}(S', S'') \}
\]

for any $S$, $S'$, $S'' \in \Surf(K)$, and so does $\tilde{\mu}_{\st}$ when restricted
to $\Surf_g(K)$.

If one of $S$, $S' \in \Surf(K)$ is a disk,
and $S_1, \dots, S_k$ is a sequence of surfaces connecting $S$ and $S'$,
as in Definition <ref>, then

\[
\min\{ g(S_1), \dots, g(S_k) \} = 0,
\]

so $\mu_{\st}(S, S')$ and $\overline{\mu}_{\st}(S, S')$
are both obtained by minimizing $\max\{g(S_1), \dots, g(S_k)\}$.
Hence $\mu_{\st}$ and $\overline{\mu}_{\st}$ agree on $\cD(K)/\{\text{2-knots}\}$, and
$(\cD(K)/\{\text{2-knots}\}, \mu_{\st})$ is an ultrametric space;
i.e., a metric space that satisfies the ultrametric inequality.
Our invariants from Heegaard Floer homology naturally give bounds on $\mu_{\st}$,
hence we will not study $\overline{\mu}_{\st}$ in the rest of this paper.

If $K$ is a slice knot and $S \in \Surf(K)$,
we define the destabilizing genus $g_{\dest}(S)$ of $S$
to be the minimum of

\[
\max\{ g(S_1), \dots, g(S_n) \}
\]

over sequences of properly embedded surfaces $S_1, \dots, S_n$ in $B^4$ such that

* $S_{i+1}$ is obtained from $S_i$ via stabilization or destabilization for $i \in \{1, \dots, n-1\}$,

* $S_1 = S$, and $S_n$ is a slice disk of $K$.

By definition, $g_{\dest}(S) \ge g(S)$. Furthermore, if $D$ is a slice disks of $K$,
then $g_{\dest}(S) \le \tilde{\mu}_{\st}(S, D)$, and hence $g_{\dest}(S)$ is finite.
In fact, $g_{\dest}(S)$ is the distance of $S$ from $\cD(K)/\{\text{2-knots}\}$ in the pseudometric space
$(\Surf(K)/\{\text{2-knots}\}, \tilde{\mu}_{\st})$.

Let $S_1$, $S_2 \in \Surf(K)$. Then

\[
\mu_{\st}(S_1, S_2) \le 2g(K) + \max\{g(S_1), g(S_2)\},
\]

where $g(K)$ is the Seifert genus of $K$.

Let $S$ be a minimal genus Seifert surface for $K$, and choose an open ball $B \subset \Int(S)$.
Consider the product $S \times I \subset \Sphere^3$, where we identify
$S$ with $S \times \{0\}$. For $i \in \{1, 2\}$, isotope $S_i$ near $\d S_i$ such that
it becomes a surface $S_i'$ bounding $K \times \{1\}$. We let

\[
\S_i := ((S \setminus B) \times \{0\}) \cup (\d B \times I) \cup ((S \setminus B) \times \{1\}) \cup S_i'.
\]

Then $\S_i$ is a surface of genus $2g(K) + g(S_i)$ that can be obtained from $S_i$
by $2g(K)$ 1-handle stabilizations in $\Sphere^3$.

If we push $\S_i \setminus (S \times \{0\})$ into $\Int(B^4)$ relative to $\d B \times \{0\}$,
we obtain a surface $\S_i'$ that is a stabilization of $S$ with $m = 1$.
The sequence of surfaces $S_1$, $\S_1'$, $S$, $\S_2'$, $S_2$ satisfies the requirements of
Definition <ref>, and has maximal genus $2g(K) + \max\{ g(S_1), g(S_2) \}$.

If $K$ is a slice knot and $S \in \Surf_g(K)$, then

\[
g_{\dest}(S) \le 2g(K) + g.
\]

Let $K$ be a slice knot, and $S$, $S' \in \Surf(K)$. Then

\[
\tilde{\mu}_{\st}(S, S') \ge  |g_{\dest}(S) - g_{\dest}(S')|.
\]

Since the claim is symmetric in $S$ and $S'$, we can assume that $g_{\dest}(S) \le g_{\dest}(S')$.
Let $D$ be a slice disk for $K$ such that $\tilde{\mu}_{\st}(D, S) = g_{\dest}(S)$.

\[
g_{\dest}(S) + \tilde{\mu}_{\st}(S, S') =
\tilde{\mu}_{\st}(D, S) + \tilde{\mu}_{\st}(S, S') \ge \tilde{\mu}_{\st}(D, S') \ge g_{\dest}(S')
\]

by the triangle inequality, and the result follows.

§.§ An upper bound on the distance between 1-roll-spun and 1-twist-spun slice disks

Let $t^n r^m$ denote the $n$-twist-$m$-roll-spinning diffeomorphism of $K$.
We will show the following:

If $K$ is a knot in $\Sphere^3$, then the deform-spun slice disks $D_{K, r}$,
$D_{K, t}\in \Surf_0(-K \# K)$ satisfy

\[
\mu_{st}(D_{K, r}, D_{K, t}) \le 2.
\]

Let $B_0 \subset \Sphere^3$ denote a 3-ball that intersects $K$ in an unknotted arc.
We consider the knotted ball-arc pair $(B,a)$, where $B := \Sphere^3 \setminus \Int(B_0)$ and $a = K \cap B$.
We present both slice disks $D_{K, r}$ and $D_{K, t}$ as movies
of ball-arc pairs which start and end at $(B, a)$.

The slice disks $D_{K, t^{\wr(\cD)}r}$ and $D_{K, t}$ of $-K \# K$.

We begin with describing a movie for $D_{K, t^n r}$ for any $n \in \N$.
We pick a diagram $\cD$ for $K$ with $\wr(\cD) = n$. We view the diagram $\cD$ as nearly being
embedded in a plane $P$. The movie for $D_{K, t^n r}$ consists of
rotating the diagram $\cD$ about an axis perpendicular to $P$ (and shifting
along the axis perpendicular to $P$ slightly), while translating $\cD$
in the plane so that the image of $K$ intersects $B$ in an unknotted arc.
The rolling coefficient of this movie is clearly 1. The twisting coefficient
is equal to the difference between the blackboard framing and the Seifert
framing. Since the difference between the blackboard framing and the Seifert
framing is $\wr(\cD)$, it follows that this movie represents $D_{K, t^{\wr(\cD)} r}$.

Next, we describe a movie for the slice disk $D_{K, t}$. We pick a line
$\ell$ in $\R^3$ which coincides with $K$ inside $B_0$, and is disjoint
from $K$ outside a small neighborhood of $B_0$. The movie for
$D_{K, t}$ is obtained by rotating $a$ in a full twist about $\ell$.
Schematics of the movies for $D_{K, t^{\wr(\cD)} r}$ and $D_{K, t}$
are shown in  Figure <ref>.

We now present a stabilization sequence from $D_{K, t^{\wr(\cD)}r}$ to
$D_{K,t}$ that has maximal genus two. Let us write $\{a_s : s \in I\}$ for the movie
of arcs corresponding to $D_{K, t^{\wr(\cD)}r}$.
Suppose that $a_s = \phi_s(K) \cap B$ for a 1-parameter family of rigid motions
$\phi_s \colon \Sphere^3 \to \Sphere^3$ for $s \in I$ that nearly preserve the plane $P$.
We give $K$ a parametrization $\gamma(s)$ such that $\phi_s(\gamma(s))$ is the
center point of the ball $B_0 = \Sphere^3 \setminus \Int(B)$ (note that $B_0$ is the
region inside the red ball in Figure <ref>).

Attaching a 1-handle to the disk $D_{K, t}$ by adding a pair
of bands to the beginning of the movie for $D_{K,t}$.

Let $0 < s_1 < \dots < s_n < 1$ be the times such that $\gamma(s_i)$
is the lower point of a crossing of $\cD$, which we denote $c_i$.
Let $\Cr(\cD)$ be the set $\{c_1, \dots, c_n\}$ of crossings of $\cD$.
As $s$ passes $s_i$ for some $i \in \{1, \dots, n\}$,
the overstrand of the crossing $c_i$ passes over $\d B$ in the movie $a_s$.

As a first step, we attach a 1-handle to $D_{K, t^{\wr(\cD)}r}$, by adding
two bands to the beginning of the movie $a_s$,  as in Figure <ref>.
We can move the second band to the end of the movie. This breaks each arc $a_s$ into a knot $K_s$
disjoint from $\d B$, which we can view as a copy of our original knot $K$,
as well as a small, boundary-parallel arc attached to $\d B$.
We now wish to continuously pull the family of knots $K_s$ downward,
such that they do not pass over $\d B$ for any $s$ (or, phrased another
way, such that there is a path from $\d B$ to $\infty \in \Sphere^3$
disjoint from $K_s$ for all $s \in I$).

Given $\ve{c} \subset \Cr(\cD)$, we let $S_{\ve{c}} \in \Surf_1(-K \# K)$
denote the genus one surface obtained by modifying the movie $\{a_s : s \in I\}$
such that the upper strand of $c_i$ passes over $\d B$ if $c_i \in \ve{c}$, and the upper
strand of $c_i$ passes under $\d B$ if $c_i \not\in \ve{c}$; see Figure <ref>.

On the top row, we show a portion of the surface $S_{\ve{c}}$
when $c_i \in \ve{c}$. The upper strand of a crossing $c_i$
passes over $\d B$. On the bottom row, we show a portion of the
movie for $S_{\ve{c}}$ when $c_i \not\in \ve{c}$. The upper strand of
the crossing $c_i$ passes underneath $\d B$.

Let $c_i$ and $c_j$ be consecutive crossings in $\ve{c}$ of opposite sign.
We claim that $S_{\ve{c}}$ and $S_{\ve{c} \setminus \{c_i, c_j\}}$
become isotopic after a single stabilization. The stabilization is
obtained by attaching a band connecting the upper strands of the crossings $c_i$ and $c_j$,
followed by attaching the dual band; see Figure <ref>.

A movie for a common stabilization of $S_{\ve{c}}$ and
$S_{\ve{c} \setminus \{c_i, c_j\}}$, when $c_i$ and $c_j$ are
consecutive crossings in $\ve{c}$ of opposite sign.
In the movie, a band is added between the
first and the second frames. An isotopy connects the second and
third frames. The third and fourth frames are related by attaching the dual band.

In the case that $\wr(\cD)=0$, the number of positive crossings is
equal to the number of negative crossings, so we can eliminate all
crossings from $\ve{c}$ pairwise via the stabilization sequence described above. Hence

\[
\mu_{st}(S_{\Cr(\cD)}, S_{\emptyset}) \le 2.
\]

We note that $S_{\Cr(\cD)}$ is a genus one stabilization of $D_{K, r}$,
while the surface $S_{\emptyset}$ is described by a movie that
starts at $(B, a)$, then has a copy of $K$ break off and move away
from $\d B$. This copy of $K$ rotates $\tw(\cD)$-many times near a plane far
away from $\d B$, where $\tw(\cD)$ denotes
the twisting number of $\cD$, and then $K$ is reattached
to the arc on $\d B$ via a band.

It is a general fact that $\tw(\cD) + \wr(\cD)$ is always odd for
any diagram of a knot (as can be verified by noting that Reidemeister
moves and crossing changes do not change the quantity $\tw(\cD) + \wr(\cD)$
modulo 2, and that $\tw + \wr = 1$ for a trivial diagram of an unknot).
Since we picked $\cD$ to satisfy $\wr(\cD) = 0$, we conclude that

\[
\tw(\cD) \equiv 1 \pmod 2.
\]

Since a path of rotations about a line in $\R^3$ induces the
generator of $\pi_1(\SO(3)) \iso \Z_2$, we can assume that the
copy of $K$ which breaks off rotates in a plane exactly once.

Hence, our movie for $S_{\emptyset}$ becomes a copy of $K$ which
breaks off of $(B, a)$ as in Figure <ref>, then makes one
full rotation in a plane (away from $\d B$), and is finally
reconnected to the arc attached to $\d B$.
A continuous deformation transforms this final movie into a
stabilization of the movie for $D_{K, t}$ shown in Figure <ref>.

We conclude this section with the following conjecture (compare Proposition <ref>, below).

If $K$ is a knot in $\Sphere^3$, then the deform-spun slice disks $D_{K, r}$,
$D_{K, t} \in \Surf_0(-K \# K)$ satisfy

\[
\mu_{st}(D_{K, r}, D_{K, t}) \le 1.
\]

§ BACKGROUND ON THE LINK FLOER TQFT

In this section, we recall some previous results about the link Floer TQFT which we will need
to compute the effect of stabilization on the link cobordism maps in Section <ref>,
to determine the map induced by the summand-swapping diffeomorphism in Section <ref>,
and to prove the trace formula in Section <ref>.

§.§ The full link Floer TQFT

We first recall the category whose objects are multi-based links, and
whose morphisms are decorated link cobordisms, following the notation
of the second author <cit.>; see also the equivalent construction of the first author <cit.>.

A multi-based link $\bL = (L,\ws,\zs)$ in a 3-manifold $Y$ is an oriented link
$L \subset Y$, together with two disjoint collections of basepoints $\ws$, $\zs \subset L$ such that

* each component of $L$ has at least two basepoints,

* the basepoints along a link component of $L$ alternate between $\ws$ and $\zs$, as one traverses the link.

Let $Y_1$ and $Y_2$ be 3-manifolds containing multi-based links $\bL_1 = (L_1, \ws_1, \zs_1)$
and $\bL_2 = (L_2, \ws_2, \zs_2)$, respectively. A decorated link cobordism from
$(Y_1, \bL_1)$ to $(Y_2, \bL_2)$ is a pair $(W,\cF)=(W,(S,\cA))$, where

* $W$ is an oriented cobordism from $Y_1$ to $Y_2$,

* $S$ is a properly embedded, oriented surface in $W$ with $\d S = -L_1 \cup L_2$, and

* $\cA$ is a properly embedded 1-manifold in $S$
that divides $S$ into two subsurfaces, $S_{\ws}$ and $S_{\zs}$,
that meet along $\cA$, such that $\ws_1$, $\ws_2 \subset S_{\ws}$
and $\zs_1$, $\zs_2 \subset S_{\zs}$.

The first author <cit.> showed that decorated link cobordisms induce functorial maps
on the hat version of link Floer homology.
The second author <cit.> extended this to the full infinity
complex, denoted $\cCFL^\infty$, which is a minor variation of the
infinity complexes of Ozsváth–Szabó <cit.> and
Rasmussen <cit.>. We now review the construction of $\cCFL^\infty$.

Let $\cR^-$ denote the ring $\bF_2[U,V]$, and let $\cR^\infty$ denote the
ring $\bF_2[U,V,U^{-1},V^{-1}]$, obtained by inverting $U$ and $V$ in $\cR^-$.
Suppose that $\bL=(L,\ws,\zs)$ is a multi-based link in $Y$. Given a
multi-pointed diagram $(\Sigma,\as,\bs,\ws,\zs)$ for $(Y,\bL)$ (see
<cit.>*Definition 3.1), the complex $\cCFL^\infty(Y,\bL,\frs)$ is
the free module over $\cR^{\infty}$ generated by intersection points $\xs\in
\bT_{\a}\cap \bT_{\b}$ with $\frs_{\ws}(\xs)=\frs$. Over $\bF_2$, the generators are the monomials

\[
U^i V^j \cdot \xs,
\]

where $i$, $j \in \Z$.

The module $\cCFL^\infty(Y,\bL,\frs)$ has a filtration $\cG$ over
$\Z\oplus \Z$, where the subset $\cG_{(n,m)}\subset \cCFL^\infty(Y,\bL,\frs)$
is generated over $\bF_2$ by monomials $U^i V^j \cdot \xs$ with $i\ge n$ and $j\ge m$.
We denote the $\cR^-$-submodule $\cG_{(0,0)}$ by $\cCFL^-(Y,\bL,\frs)$, and
call it the full minus complex. It is generated over $\bF_2$ by monomials $U^i
V^j  \cdot \xs$ with $i$, $j \ge 0$.

There is a filtered, $\cR^\infty$-equivariant endomorphism $\d$ of $^∞(Y,,)$,
defined by the formula
\[
\d \xs = \sum_{\ys\in \bT_{\a}\cap \bT_{\b}}\sum_{\substack{\phi\in \pi_2(\xs,\ys)\\
\mu(\phi) = 1}} \# \hat{\cM}(\phi) \cdot U^{n_{\ws}(\phi)} V^{n_{\zs}(\phi)} \cdot \ys,
\]
which satisfies $∘̣=̣ 0$.

When $[L] = 0$ in $H_1(Y)$ and $$ is torsion,
the chain complex $(^∞(Y,,),)̣$ has several gradings.
Ozsv\'{a}th and Szab\'{o} defined a homological grading and an Alexander
grading. It is convenient for our purposes to repackage their two gradings
into three gradings that satisfy a linear dependency. Namely, there are two
homological gradings, $_$ and $_$, and an Alexander grading
$A$, which together satisfy
\[
A = \frac{1}{2}(\gr_{\ws}-\gr_{\zs}).
\]
Note that $V$ is $+1$ graded with respect to $A$, and $U$ is $-1$ graded.

When $= (K,w,z)$ is a doubly-based knot in $^3$,
we will write $^∞()$ for $^∞(^3, ,_0)$, where $_0$ is the unique $^c$ structure on $^3$.

The second author \cite{ZemCFLTQFT} constructed cobordism maps for the full knot
and link Floer complexes. Given a decorated link cobordism
$(W, )$ from $(Y_1, _1)$ to $(Y_2, _2)$ and a $^c$ structure $∈^c(W)$,
there is an induced $^∞$-equivariant, filtered chain map
\[
F_{W,\cF,\frs} \colon \cCFL^\infty(Y_1,\bL_1,\frs|_{Y_1}) \to \cCFL^\infty(Y_2,\bL_2,\frs|_{Y_2}),
\]
well-defined up to filtered, $^∞$-equivariant chain homotopy.

\subsection{Basepoint actions on link Floer homology}
\label{sec:basepointactions}

Let $= (L, , )$ be a multi-based link in the 3-manifold $Y$, and fix $∈^c(Y)$.
We recall that, for each $w∈$ and $z∈$, there are distinguished endomorphisms $Φ_w$ and $Ψ_z$ of $^∞(Y,,)$. If $= (Σ,,,,)$ is a
diagram for  $(Y, )$, then $Φ_w$ and $Ψ_z$ can be defined via the formulas
\begin{equation}
\Phi_w(\xs) = \sum_{\ys \in \bT_\a \cap \bT_\b} \sum_{\substack{\phi \in \pi_2(\xs,\ys)\\
\mu(\phi) = 1}} n_w(\phi) \# \hat{\cM}(\phi) \cdot U^{n_{\ws}(\phi)-1} V^{n_{\zs}(\phi)} \cdot \ys,
\label{eq:defPhi}
\end{equation}
\begin{equation}
\Psi_z(\xs) = \sum_{\ys\in \bT_\a \cap \bT_\b} \sum_{\substack{\phi \in \pi_2(\xs,\ys)\\
\mu(\phi) = 1}} n_z(\phi) \# \hat{\cM}(\phi) \cdot U^{n_{\ws}(\phi)} V^{n_{\zs}(\phi)-1} \cdot \ys
\label{eq:defPsi}
\end{equation}
for $∈_ ∩_$ with $_() = $.
According to \cite{ZemConnectedSums}*{Lemma~4.1}, the endomorphisms $Φ_w$
and $Ψ_z$ are the link cobordism maps induced by the two decorations of $(I×Y,I×L)$ shown in Figure~\ref{fig::6}.
When we are working with doubly-based knots, we will often write $Φ$ and $Ψ$ for the maps~$Φ_w$
and~$Ψ_z$, respectively.

\begin{figure}[ht!]
\centering
\input{fig6.pdf_tex}
\caption{The two decorated link cobordisms for $\Phi_w$ and $\Psi_z$.}\label{fig::6}
\end{figure}

According to \cite{ZemCFLTQFT}*{Lemma~4.9}, the basepoint actions satisfy
\begin{equation}
\Phi_w^2 \simeq \Psi_{z}^2 \simeq 0.
\label{eqn:R7}
\end{equation}
Note that the dividing sets on the decorated link cobordisms corresponding to $Φ_w^2$ and $Ψ_z^2$
both contain a closed curve that bounds a disk in either $S_$ or $S_$.

\subsection{Quasi-stabilizations and basepoint moving maps}
\label{sec:backgroundquasibasepoint}

We now review the quasi-stabilization maps. Suppose $= (L,,)$ is a
multi-based link in $Y$, and suppose that $w$ and $z$ are two new
basepoints contained in a single component of $L ∖(∪)$. Let us assume
that~$w$ immediately follows~$z$ with respect to the orientation of $L$,
and write
\[
\bL_{w,z}^+ := (L, \ws \cup \{w\}, \zs \cup \{z\}).
\]
There are two \emph{positive quasi-stabilization} maps
\[
S_{w,z}^+ \text{, } T_{w,z}^+ \colon \cCFL^\infty(Y, \bL, \frs) \to \cCFL^\infty(Y, \bL^+_{w,z}, \frs),
\]
as well as two \emph{negative quasi-stabilization} maps $S_w,z^-$ and $T_w,z^-$, defined in the opposite direction.
If instead $z$ follows $w$ with respect to the orientation of $L$, then we
obtain similar maps $S_z, w^±$ and $T_z,w^±$.

We briefly summarize the construction of the quasi-stabilization maps.
Additional details may be found in \cite{ZemCFLTQFT}*{Section~4}. Suppose
$=(Σ,,,,)$ is a diagram for $(Y,)$. Let $w'$ and $z'$
denote the basepoints adjacent to $w$ and $z$ on $$. There is a component
$A$ of $Σ∖$ which contains~$w'$ and~$z'$. We pick a simple
closed curve $α_s⊂A$ that cuts $A$ into two components, one of
which contains $w'$ and the other $z'$. We add another curve, $β_0$,
that bounds a small disk on $Σ$, which is cut into two bigons by
$α_s$, and is disjoint from the other $$ curves. Inside one bigon, we
place $w$. In the other bigon, we place $z$. See Figure~\ref{fig::23} for an

We write $θ^$ and $θ^$ for the higher $_$- and
$_$-graded intersection points of $α_s ∩β_0$,
respectively. The maps $S_w,z^+$ and $T_w,z^+$ are defined via the
\[
S_{w,z}^+(\xs) := \xs \times \theta^{\ws} \quad \text{and}
\quad T_{w,z}^+(\xs) := \xs \times \theta^{\zs}
\]
for $∈_ ∩_$, extended $^∞$-equivariantly.
The negative quasi-stabilization maps are defined dually, via the equations
\[
S_{w,z}^-(\xs\times \theta^{\ws})=0, \quad S_{w,z}^-(\xs\times \theta^{\zs}) =
\xs, \quad T_{w,z}^-(\xs\times \theta^{\ws})=\xs,\quad \text{and} \quad T_{w,z}^-(\xs\times \theta^{\zs}) = 0,
\]
extended $^∞$-equivariantly.
The decorations on $(I ×Y, I ×L)$ inducing the
quasi-stabilization maps $S_w,z^+$, $S_w,z^-$, $T_w,z^+$, and
$T_w,z^-$ are shown in Figure~\ref{fig::15}.

\begin{figure}[ht!]
\centering
\input{fig15.pdf_tex}
\caption{The decorated link cobordisms for the quasi-stabilization maps.}
\label{fig::15}
\end{figure}

We will need the following relations between the quasi-stabilization maps and the basepoint actions:
\begin{align}
T_{w,z}^+ &\simeq \Psi_z S_{w,z}^+, \label{eq:R1} \\
T_{w,z}^- &\simeq S_{w,z}^-\Psi_z, \label{eq:R2} \\
\Phi_w    &\simeq S_{w,z}^+S_{w,z}^-, \label{eq:R3} \\
\Psi_z    &\simeq T_{w,z}^+T_{w,z}^-. \label{eq:R4}
\end{align}
Proofs of equations~\eqref{eq:R1}--\eqref{eq:R4} can be found in \cite{ZemCFLTQFT}*{Lemmas~4.10 and 4.11}.
Examples of the corresponding dividing sets for the relations in equations~\eqref{eq:R1}--\eqref{eq:R4}
appear in Figure~\ref{fig::16}.

\begin{figure}[ht!]
\centering
\input{fig16.pdf_tex}
\caption{Dividing set manipulations corresponding to the relations
$T_{w,z}^+ \simeq \Psi_z S_{w,z}^+$ and $\Phi_w \simeq S_{w,z}^+ S_{w,z}^-$.}
\label{fig::16}
\end{figure}

Next, we review the connection between the basepoint moving maps
and the quasi-stabilization maps.
We first focus on using the quasi-stabilization maps to move a single
basepoint, while fixing all other basepoints. Suppose that $(L, _0 ∪{w'}, )$
is a multi-based link in $Y$. Suppose that $(w,z)$ are a new
pair of basepoints in a single component of $L ∖(_0 ∪{w'} ∪)$,
such that $z$ is adjacent to~$w'$. Suppose further that, according to
the orientation of $L$, the three basepoints appear in the order $w'$, $z$, $w$.
We can construct a diffeomorphism
\[
\tau^{w\from w'} \colon (Y,L,\ws_0\cup \{w'\},\zs) \to (Y,L,\ws_0\cup \{w\}, \zs),
\]
by moving $w'$ to $w$ along the arc connecting them, but fixing all of $Y$ outside a neighborhood of this arc.
According to \cite{ZemCFLTQFT}*{Lemma~4.24},
\begin{equation}
T^+_{w,z}\simeq T_{z,w'}^+ \tau^{w \from w'}_*.\label{eq:movebasepointandquasistabilization}
\end{equation}
Equation~\eqref{eq:movebasepointandquasistabilization} has a simple description in terms of dividing sets and link cobordisms,
shown in Figure~\ref{fig::25}.

\begin{figure}[ht!]
\centering
\input{fig25.pdf_tex}
\caption{The interpretation of equation~\eqref{eq:movebasepointandquasistabilization}
in terms of decorated link cobordisms.}
\label{fig::25}
\end{figure}

Using the quasi-stabilization maps, we can also move a pair of adjacent
basepoints at the same time. Suppose that $= (L,_0,_0)$ is a
multi-based link (though we allow the case where $_0$ and $_0$
intersect a single  component of $L$ trivially). Suppose that $(w',z',w,z)$
are four new consecutive basepoints, ordered right-to-left, on a single
component of $L ∖(_0 ∪_0)$. There is a diffeomorphism
\[
\rho^{(w',z') \from (w,z)} \colon (Y ,L, \ws_0 \cup \{w\}, \zs_0 \cup \{z\}) \to
(Y, L, \ws_0 \cup \{w'\}, \zs_0 \cup \{z'\}),
\]
obtained by moving the pair $(w,z)$ to $(w',z')$, but fixing everything outside a neighborhood
of an interval containing the four basepoints $(w',z',w,z)$.
According to \cite{ZemCFLTQFT}*{Lemma~4.27}, there is a chain homotopy
\begin{equation}
\rho^{(w',z') \from (w,z)}_* \simeq S_{w,z}^- T_{w',z'}^+.
\label{eq:R5}
\end{equation}
Equation~\eqref{eq:R5} can be interpreted in terms of the manipulation of
dividing sets shown in Figure~\ref{fig::17}.

\begin{figure}[ht!]
\centering
\input{fig17.pdf_tex}
\caption{The interpretation of the relation $\rho^{(w',z') \from (w,z)}_*\simeq S_{w,z}^- T_{w',z'}^+$
in terms of dividing sets.}
\label{fig::17}
\end{figure}

Another useful relation  is the following:
\begin{equation}
T_{w,z}^+ S_{w,z}^- + S_{w,z}^+ T_{w,z}^- + \id \simeq 0
\label{eq:R6}.
\end{equation}
A proof of equation~\eqref{eq:R6} can be found in \cite{ZemCFLTQFT}*{Lemma~4.13}.
For a pictorial description, see Figure~\ref{fig::18}.
Equation~\eqref{eq:R6} is an example of the \emph{bypass relation},
which is often helpful when doing computations in the link Floer TQFT.

\begin{figure}[ht!]
\centering
\input{fig18.pdf_tex}
\caption{A pictorial description of the bypass relation, equation~\eqref{eq:R6}.}
\label{fig::18}
\end{figure}

\subsection{Cobordism maps for saddles}

Next, we discuss the maps for saddle cobordisms.
Suppose that $= (L,,)$ is a multi-based link in $Y$,
and that $B$ is an embedded band for $L$ that has both ends in subarcs of $L ∖(∪)$
that run from $$ to $$, or has both ends in subarcs that run from $$ to $$.
Assuming that $(B)$, the result of band surgery, is also a valid multi-based link, there is a map
\[
F_B^{\zs} \colon \cCFL^\infty(Y,\bL,\frs) \to \cCFL^\infty(Y,\bL(B),\frs),
\]
described in \cite{ZemCFLTQFT}*{Section~6}.
The map $F_B^$ corresponds to a saddle link cobordism,
with an index~1 critical point occurring in the type-$$ subregion.
There is another map $F_B^$, with the same domain and codomain as $F_B^$,
that corresponds to adding a band to the type-$$ subregion.

The relation between the band maps and the basepoint maps
is studied in \cite{ZemCFLTQFT}*{Section~9.1}. According to \cite{ZemCFLTQFT}*{Lemma~9.1},
\begin{equation}
F_{B}^{\zs} \Phi_w + \Phi_w F_{B}^{\zs} \simeq 0.
\label{eq:R7}
\end{equation}
In contrast, the map $F_B^$ does not always commute with $Ψ_z$.
Instead, if $z$ is either of the two $$-basepoints adjacent to the ends of $B$, then
\begin{equation}
F_B^{\zs} \Psi_z + \Psi_zF_B^{\zs} \simeq F_{B}^{\ws},
\label{eq:commutatorFBzandPsi}
\end{equation}
according to \cite{ZemCFLTQFT}*{Proposition~9.3}.
In fact, the three dividing sets corresponding to the maps in equation~\eqref{eq:commutatorFBzandPsi}
can be interpreted as a bypass relation on a saddle cobordism; see \cite{ZemCFLTQFT}*{Figure~9.1}.

Note that, if $z$ and $z'$ are the two $$-basepoints adjacent to the ends of $B$,
then equation~\eqref{eq:commutatorFBzandPsi} holds for both $z$ and $z'$.
As a consequence, if we sum both relations, we obtain
\begin{equation}
F_B^{\zs}(\Psi_z + \Psi_{z'}) + (\Psi_z + \Psi_{z'}) F_B^{\zs} \simeq 0.
\label{eq:R8}
\end{equation}

\subsection{Birth cobordisms and quasi-stabilizations}
\label{sec:birthsandquasis}

We recall from \cite{ZemCFLTQFT}*{Section~7.1} the birth cobordism map.
Suppose $=(U,w,z)$ is a doubly-based unknot, which is unlinked from the
multi-based link in $Y$, and is given a distinguished Seifert disk $D$. In
this situation, there is a well defined birth map
\[
\cB_{\bU,D}^+\colon \cCFL^\infty(Y,\bL,\frs)\to \cCFL^\infty(Y,\bL\cup \bU,\frs).
\]
The map $_,D^+$ corresponds to a birth cobordism in $I×Y$,
where the disk portion of the link cobordism surface is decorated with a
single dividing arc.

The map $_,D^+$ can be computed as follows. We pick a Heegaard
diagram $(Σ,,,,)$ for $(Y,)$ such that $w,z∈Σ$,
and $D∩Σ$ consists of an embedded arc in $Σ∖(∪)$ that connects $w$ and $z$. We add two new curves, $α_0$ and
$β_0$, that bound a small disk containing $D∩Σ$, and intersect
in a pair of points. Let $θ_α_0,β_0^+∈α_0∩β_0$ denote the higher  Maslov graded intersection point (the designation
is the same for both $_$ and $_$). The map $_,D^+$ is
defined via the formula
\[
\cB_{\bU,D}^+(\xs)=\xs\times \theta^+_{\alpha_0,\beta_0}
\]
for $∈_ ∩_$, and extended $^∞$-equivariantly.
There is also a death map $_,D^-$, defined in the opposite direction,
though it will not make an appearance in this paper.

Suppose $L$ is a link in $Y$, and $U$ is an unknot that bounds a Seifert
disk $D$,  disjoint from $L$. Suppose further that $B$ is a band connecting
$U$ and $L$, which is disjoint from the interior of $D$. Let
\[
\phi\colon  (Y,L)\to (Y,(L\cup U)(B))
\]
denote a diffeomorphism which is the identity outside a neighborhood of $B∪D$.

Since a birth cobordism adds two basepoints, the composition of a birth
cobordism map and a band map is not simply the diffeomorphism map $ϕ_*$.
Instead the composition is a quasi-stabilization. More precisely, if
$=(U,w,z)$, and $B$ is an $α$-band (i.e., has both ends on strands of
$L$ that lie in the $α$-handlebody), then
\begin{equation}
F_{B}^{\ws}\cB^+_{\bU,D}\simeq T_{w,z}^+ \phi_*.
\label{eq:band-birth=quasi}
\end{equation}
A proof of equation~\eqref{eq:band-birth=quasi} can be found in
\cite{ZemCFLTQFT}*{Proposition~8.5}. Note that there are other variations of
equation~\eqref{eq:band-birth=quasi}. For example, if $B$ is instead a $β$-band, then
\begin{equation}
F_{B}^{\ws} \cB^+_{\bU,D} \simeq T_{z,w}^+ \phi_*.
\label{eq:band-birth=quasi2}
\end{equation}
A schematic illustrating equation~\eqref{eq:band-birth=quasi} is shown in Figure~\ref{fig::26}.

\begin{figure}[ht!]
\centering
\input{fig26.pdf_tex}
\caption{A manipulation demonstrating equation~\eqref{eq:band-birth=quasi}.
The composition of a birth cobordism followed by a band is (up to diffeomorphism) a quasi-stabilization.}
\label{fig::26}
\end{figure}

\subsection{4-dimensional connected sums of link cobordisms}

Suppose $(W_1,_1)$ and $(W_2,_2)$ are two link cobordisms, and pick
two embedded disks, $D_1$ and $D_2$, on $_1$ and $_2$, respectively,
such that $D_i$ intersects the dividing set of $_i$ in a single arc. By
removing 4-balls from $W_i$ that intersect $_i$ along $D_i$, and gluing
together the new boundary components, we obtain the connected sum link
cobordism $(W_1# W_2,_1# _2)$.  Using a handle cancellation argument
in the connected sum region, one can prove that
\begin{equation}
F_{W_1\# W_2, \cF_1\# \cF_2,\frs_1\# \frs_2}\simeq F_{W_1,\cF_1,\frs_1}\otimes F_{W_2, \cF_2, \frs_2}.
\label{eq:connectedsum=tensorprod}
\end{equation}
See \cite{ZemConnectedSums}*{Proposition~5.2} for a detailed proof. To make
use of equation~\eqref{eq:connectedsum=tensorprod}, it will be convenient to
have a more explicit description of the cobordism map for $(W_1# W_2,_1#
_2)$. To this end, suppose  the following:
\begin{enumerate}
\item $(Y_1,\bL_1)$ and $(Y_2,\bL_2)$ are two 3-manifolds with multi-based links.
\item $\bS_0\subset Y_1\sqcup Y_2$ is a framed 0-sphere with one foot in $Y_1$ and the other in $Y_2$.
\item  $\bS_2\subset Y_1\# Y_2$ is the dual framed 2-sphere.
\item $B$ is a band in $Y_1\# Y_2$ that connects $\bL_1$ and $\bL_2$, and intersects $\bS_2$ in a single arc.
\item $B$ is adjacent to the basepoints $w_1$ and $z_1$ on $\bL_1$, as well as $w_2$ and $z_2$, on $\bL_2$.
\item $B'$ is the band in $Y_1\# Y_2$ attached to $\bL_1 \# \bL_2$ dual to $B$.
\end{enumerate}
When $(W_i, _i)$ is the identity cobordism of $(Y_i, _i)$ for $i ∈{1, 2}$,
equation~\eqref{eq:connectedsum=tensorprod} can be rewritten as
\begin{equation}
F_{\bS_2} F_{B'}^{\ws} \Phi_w F_{B}^{\ws} F_{\bS_0}\simeq \id,
\label{eq:connectedsum=tensorprod2}
\end{equation}
where $w ∈{w_1,w_2}$. Figure~\ref{fig::27} shows equation~\eqref{eq:connectedsum=tensorprod2}
in terms of dividing sets.

\begin{figure}[ht!]
\centering
\input{fig27.pdf_tex}
\caption{A schematic of equation~\eqref{eq:connectedsum=tensorprod2}, the
effect of taking the connected sum of two link cobordisms. There is
additionally a 4-dimensional 1-handle and 3-handle, which are not shown.}
\label{fig::27}
\end{figure}

\section{Heegaard Floer invariants of surfaces}\label{sec:invariants}

\subsection{Variations on the knot Floer complex}
\label{sec:variations}
In this section, we describe several variations on the full infinity complex $^∞(Y,,)$, which we will use to define our invariants. We focus on the case that $Y=^3$ and $== (K, w, z)$ is a doubly-based knot.

The first variation we consider is the  \emph{standard infinity complex}, denoted $^∞()$. It is defined as the homogeneous subset of $^∞()$ in Alexander grading zero; i.e., the $_2$-vector space generated by monomials
\[
U^i V^j\cdot \xs
\]
with $A()+j-i=0$. Since the actions of $U$ and $V$ are $-1$ and $+1$ graded with respect to the Alexander grading, the chain complex $^∞()$ is not an $_2[U,V]$-module. However the product $Û := UV$ is 0-graded with respect to $A$, and we view $^∞()$ as an $_2[Û,Û^-1]$-module. The complex $^∞()$ contains essentially the same information as $^∞()$. We will use $^∞()$ to define our invariants $V_k(S,S')$ and $(S)$.

There are two \emph{small minus complexes},
\[
\CFK^-_{U=0}(\bK):=\cCFL^-(\bK)\otimes_{\cR^-} \cR^-/(U)\,\text{ and }\, \CFK^-_{V=0}(\bK):=\cCFL^-(\bK)\otimes_{\cR^-} \cR^-/(V),
\]
which are modules over $_2[V]$ and $_2[U]$, respectively.
By inverting $V$ or $U$, respectively, we obtain the \emph{small infinity complexes}
\[
\CFK^\infty_{U=0}(\bK) \quad \text{and} \quad \CFK^\infty_{V=0}(\bK),
\]
whose homologies are canonically isomorphic to $_2[V,V^-1]$ and $_2[U,U^-1]$, respectively.

By setting $V=1$ in the complex $^-_U=0()$, we obtain the \emph{Alexander filtered complex}
\[
\hCFKfz(\bK),
\]
described by Ozsv\'{a}th and Szab\'{o}~\cite{OS4ballgenus}.
It has an increasing filtration over $$; i.e., we have an increasing sequence of subcomplexes
\[
\hCFKfz_i(\bK) \subset \hCFKfz_{i+1}(\bK)
\]
for $i ∈$. The subspace $_i()$ is generated by the set of
intersection points $∈_ ∩_$ with $A() ≤i$.
The differential counts holomorphic disks which are allowed to pass over $z$, but not $w$.
Note that
\[
\hCFKfz_i(\bK) = \hCFKfz(\bK) \iso \hat{\CF}(\Sphere^3)
\]
for sufficiently large $i$, while $_i() = {0}$ for sufficiently negative $i$. In particular, the total homology of $()$ is isomorphic to $(^3)_2$.

Symmetrically, one can filter using the $w$ basepoint to get a $$-filtered chain complex $()$. The filtration $()$ is \emph{decreasing}; i.e.,
\[
\hCFKfw_{i+1}(\bK)\subset \hCFKfw_{i}(\bK),
\]
where $_i()$ is generated over $_2$ by intersection points $$ with $A()≥i$.

The  chain complex $^-_U=0()$ and the filtered chain complex $()$ contain essentially equivalent information. Furthermore, there is a conjugation symmetry of knot Floer homology that allows one to recover $^-_V=0()$ from $^-_U=0()$, and vice versa, and similarly recover $()$ from $()$.
Our invariant $τ(S,S')$ will be defined in Section~\ref{subsec:taudef} using $()$,
while $κ(S,S')$ and $κ_0(S)$ in Section~\ref{sec:kappa}
in terms of $^-_U=0()$.

Another variation we use to construct our invariants is the \emph{$t$-modified complex}, denoted $^-()$, due to Ozsv\'{a}th--Stipsicz--Szab\'{o} \cite{OSSUpsilon}. If $t=m/n∈[0,2]$ is a rational number with $m$ and $n$ relatively prime integers and $n≠0$, then we consider the ring $_2[v^1/n]$, where $v$ has grading $-1$. The ring $_2[v^1/n]$ can be given an action of $_2[U,V]$ by having $U$ act by  $v^2-t$ and $V$ act by $v^t$. The chain complex $^-()$ is defined as the tensor product
\[
\tCFK^-(\bK):=\cCFL^-(\bK)\otimes_{\bF_2[U,V]} \bF_2[v^{1/n}].
\]
The invariant $Υ_(S,S')(t)$ will be defined in Section~\ref{sec:upsilon} using $^-()$.

A final variation is the \emph{hat complex}
\[
\hat{\CFK}(\bK):=\cCFL^-(\bK)\otimes_{\cR^-} \cR^-/(U,V).
\]
We will not discuss $()$ extensively in this paper,
since it does not contain enough information to compute most of our invariants.

\subsection{The principal invariants of a surface bounding a knot}\label{sec:invariant}
We now describe two generalizations of the slice disk invariant
$t_D ∈()$, defined by Marengon and the first author~\cite{JMConcordance},
to higher genus surfaces in the full infinity complex.

\begin{define}
Let $\bK=(K,w,z)$ be a multi-based knot in $\Sphere^3$, and let $S \in \Surf_g(K)$ be a surface in $B^4$ bounding $K$.
If $\cA$ is a decoration on $S$ consisting of a single arc which divides $S$ into two connected subsurfaces, then we say that the map $F_{B^4,(S,\cA)}$ is a \emph{principal invariant} of the surface $S$.

Let $\cA_{\zs}$ denote the decoration on $S$ consisting of a single arc
such that $g(S_{\zs})=g(S)$ and $g(S_{\ws})=0$; see Figure~\ref{fig::30}. We  define
\[
\ve{t}^\infty_{S, \zs} := F_{B^4, (S, \cA_{\zs})} \colon \cR^\infty \to \cCFL^\infty(\bK).
\]
Similarly, if $\cA_{\ws}$ denotes the decoration on $S$ with $g(S_{\ws})=g(S)$ and $g(S_{\zs})=0$,  we define
\[
\ve{t}^\infty_{S, \ws} := F_{B^4, (S, \cA_{\ws})} \colon \cR^\infty \to \cCFL^\infty(\bK).
\]
We call $\ve{t}^\infty_{S, \ws}$ and $\ve{t}^\infty_{S, \zs}$ the \emph{extremal principal invariants}
of the surface~$S$.
\end{define}

Both $t^∞_S, $ and $t^∞_S, $ are filtered, $^∞$-equivariant
chain map that are well-defined up to filtered, $^∞$-equivariant chain homotopy.
Furthermore, both of them induce isomorphisms on homology by \cite[Theorem~9.9]{ZemAbsoluteGradings}.
By \cite[Theorem~1.4]{ZemAbsoluteGradings}, the map $t_S, ^∞$
decreases the Alexander grading by $g(S)$, while $t_S, ^∞$ increases it by $g(S)$.
When $D ∈(K)$ is a slice disk for $K$, then $_ = _$,
and we denote $t^∞_D,  = t^∞_D, $ by $t^∞_D$.

\begin{figure}
\centering
\input{fig30.pdf_tex}
\caption{The dividing set $\cA_{\zs}$ used to define the map $\ve{t}_{\Sigma,\zs}^\infty$.}
\label{fig::30}
\end{figure}

Although the chain complexes $^∞()$ and $^-()$ contain equivalent information,
it is convenient to also define maps
\[
\ve{t}^-_{S, \ws} \text{, } \ve{t}^-_{S, \zs} \colon \cR^-\to \cCFL^-(\bK)
\]
as the cobordism maps on the full minus complexes. For a slice disk $D$, we again have
$t^-_D,  = t^-_D, $, which we denote by $t^-_D$.
By \cite{JuhaszZemkeContactHandles}*{Theorem~1.4}, when we set $U = V = 0$, the map $t_D^-$,
defined using the maps from \cite{ZemCFLTQFT}, becomes $t_D$, defined in \cite{JMConcordance}.

We note that the elements $[t_S,^-(1)]$ and $[t_S,^-(1)]$
in the homology group $^-():=H_*(^-())$ contain exactly the
same information as the $^-$-equivariant, $⊕$-filtered chain
homotopy types of the maps
\[
\ve{t}_{S,\ws}^-,\, \ve{t}_{S,\zs}^-\colon \cR^-\to \cCFL^-(\bK),
\]
since two filtered, equivariant maps from $^-$ to $^-()$ are
filtered, equivariantly chain homotopic if and only if their values on $1∈^-$ differ by $η̣$ for some  $η∈^-()$. Nonetheless,
we will usually not view $t_S,^-$ and $t_S,^-$ as elements of
$^-()$ because, on its own, the group $^-()$ is not sufficient to define our invariants.

We now describe some basic properties of the invariants $t_S,^∞$ and $t_S,^∞$.
Let $r$ denote the rolling automorphism of $(^3, )$ that consists of a Dehn twist about $K$;
see Definition~\ref{def:rolling}. Then
\begin{equation}
r_* \circ \ve{t}_{S, \zs}^\infty \simeq \ve{t}_{S, \zs}^\infty,
\label{eq:SarkarmaptSz}
\end{equation}
and similarly for $t_S,^∞$,
as can be seen via a simple manipulation of the dividing sets.

\begin{define}
Let $C$ and $C'$ be free, $\Z \oplus \Z$-filtered chain complexes over $\cR^\infty$.
We say that a map $f \colon C \to C'$ is \emph{skew-equivariant} and \emph{skew-filtered} if
\[
f \circ U = V \circ f, \quad f \circ V = U \circ f, \quad \text{and} \quad f(\cG_{i,j}(C)) \subset \cG_{j,i}(C').
\]
Given skew-equivariant and skew-filtered chain maps $f \text{, } g \colon C \to C'$,
we say that they are \emph{skew-equivariant and skew-filtered chain homotopic}, and write $f \eqsim g$,
if they are chain homotopic through a skew-equivariant and skew-filtered chain homotopy.
\end{define}

There is a skew-equivariant and skew-filtered homotopy automorphism
\[
\iota_K \colon \cCFL^\infty(\bK) \to \cCFL^\infty(\bK),
\]
defined as the composition of a tautological conjugation automorphism of
$^∞()$, and the map induced by a half Dehn twist about $K$
that switches $w$ and $z$; see \cite{HMInvolutive}*{Section~6.1}.
The map $ι_K$ satisfies $ι_K^2 ≃r_*$.

Using the conjugation formula for the link cobordism maps \cite{ZemConnectedSums}*{Theorem~1.3},
as well as a simple manipulation of dividing sets, it is not hard to see that
\begin{equation}
\ve{t}_{S ,\ws}^\infty \circ \iota_{\cR^\infty} \eqsim \iota_K \circ \ve{t}_{S,\zs}^\infty,
\label{eq:conjugationandtSigmaw}
\end{equation}
where $ι_^∞ ^∞→^∞$ denotes the unique
involution that switches $U$ and $V$.
Using equation~\eqref{eq:SarkarmaptSz}, we can rewrite equation~\eqref{eq:conjugationandtSigmaw} as
\begin{equation}
\iota_K \circ \ve{t}_{S, \ws}^{\infty} \circ \iota_{\cR^\infty} \simeq \ve{t}_{S, \zs}^{\infty}.
\label{eq:equationiotaupsilon}
\end{equation}
Hence, together with the conjugation automorphism, $t_S,^∞$ and $t_S,^∞$
provide essentially equivalent information.

In the following subsections, we introduce several invariants of a pair of surface $S$, $S' ∈(K)$,
in order to give lower bounds on their stabilization distance. These are all derived from the principal invariants
for $S$ and $S'$. Furthermore, our invariants $τ$, $ν$, $V_k$, and $Υ$
are constructed as algebraic analogues of the homonymous knot invariants from knot Floer homology.
Hence, we shall sometimes call these \emph{secondary versions} of their knot invariant counterparts.

\subsection{The tau invariant}\label{subsec:taudef}

Let $= (K, w, z)$ be a doubly-based knot in $^3$.
We now describe a map
\[
\tau \colon \Surf(K) \times \Surf(K) \to \Z^{\ge 0}.
\]
It is a secondary version of the concordance invariant defined by Ozsv\'{a}th and Szab\'{o} \cite{OS4ballgenus}.
Let $(,,,w,z)$ be a doubly-pointed diagram representing $$. We will define
the invariant $τ(S, S')$ for $S$, $S' ∈(K)$ in terms of the Alexander filtered
complex $()$ described in Section~\ref{sec:variations}.

Following Section~\ref{sec:invariant}, a surface $S ∈_g(K)$ induces a chain map
$t̂_S,  _2 →()$ whose  image is contained in $_g()$.

\begin{define}
Let $\bK = (K, w, z)$ be a doubly-based knot in $\Sphere^3$.
Given surfaces $S \in \Surf_g(K)$ and $S' \in \Surf_{g'}(K)$, we define the invariant
\[
\tau(S, S') = \min \left\{ n \ge \max\{g, g'\} : \left[\hat{t}_{S, \zs}(1) \right] =
\left[\hat{t}_{S', \zs}(1)\right] \in H_*\left(\hCFKfz_n(\bK)\right) \right\}.
\]
\end{define}

It is straightforward to see that $τ(S, S')$ is independent of the choice of basepoints on $K$.

\begin{lem}
Let $S_1$, $S_1'$, $S_2$, $S_2' \in \Surf(K)$ be surfaces such that
$[S_1] = [S_2]$ and $[S_1'] = [S_2']$ in $\Surf(K)/\{\text{2-knots}\}$.
\[
\tau(S_1, S_1') = \tau(S_2, S_2').
\]
\end{lem}

\begin{proof}
This follows from the observation that $\hat{t}_{S, \zs}$
is unchanged, up to filtered chain homotopy,
if we take the connected sum of $S$ with a 2-knot,
which can be shown by adapting the proof of \cite[Lemma~4.2]{SliceDisks}.
\end{proof}

We recall that the concordance invariant $τ(K)$ may be
computed from the $_2[V]$-module
\[
\CFK^-_{U=0}(\bK) := \cCFL^-(\bK) \otimes_{\cR^-} \cR^-/(U),
\]
obtained from $^-()$ by setting $U = 0$; see Ozsv\'ath--Szab\'o \cite{OSTLegendrian}*{Lemma~A.2}.
Analogously, we can reformulate $τ(S, S')$ in terms of $^-_U=0()$.

Let us write $t_S, ^-$ and $t_S', ^-$ for the maps from $_2[V]$ to $_U=0^-()$
induced by $t_S, ^-$ and $t_S', ^-$, respectively.

\begin{lem}\label{lem:slightreformulationoftau}
Let $\bK = (K, w, z)$ be a doubly-based knot in $\Sphere^3$.
If $S \in \Surf_g(K)$ and $S' \in \Surf_{g'}(K)$, then
\[
\tau(S, S') = \min\{\, n \ge \max\{g, g'\} : V^{n - g} \cdot [t_{S, \zs}^-(1)] =
V^{n- g'} \cdot [t_{S', \zs}^-(1)] \in \HFK^-_{U=0}(\bK) \,\}.
\]
\end{lem}

\begin{proof}
Let us write $\zeta(S, S')$ for the right-hand side. If $n = \zeta(S, S')$, then
\[
V^{n - g} \cdot t_{S, \zs}^-(1) + V^{n - g'} \cdot t_{S', \zs}^-(1) = \d x
\]
for some $x \in \CFK_{U=0}^-(\bK)$. Note that $\d$ preserves the Alexander
grading, $V$ increases it by one, $A(t_{S, \zs}^-(1)) = g$,
and $A(t_{S', \zs}^-(1)) = g'$. Hence, we can assume that $A(x) = n$.
Consequently, using the identification
\[
\hCFKfz(\bK) \iso \CFK_{U=0}^-(\bK) \otimes_{\bF_2[V]} \bF_2[V]/(V-1),
\]
we have $x \otimes 1 \in \hCFKfz_n(\bK)$, and
\[
\hat{t}_{S, \zs}(1) + \hat{t}_{S', \zs}(1) = \d (x \otimes 1) \in \hCFKfz_n(\bK).
\]
It follows that
\[
\tau(S, S') \le \zeta(S, S').
\]

Conversely, if $n = \tau(S, S')$, then
$\hat{t}_{S, \zs}(1) + \hat{t}_{S', \zs}(1) = \d x$ for some $x \in \hCFKfz_n(\bK)$. Since $\hCFKfz(\bK)$ is
obtained by setting $V = 1$ in $\CFK_{U=0}^-(\bK)$, we can uniquely fill in negative powers of $V$ to lift $x$
to an element $\tilde{x} \in V^{-n} \CFK_{U=0}^-(\bK)$ satisfying
\[
V^{-g} \cdot t_{S, \zs}^-(1) + V^{-g'} \cdot t_{S', \zs}^-(1) = \d \tilde{x}.
\]
\[
V^{n - g} \cdot t_{S, \zs}^-(1) + V^{n - g'} \cdot t_{S, \zs}^-(1) =
\d(V^n \cdot \tilde{x})
\]
for $V^n \cdot \tilde{x} \in \CFK_{U=0}^-(\bK)$. It follows that
\[
\zeta(S, S') \le \tau(S, S'),
\]
which concludes the proof.
\end{proof}

Let $K_1$ and $K_2$ be knots in $\Sphere^3$.
Given surfaces $S_1 \in \Surf(K_1)$ and $S_2 \in \Surf(K_2)$, their boundary
connected sum $S_1 \natural S_2$ is an element of $\Surf(K_1 \# K_2)$.

\begin{prop}
Let $K_1$ and $K_2$ be knots in $\Sphere^3$.
If $S_1$, $S_1' \in \Surf(K_1)$ and $S_2$, $S_2' \in \Surf(K_2)$
are surfaces of genera $g_1$, $g_1'$, $g_2$, and $g_2'$, respectively, then
\[
\tau(S_1 \natural S_2, S_1' \natural S_2') \le
\max \left\{\tau(S_1, S_1') + \max\{g_2, g_2'\},
\tau(S_2, S_2') + \max\{g_1, g_1'\} \right\}.
\]
Furthermore, when $g_1 = g_1' = g_2 = g_2' = 0$, then equality holds.
\end{prop}

\begin{proof}
Using the connected sum formula \cite[Proposition~5.1]{ZemConnectedSums}, there is chain homotopy equivalence
\[
F \colon \CFK_{U=0}^-(\bK_1) \otimes_{\bF_2[V]} \CFK_{U=0}^-(\bK_2) \to \CFK_{U=0}^-(\bK_1 \# \bK_2),
\]
where the decoration on $\bK_1 \# \bK_2$ is the decoration of $\bK_1$.
Furthermore, the map $F$ can be taken to be the
link cobordism map for a 1-handle cobordism containing a band. By the functoriality
of the link cobordism maps,
\[
F \circ(t_{S_1, \zs}^- \otimes t_{S_2, \zs}^-) \simeq t_{S_1 \natural S_2, \zs}^-,
\]
and similarly for $S_1'$ and $S_2'$.

By the K\"{u}nneth theorem for tensor products over a PID, there is a short exact sequence
\[
\begin{split}
0 \to \HFK^-_{U=0}(\bK_1) \otimes_{\bF_2[V]} \HFK^-_{U=0}(\bK_2) \xrightarrow{G} \HFK^-_{U=0}(\bK_1 \# \bK_2) \to \\
\Tor_{\bF_2[V]}^1 (\HFK^-(\bK_1), \HFK^-_{U=0}(\bK_2)) \to 0,
\end{split}
\]
where $G$ is the composition of the natural map
\[
\HFK^-_{U=0}(\bK_1) \otimes_{\bF_2[V]} \HFK^-_{U=0}(\bK_2) \to H_*(\CFK_{U=0}^-(\bK_1) \otimes_{\bF_2[V]} \CFK_{U=0}^-(\bK_2))
\]
and $F_*$. The map $G$ sends $[t_{S_1, \zs}^-(1)] \otimes [t_{S_2, \zs}^-(1)]$ to $[t_{S_1 \natural S_2, \zs}^-(1)]$.

For $i \in \{1, 2\}$, the $\bF_2[V]$-module $\HFK^-_{U=0}(\bK_i)$
splits (non-canonically) as $\bF_2[V] \oplus T_i$, where $T_i$ is
a torsion $\bF_2[V]$-module, and
$t_{S_i, \zs}^-(1) = V^{g_i} \oplus s_i$ and $t_{S_i', \zs}^-(1) = V^{g_i'} \oplus s_i'$ for some $s_i$, $s_i' \in T_i$.
\begin{equation}\label{eqn:def-n}
n = \max \left\{\tau(S_1, S_1') + \max\{g_2, g_2'\},
\tau(S_2, S_2') + \max\{g_1, g_1'\} \right\}.
\end{equation}
Then we claim that
\begin{equation}\label{eqn:sumineq}
V^{n - g_1 - g_2} \cdot ([t_{S_1, \zs}^-(1)] \otimes [t_{S_2, \zs}^-(1)]) =
V^{n - g_1' - g_2'} \cdot ([t_{S_1', \zs}^-(1)] \otimes [t_{S_2', \zs}^-(1)])
\end{equation}
as elements of $\HFK^-(\bK_1) \otimes \HFK^-(\bK_2)$. This is equivalent to
\[
\begin{split}
&V^{n - g_1 - g_2}((V^{g_1} \otimes V^{g_2}) \oplus (s_1 \otimes V^{g_2}) \oplus (V^{g_1} \otimes s_2) \oplus (s_1 \otimes s_2)) = \\
&V^{n - g_1' - g_2'}((V^{g_1'} \otimes V^{g_2'}) \oplus (s_1' \otimes V^{g_2'}) \oplus (V^{g_1'} \otimes s_2')
\oplus (s_1' \otimes s_2')).
\end{split}
\]
For $i \in \{1, 2\}$ and $k \ge \tau(S_i, S_i')$,
by Lemma~\ref{lem:slightreformulationoftau}, we have
\[
V^k \oplus (V^{k-g_i} s_i) = V^{k - g_i} \cdot [t_{S_i, \zs}^-(1)] =
V^{k - g_i'} \cdot [t_{S_i', \zs}^-(1)] = V^k \oplus (V^{k-g_i'} s_i').
\]
Together with equation~\eqref{eqn:def-n},
this implies that $V^{n-g_i} s_i = V^{n-g_i'} s_i'$ and
\[
V^{n - g_1 - g_2} (s_1 \otimes s_2) = V^{n - g_1' - g_2'} (s_1' \otimes s_2'),
\]
so equation~\eqref{eqn:sumineq} holds.
The result follows by applying $G$ to equation~\eqref{eqn:sumineq},
and invoking Lemma~\ref{lem:slightreformulationoftau}.

When $g_1 = g_1' = g_2 = g_2' = 0$, then we choose
\[
n = \tau(S_1 \natural S_2, S_1' \natural S_2').
\]
By Lemma~\ref{lem:slightreformulationoftau},
$n$ satisfies equation~\eqref{eqn:sumineq}, which implies
that $V^n s_i = V^n s_i'$ for $i \in \{1, 2\}$, and hence
\[
V^n \cdot [t^-_{S_i, \zs}] = V^n \cdot [t^-_{S_i', \zs}].
\]
So $n \ge \max\{\tau(S_1, S_1'), \tau(S_2, S_2')\}$, and equality holds, as claimed.
\end{proof}

\subsection{An infinitesimal refinement of tau}\label{sec:tau+}
In this section, we describe a refinement of $\tau(S, S')$,
inspired by work of Ozsv\'{a}th--Szab\'{o}~\cite{OSRationalSurgeries},
Hom--Wu~\cite{HomWuNu+}, and Hom~\cite{HomEpsilon},

Let $\bar{\Z}$ denote $\Z \cup \{-\infty,\infty\}$,
and write $\bar{\Z}^{\le 0} = [-\infty, 0] \cap \bar{\Z}$.
Given a knot $K$ in $\Sphere^3$, we will define a symmetric map
\[
\tau^+ \colon \Surf(K) \times \Surf(K) \to \N \times \bar{\Z}^{\le 0}.
\]
The invariant $\tau^+(S, S')$ takes the form
\[
\tau^+(S, S') := (\tau(S, S'), \tau'(S, S')),
\]
where $\tau(S, S')$ is the integer defined in Section~\ref{subsec:taudef},
and $\tau'(S, S')$ is an element of $\bar{\Z}^{\le 0}$ that we define shortly.
We will think of $\tau'$ as a second-order version of $\tau$, or an infinitesimal refinement.

To define $\tau'$, we introduce some notation. If $(i,j) \in \Z \oplus \Z$, let
\[
R_{i,j} := \{\, (m,n) \in \Z \oplus \Z : m \ge i \text{, } n \ge j \,\}.
\]
If $\cS \subset \Z \oplus \Z$, let $H(\cS)$ denote
the \emph{filtered hull} of $\cS$; i.e.,
\[
H(\cS) = \bigcup_{(i,j) \in \cS} R_{i,j}.
\]
We say that $\cS$ is a \emph{filtered} shape if
\[
\cS = H(\cS).
\]

Let $w$ and $z$ be basepoints on $K$, and write $\bK = (K, w, z)$.
If $\cS \subset \Z \oplus \Z$, let $C(\bK,\cS)$ denote the subspace of $\CFK^\infty(\bK)$
generated over $\bF_2$ by monomials $U^i V^j \cdot \xs$ with $A(\xs) + j - i = 0$
and $(i,j) \in \cS$. If $\cS$ is a filtered shape, then $C(\bK,\cS)$ is a subcomplex of $\CFK^\infty(\bK)$.

More generally, we say $\cS \subset \Z \times \Z$ is a \emph{sub-quotient} shape if, whenever
$(i,j)$, $(m,n) \in \cS$ with $i \le m$ and $j \le n$,
then the entire rectangle spanned by the points $(i,j)$ and $(m,n)$ is
contained in~$\cS$.
If $\cS$ is a sub-quotient shape, then $C(\bK,\cS)$ is in fact a sub-quotient
complex of $\CFK^\infty(\bK)$; i.e., there are subcomplexes
$C_{\text{in}}(\bK, \cS) \subset C_{\text{out}}(\bK, \cS) \subset \CFK^\infty(\bK)$ such that
$C_{\text{out}}(\bK, \cS)/C_{\text{in}}(\bK, \cS)$
is chain isomorphic to $C(\bK, \cS)$. Indeed, the two sub-complexes of $\CFK^\infty(\bK)$ are
\[
\begin{split}
C_{\text{out}}(\bK, \cS) &:= C\left(\bK, H(\cS)\right) \text{, and}\\
C_{\text{in}}(\bK, \cS) &:= C\left(\bK, H(\cS) \setminus \cS\right).
\end{split}
\]
We note that, if $\cS \subset \Z \oplus \Z$ is an arbitrary subset, then its filtered hull
$H(\cS)$ is automatically a filtered shape. It is an easy exercise to show that,
if $\cS$ is a sub-quotient shape, then $H(\cS) \setminus \cS$ is filtered. Hence
$C_{\text{in}}(\bK, \cS)$ and $C_{\text{out}}(\bK, \cS)$ are both subcomplexes of $\CFK^\infty(\bK)$.

We note that the map $\ve{t}^-_{S,\zs}$ naturally has image in the $g(S)$ Alexander graded subspace of $\cCFL^-(\bK)$.
Hence, there is a well-defined map
\[
V^{-g(S)}\cdot \ve{t}_{S,\zs}^-\colon \bF_2[\hat{U}]\to C(\bK, R_{0,-g(S)}).
\]
Furthermore, if $\cS$ is a sub-quotient shape of $\Z\oplus \Z$
such that $R_{0,-g(S)}\subset H(\cS)$, then there is a natural map
\[
q\colon C(\bK,R_{0,-g(S)})\to C(\bK,\cS),
\]
which is the composition of the inclusion map $C(\bK,R_{0,-g(S)}) \to C_{\text{out}}(\bK,\cS)$,
followed by the quotient map $C_{\text{out}}(\bK,\cS) \to C_{\text{out}}(\bK,\cS)/C_{\text{in}}(\bK,\cS)\iso C(\bK,\cS)$.
In particular,  $[V^{-g(S)}\cdot \ve{t}_{S,\zs}^-(1)]$ determines a well-defined element of $H_*(C(\bK,\cS))$.

\[
\I{n} := \{0\} \times ([-n, \infty) \cap \Z),
\]
which is a sub-quotient shape.
Noting that $C(\bK,\I{n})$ is chain isomorphic to $\hCFKfz_n(\bK)$, we obtains the following:

\begin{lem}\label{lem:tausubquotientshapes}
Given a doubly-based knot $\bK = (K, w, z)$ and surfaces $S$, $S' \in \Surf(K)$, we have
\[
\tau(S,S') = \min\left\{ n \ge \max\{g(S),g(S')\} :
[V^{-g(S)}\cdot \ve{t}^-_{S, \zs}(1)] = [V^{-g(S')} \cdot \ve{t}^{-}_{S', \zs}(1)] \in H_*(C(\bK, \I{n}))\right\}.
\]
\end{lem}

We are now ready to define the refinement $\tau'(S, S')$.

\begin{define}
For $n$, $m \in \Z$, let
\[
\L{m}{n} := \I{n} \cup ([0,m] \cap \Z) \times \{-n\}.
\]
This is an L-shaped subset of $\Z \times \Z$, and hence a sub-quotient shape.

Let $\bK = (K, w, z)$ be a doubly-pointed knot in $\Sphere^3$,
let $S$, $S' \in \Surf(K)$, and write $\tau = \tau(S, S')$.
Then we define
\[
\tau'(S, S') = -\sup \{\, m \in \Z : [V^{-g(S)}\cdot\ve{t}^{-}_{S, \zs}(1) - V^{-g(S')}\cdot \ve{t}^-_{S', \zs}(1)] = 0 \in
H_* \left(C(\bK, \L{m}{\tau} \right) \,\}.
\]
\end{define}

Note that $\L{0}{\tau} = \I{\tau}$, and $[V^{-g(S)}\cdot \ve{t}^{-}_{S, \zs}(1) - V^{-g(S')}\cdot \ve{t}^-_{S', \zs}(1)] = 0$
in $H_*(C(\bK, \I{\tau}))$ by the definition of $\tau$. It follows that $\tau'(S, S') \le 0$.
However, if $x \in C(\bK, \I{\tau})$ satisfies
\[
\d x = V^{-g(S)}\cdot \ve{t}^{-}_{S, \zs}(1) - V^{-g(S')} \cdot \ve{t}^-_{S', \zs}(1)
\]
in $C(\bK, \I{\tau})$, then $\d x$ might have some nonzero terms in $([0,m] \cap \Z) \times \{-\tau\}$ for $m > 0$.
Hence $[V^{-g(S)}\cdot \ve{t}^{-}_{S, \zs}(1) - V^{-g(S')}\cdot \ve{t}^-_{S', \zs}(1)]$ might not be zero
in $H_* \left(C(\bK, \L{m}{\tau})\right)$ for $m > 0$,
and this is what $\tau'(S, S')$ measures.

The invariant $\tau^+(S,S')$ was inspired by the concordance invariant $\nu(K)$,
due to Ozsv\'{a}th and Szab\'{o} \cite{OSRationalSurgeries}*{Definition~9.1},
which gives an improved 4-ball genus bound over $\tau(K)$ by at most 1. We now extract
an analogue of $\nu$ from $\tau^+$ for pairs of surfaces in $\Surf(K)$,
though there is some information lost when doing this as
$\tau'$ can take any value in $\bar{\Z}^{\le 0}$.

\begin{define}
Let $K$ be a knot in $\Sphere^3$ and $S$, $S' \in \Surf(K)$. Then let
\[
\nu(S, S') =
\begin{cases}
\tau(S, S') &\text{if } \tau'(S, S') = -\infty, \\
\tau(S, S') + 1 &\text{otherwise.}
\end{cases}
\]
\end{define}

We will see that $\nu$ gives a lower bound on the stabilization and double point
distances in Proposition~\ref{prop:tau+}.

\subsection{A sequence of local $h$-invariants}\label{sec:V}

Let $K$ be a knot in $\Sphere^3$ and $S$, $S' \in \Surf_g(K)$.
Modeled on the invariants $V_k(K)$ of large surgeries from knot Floer
homology, also referred to as Rasmussen's \emph{local $h$-invariants}
\cite{RasmussenKnots},  we describe a sequence of integer invariant $V_k(S,S')$
for $k\ge g$, such that
\[
V_{g}(S,S')\ge V_{g+1}(S,S')\ge \cdots\ge 0,
\]
and such that $V_k(S, S') = 0$ for $k$ sufficiently large.

\begin{define}
Let $\bK = (K, w, z)$ be a doubly-based knot in $\Sphere^3$, and let $S$, $S' \in \Surf_g(K)$.
To define $V_k(S,S')$, we consider the subcomplex
\[
A_k^-(\bK) := C(\bK,R_{0, -k})
\]
of $\CFK^\infty(\bK)$.
We think of the complexes $A_k^-(\bK)$ as modules over the ring $\bF_2[\hat{U}]$, where $\hat{U} = UV$.

The map $\ve{t}_{S,\zs}^-$ increases the Alexander grading by $g(S)$. If $k\ge g(S)$,
then $V^{-g(S)}\cdot \ve{t}_{S,\zs}^-(1)$ has Alexander grading $0$, and determines a
well-defined element of $H_*(A_k^-(\bK))$. We define the invariant
\[
V_k(S,S') := \min \left\{\, n \in \N : \hat{U}^n \cdot [V^{-g}\cdot \ve{t}^-_{S,\zs}(1)] =
\hat{U}^n \cdot [V^{-g}\cdot \ve{t}^-_{S',\zs}(1)]\in H_*(A_k^-(\bK)) \,\right\}.
\]
\end{define}

\begin{rem}
The above definition of $V_k$ can be easily adapted to surfaces of different genera;
however, we specialize to the case when $g(S) = g(S')$ since our topological applications for $V_k$
only hold when this is the case.
\end{rem}

We now show that the invariants $V_k$ of pairs of surfaces in $\Surf_g(K)$ satisfy many of
the same properties as Rasmussen's local $h$-invariants. The reader should
compare the following to \cite{RasmussenKnots}*{Proposition~7.6}:

\begin{lem}\label{lem:monotonicity}
If $k \ge g$, then
\[
V_{k}(S,S') \ge V_{k+1}(S,S') \ge V_{k}(S,S')-1.
\]
\end{lem}

\begin{proof}
There is a natural, grading-preserving inclusion of chain complexes
\[
i_{k} \colon A_{k}^-(\bK) \hookrightarrow A_{k+1}^-(\bK),
\]
which becomes an isomorphism on homology after we invert $\hat{U}$, and satisfies
\[
(i_{k})_*([ V^{-g} \cdot \ve{t}_{S,\zs}^-(1)])= [V^{-g}\cdot  \ve{t}_{S,\zs}^-(1)].
\]
\[
V_{k+1}(S,S') \le V_{k}(S,S').
\]

Multiplication by $\hat{U}$ induces a $-2$-graded inclusion $A_{k+1}^-(\bK) \hookrightarrow A_k^-(\bK)$ of chain complexes,
which becomes an isomorphism on homology after we invert $\hat{U}$. The map sends
$[V^{-g}\cdot \ve{t}_{S,\zs}(1)]\in H_*(A_{k+1}^-(\bK))$
to $\hat{U} \cdot [V^{-g}\cdot \ve{t}_{S,\zs}^-(1)]\in H_*(A_k^-(\bK))$, and similarly for $S'$. Hence, if
\[
\begin{split}
\hat{U}^n \cdot [V^{-g}\cdot \ve{t}_{S,\zs}^-(1)] &= \hat{U}^n \cdot [V^{-g}\cdot \ve{t}_{S',\zs}^-(1)]\in H_*(A_{k+1}^-(\bK)) \text{, then} \\
\hat{U}^{n+1} \cdot [V^{-g}\cdot \ve{t}_{S,\zs}^-(1)] &= \hat{U}^{n+1} \cdot [V^{-g}\cdot \ve{t}_{S',\zs}^-(1)]\in H_*(A_{k}^-(\bK)).
\end{split}
\]
We conclude that $V_k(S,S') \le V_{k+1}(S,S')+1$.
\end{proof}

The reader should compare the following to \cite{RasmussenKnots}*{Proposition~7.7}:

\begin{lem}
If $S$, $S'\in \Surf_g(K)$ and $g \le k < \tau(S,S')$, then $0 < V_{k}(S,S')$.
\end{lem}

\begin{proof}
By Lemma~\ref{lem:slightreformulationoftau},
we can describe $\tau(S, S')$ as the minimal $n$
such that $V^{n-g} \cdot [t_{S,\zs}^-(1)] = V^{n-g}\cdot  [t_{S',\zs}^-(1)]$ in
\[
\HFK^-_{U=0}(\bK) = H_*\left( \cCFL^-(\bK) \otimes \bF_2[U,V]/(U) \right).
\]
Note that multiplication by $V^k$ determines an inclusion of chain complexes
\[
A_k^-(\bK) \hookrightarrow \cCFL^-(\bK),
\]
which we compose with the natural map $\cCFL^-(\bK) \to \CFK_{U=0}^-(\bK)$ given by
$\xs \mapsto \xs \otimes 1$, corresponding to setting $U = 0$.
The induced map on homology sends $[V^{-g}\cdot \ve{t}_{S,\zs}^-(1)]$, $[V^{-g}\cdot \ve{t}_{S',\zs}^-(1)] \in H_*(A_k^-(\bK))$ to
$V^{k-g} \cdot [t_{S,\zs}^-(1)]$, $V^{k-g} \cdot [t_{S',\zs}^-(1)] \in \HFK^-_{U=0}(\bK)$, respectively.
If $g\le k < \tau(S, S')$, then
\[
V^{k-g} \cdot [t_{S,\zs}^-(1)] \neq V^{k-g} \cdot [t_{S',\zs}^-(1)] \in \HFK^-(\bK),
\]
and, consequently,
\[
[V^{-g}\cdot \ve{t}_{S,\zs}^-(1)] \neq [V^{-g}\cdot \ve{t}_{S',\zs}^-(1)] \in H_*(A_k^-(\bK)).
\]
Hence $V_k(S, S') > 0$, as claimed.
\end{proof}

\subsection{The upsilon invariant}\label{sec:upsilon}
Let $K$ be a knot in $\Sphere^3$, and let $S$, $S' \in \Surf(K)$.
We now describe our invariant
\[
\Upsilon_{(S, S')} \colon [0,2] \to \R^{\ge 0}.
\]
It is a secondary version of Ozsv\'{a}th, Stipsicz, and Szab\'{o}'s \cite{OSSUpsilon} invariant~$\Upsilon_K(t)$.

We recall the \emph{$t$-modified} version of knot Floer homology, described
by Ozsv\'{a}th, Stipsicz, and Szab\'{o}. Suppose that $t = \frac{m}{n}\in [0,2]$ is a
rational number with $m \in \N$ and $n \in \Z_+$ relatively prime.
We define $\tCFK^-(\bK)$ to be the free $\bF_2[v^{1/n}]$-module generated by $\bT_{\a} \cap \bT_{\b}$, where $v$ is a formal variable.
Similarly, let $\tCFK^\infty(\bK)$ be the free $\bF_2[v^{1/n}, v^{-1/n}]$-module generated
by $\bT_{\a} \cap \bT_{\b}$. The modules $\tCFK^-(\bK)$ and $\tCFK^\infty(\bK)$ are equipped with a
differential $\d$ that satisfies
\[
\d \xs := \sum_{\ys \in \bT_{\a} \cap \bT_{\b}} \sum_{\substack{\phi \in \pi_2(\xs,\ys)\\
\mu(\phi) = 1}} \# \hat{\cM}(\phi) \cdot v^{t n_{z}(\phi) + (2-t) n_{w}(\phi)} \cdot \ys
\]
for $\x \in \T_\a \cap \T_\b$.

We note that $\tCFK^-(\bK)$ and $\tCFK^\infty(\bK)$ can easily be expressed in
terms of $\cCFL^-(\bK)$ and $\cCFL^\infty(\bK)$, respectively, as we now describe.
We give $\bF_2[v^{1/n}]$ the structure of an $\bF_2[U,V]$-module, where
$U$ acts by $v^{2-t}$ and $V$ acts by $v^{t}$. With this action, we have
a canonical isomorphism
\[
\tCFK^-(\bK) \iso \cCFL^-(\bK) \otimes_{\bF_2[U,V]} \bF_2[v^{1/n}],
\]
as well as a similar isomorphism involving $\tCFK^\infty(\bK)$ and
$\cCFL^\infty(\bK)$. Note that, in particular, if $(W,\cF) \colon
(\Sphere^3, \bK_1) \to (\Sphere^3, \bK_2)$ is a decorated link cobordism, then
the map $F_{W,\cF,\frs}$ determines a $t$-modified version
\[
\tF_{W,\cF,\frs} := F_{W,\cF,\frs} \otimes \id_{\bF_2[v^{1/n}]} \colon \tCFK^-(\bK_1) \to \tCFK^-(\bK_2).
\]
Finally, we note that there is a $t$-grading on $\cCFL^-(\bK)$, defined via the formula
\[
\gr_t(\xs) := \frac{t}{2} \cdot \gr_{\zs}(\xs) + \left(1 - \frac{t}{2} \right) \cdot \gr_{\ws}(\xs).
\]
This induces a well-defined grading on $\tCFK^-(\bK)$,
for which we also write $\gr_t$. With respect to $\gr_t$, the variable $v$ is $-1$ graded.

If $S \in \Surf_{g}(K)$ and $S' \in \Surf_{g'}(K)$, the invariants
$\ve{t}_{S,\zs}^-$ and $\ve{t}_{S',\zs}^-$ admit $t$-modified versions
$\tvt_{S,\zs}^-$ and $\tvt_{S',\zs}^-$, respectively. Furthermore,
the elements $\tvt_{S,\zs}^-(1)$ and $\tvt_{S',\zs}^-(1)$ for $1 \in \bF_2[v^{1/n}]$
have $\gr_t$-grading $-t \cdot g$ and $-t \cdot g'$, respectively.

\begin{define}
For $t = \frac{m}{n} \in [0,2]$, we define
\[
\Upsilon_{(S,S')}(t) := \min \{\,s = k/n \ge \max \{t \cdot g, t \cdot g'\} :
v^{s - t \cdot g} \cdot [\tvt^-_{S,\zs}(1)] = v^{s - t \cdot g'} \cdot
[\tvt^-_{S',\zs}(1)] \in \tHFK^-(\bK)\,\}.
\]
\end{define}

There is an alternate definition of the invariant $\Upsilon_{(S,S')}(t)$, which is more amenable to computations, and is based on Livingston's description of the corresponding knot invariant \cite{LivingstonUpsilon}.
If $t\in [0,2]$, there is a filtration $\cG_{s}^t(\bK)$ of $\CFK^\infty(\bK)$ which is indexed by a parameter $s\in \R$. The set $\cG_s^{t}(\bK)$ is the $\bF_2$-module generated by monomials $U^i V^j \cdot \xs$ with $A(\xs)+j-i=0$ and
\[
t\cdot j+(2-t)\cdot i\ge -s.
\]
If $s\ge t\cdot g(S)$, then it is straightforward to see that $[V^{-g(S)}\cdot \ve{t}_{S,\zs}^-(1)]$ is a well-defined element of $H_*(\cG_s^t(\bK))$. It is not hard to adapt \cite{LivingstonUpsilon}*{Section~14.1} to establish the following:

\begin{lem}\label{lem:livingstonreformupsilon}
If $S \in \Surf_g(K)$ and $S' \in \Surf_{g'}(K)$, then
\[
\Upsilon_{(S,S')}(t) =  \min\left\{s\ge t\cdot \max\{g,g'\} :
[V^{-g} \cdot \ve{t}_{S,\zs}^-(1)] = [V^{-g'} \cdot \ve{t}_{S',\zs}^-(1)] \in H_*(\cG_s^t(\bK)) \right\}.
\]
\end{lem}

\subsection{The kappa and kappa-nought invariants}\label{sec:kappa}
If $\bK=(K,w,z)$ is a doubly-based knot in $\bS^3$, let $\CFK^-_{U=0}(\bK)$
and $\CFK^\infty_{U=0}(\bK)$ denote the small minus and infinity knot Floer complexes
described in Section~\ref{sec:variations}.

\begin{lem}
If $g(S)>0$, then
\[
[t_{S,\ws}^\infty(1)] = 0 \in \HFK^\infty_{U=0}(\bK).
\]
\end{lem}

\begin{proof}
For $n \in \Z$, let $\CFK^\infty_{U=0}(\bK)_n$ denote the subspace of $\CFK^\infty_{U=0}(\bK)$ in Alexander grading $n$.
Explicitly, the subspace $\CFK^\infty_{U=0}(\bK)_n$ is generated by monomials of the form $V^i\cdot \xs$, where
\[
A(\xs) + i = n.
\]
We define the reduction map
\[
R_w^n\colon \CFK^\infty_{U=0}(\bK)_n\to \hat{\CF}( \Sphere^3 , w ),
\]
by the formula $R_w^n(V^i \cdot \xs)=\xs$. It is straightforward to see that
$R_w^n$ is a chain map. Furthermore, since the differential on
$\CFK^\infty_{U=0}(\bK)$ preserves the Alexander grading, the map $R_w^n$ is
a chain isomorphism. Consider the chain isomorphism
\[
R_{w} := \bigoplus_{n \in \Z} R_w^n \colon \CFK^\infty_{U=0}(\bK)\to \bigoplus_{n\in \Z} \hat{\CF}(\Sphere^3,w).
\]
In particular, since $\hat{\HF}( \Sphere^3 , w )$ is supported in $\gr_{\ws}$-grading 0, it follow that $\HFK^\infty_{U=0}(\bK)$ is as well. The map $\ve{t}_{S,\ws}^\infty(1)$ has $\gr_{\ws}$-grading $-2g(S)$ by the grading formula in \cite{ZemAbsoluteGradings}*{Theorem~1.4}. It follows that
\[
[\ve{t}_{S,\ws}^\infty(1)]=0\in \HFK^\infty_{U=0}(\bK),
\]
completing the proof.
\end{proof}

\begin{define}
Let $\bK = (K, w, z)$ be a doubly-based knot in $\Sphere^3$, and let $S \in \Surf_g(K)$ for $g > 0$.
Then we let
\[
\kappa_0(S) := \min\left\{\, n \ge g : V^{n-g} \cdot [t_{S,\ws}^-(1)] = 0 \in \HFK^-_{U=0}(\bK) \,\right\}.
\]
If $S \in \Surf_0(K)$, we set $\kappa_0(S) = 0$.
\end{define}

We note that the element $\ve{t}_{S,\ws}^-(1)$ lives in Alexander grading $-g$. Hence
\[
\kappa_0(S) = g + \min\left\{\, k \in \N : [\hat{t}_{S,\ws}(1)] = 0 \in H_*\left(\hCFKfz_{k-g}(\bK)\right) \,\right\}.
\]

\begin{define}
If $g>0$ and $S$, $S'\in \Surf_g(K)$, we define the invariant
\[
\kappa(S,S'):=\min\left\{\, n \ge g : V^{n-g} \cdot [t^-_{S,\ws}(1)] =
V^{n-g} \cdot [t^-_{S',\ws}(1)] \in \HFK^-_{U=0}(\bK) \,\right\}.
\]
\end{define}

Note that
\[
\kappa(S,S') \le \max\{\kappa_0(S), \kappa_0(S')\}.
\]
We emphasize that the invariant $\tau(S,S')$ is defined
in terms of the maps $t_{S,\zs}^-$ and $t_{S',\zs}^-$,
while $\kappa(S,S')$ is defined in terms of the maps $t_{S,\ws}^-$ and $t_{S',\ws}^-$.
Also, unlike the invariant $\tau(S,S')$, the
definition of $\kappa(S,S')$ only makes sense if $g(S) = g(S') > 0$.

\subsection{Upsilon near $0$ and $2$}

Ozsv\'{a}th, Stipsicz, and Szab\'{o} \cite{OSSUpsilon}*{Proposition~1.6} proved that the knot invariant
$\Upsilon_K(t) = -\tau(K)\cdot t$ near $t=0$. In this section, we prove a
similar result for $\Upsilon_{(S,S')}(t)$.

\begin{thm}\label{thm:upsilon-slope}
Suppose that $S$, $S'\in \Surf(K)$. For all $t\in [0,2]$ sufficiently close to 0, we have
\[
\Upsilon_{(S,S')}(t)= \tau(S,S')\cdot t.
\]
For $t$ sufficiently close to $2$, we have
\[
\Upsilon_{(S,S')}(t)=\begin{cases} (\kappa_0(S)-g(S))\cdot (2-t)+ g(S)\cdot t& \text{ if } g(S)>g(S'),\\
(\kappa(S,S')-g(S))\cdot (2-t)+g(S)\cdot t& \text{ if } g(S)=g(S').
\end{cases}
\]
\end{thm}

\begin{proof}
The argument we present is an adaptation of Livingston's proof of the
analogous fact \cite{LivingstonUpsilon}*{Theorem~13.1} for the knot invariants $\Upsilon_K(t)$,
and we use the reformulation of
$\Upsilon_{(S,S')}(t)$ in terms of filtrations on $\CFK^\infty$ from
Lemma~\ref{lem:livingstonreformupsilon}. We focus on $\Upsilon_{(S,S')}(t)$ near $t=2$ when
$g(S)>g(S')$, since the other cases are straightforward adaptations of this.
Let us write $g=g(S)$ and $g'=g(S')$.

Define the following sub-quotient shapes of $\Z\oplus \Z:$
\begin{align*}
H_{-g+1}&:=\{\,(i,j): j\ge -g+1\,\},\\
T_{-g,k}&:=\{\,(i,j): i\ge k \text{, } j=-g\,\}\\
Z_{-g, k}&:= H_{-g+1} \cup T_{-g,k}.
\end{align*}

\begin{figure}[ht!]
\centering
\input{fig34.pdf_tex}
\caption{Examples of the sub-quotient shapes $H_{-g+1}$, $Z_{-g,k}$, and $T_{-g,k}$ of $\Z \oplus \Z$.}
\label{fig::34}
\end{figure}

It is straightforward to see that, for $t$ sufficient close to $2$, the
complex $\cG_s^t(\bK)$ is always equal to $C(\bK,Z_{i,j})$ for some $i$, $j$.
Hence $[V^{-g}\cdot \ve{t}_{S,\zs}^-(1)]=[V^{-g'}\cdot \ve{t}_{S',\zs}^-]\in
H_*(\cG_{s}^t(\bK))$ if and only if
\[
C(\bK, Z_{-g,m})\subset \cG_{s}^t(\bK),
\]
\[
m := \max\left\{\, k\le 0: [V^{-g}\cdot \ve{t}_{S,\zs}^-(1)]=
[V^{-g'} \cdot \ve{t}^-_{S',\zs}(1)]\in H_*(C(\bK,Z_{-g,k}))\,\right\}.
\]
Consequently, an easy computation shows that
\begin{equation}
\Upsilon_{(S,S')}(t)=-m(2-t)+g t,\label{eq:Upsilonzfortnear2}
\end{equation}
for $t$ near $2$.

There is a short exact sequence of chain complexes
\begin{equation}
0\to C(\bK, H_{-g+1})\xrightarrow{i}  C(\bK, Z_{-g,k})\xrightarrow{q} C(\bK, T_{-g,k})\to 0.\label{eq:SES}
\end{equation}
Since $H_*(C(\bK, T_{-g,k}))$ is a torsion $\bF_2[\hat{U}]$-module, while
$H_*(C(\bK,H_{-g+1})$ is torsion-free, the connecting homomorphism of the
long exact sequence associated to equation~\eqref{eq:SES} vanishes.
Consequently, there is a short exact sequence
\begin{equation}
0\to H_*(C(\bK, H_{-g+1}))\xrightarrow{i} H_*(C(\bK, Z_{-g,k}))\xrightarrow{q} H_*(C(\bK, T_{-g,k}))\to 0.
\label{eq:LESsplits}
\end{equation}

Furthermore, since $H_*(C(\bK,H_{-g+1})) \iso \bF_2[\hat{U}]$, it follows that
$q$ is an isomorphism on the torsion submodule of $H_*(C(\bK, Z_{-g, k}))$.
Since $[V^{-g} \cdot \ve{t}_{S,\zs}^-(1)] + [V^{-g'} \cdot
\ve{t}_{S',\zs}^-(1)]$ is a torsion element of $H_*(C(\bK,Z_{-g,k}))$, it
follows that $[V^{-g} \cdot \ve{t}_{S,\zs}^-(1) + V^{-g'}\cdot  \ve{t}_{S',\zs}^-(1)]$
is zero in $H_*(C(\bK, Z_{-g,k}))$ if an only if its image under $q$ is zero
in $H_*(C(\bK, T_{-g,k}))$. Consequently,
\begin{equation}\label{eq:shiftbyUV^g}
m=\max\left\{k\le 0:  [V^{-g}\cdot \ve{t}^-_{S,\zs}(1)]=
[V^{-g'} \cdot \ve{t}^-_{S',\zs}(1)] \in H_*(C(\bK, T_{-g,k}))\right\}.
\end{equation}
Note that if $g>g'$, then  $[V^{-g'}\cdot \ve{t}^-_{S',\zs}]= 0 \in H_*(C(\bK, T_{-g,k}))$,
so equation~\eqref{eq:shiftbyUV^g} implies that
\begin{equation}\label{eq:invariantforS'vanishes}
m=\max\left\{k\le 0:  [V^{-g}\cdot \ve{t}^-_{S,\zs}(1)]=0 \in H_*(C(\bK, T_{-g,k}))\right\}.
\end{equation}

Next, we note that multiplication by $\hat{U}^g$ induces a chain isomorphism
between $C(\bK, T_{-g,k})$ and $C(\bK, T_{0, k+g})$. The group
$C(\bK, T_{0,k+g})$ is the $\bF_2$-module generated by intersection points
$\xs\in \bT_{\a}\cap \bT_{\b}$ with $A(\xs)\ge k+g$. The differential on
$C(\bK, T_{0,k+g})$ counts holomorphic disks which are allowed to go over $w$,
but not $z$. This is simply the subcomplex $\hCFKfw_{k+g}(\bK)\subset \hCFKfw(\bK)$.
Applying the conjugation symmetry of knot Floer homology to
equation~\eqref{eq:invariantforS'vanishes} implies that
\begin{equation}
\begin{split}
m &= \max\left\{k\le 0 : [U^g \cdot \ve{t}^-_{S,\zs}(1)] = 0 \in H_*(C(\bK, T_{0,k+g}))\right\}\\
&= -\min\left\{k \ge 0 : [V^g \cdot \ve{t}_{S,\ws}^-(1)] = 0 \in H_*(\hCFKfz_{k-g}(\bK))\right\}\\
&= g - \kappa_0(S).
\end{split}
\label{eq:computem}
\end{equation}
Equations~\eqref{eq:Upsilonzfortnear2} and \eqref{eq:computem} together imply that, for $t$ near 2, we have
\[
\Upsilon_{(S,S')}(t)=(\kappa_0(S) - g) \cdot (2-t) + g \cdot t.
\]
Similar arguments apply when $g = g'$, and for $t$ close to 0.
\end{proof}

\subsection{Further properties of the secondary invariants}
Our secondary invariants satisfy a monotonicity condition with respect
to stacking link cobordisms.

\begin{prop}\label{prop:concordancemonotonicity}
Suppose that $(I \times \Sphere^3, S)$ is a link cobordism from $(\Sphere^3, K_0)$
to $(\Sphere^3, K_1)$, and $S_0$, $S_0' \in \Surf(K_0)$.
Let $S_1$, $S_1' \in \Surf(K_1)$ be the surfaces obtained by
stacking $S_0$ or $S_0'$ and $S$, respectively. Then
\[
\tau(S_0, S_0') + g(S) \ge \tau(S_1, S_1').
\]
When $g(S_0) = g(S_0')$, the invariant $\kappa$ satisfies an analogous inequality; furthermore,
\[
V_{k}(S_0, S_0') \ge V_{k+g(S)}(S_1, S_1')
\]
for $k \ge g(S_0)$.
\[
\Upsilon_{(S_0, S_0')}(t) + (1 - |1-t|)\cdot g(S) \ge \Upsilon_{(S_1, S_1')}(t),
\]
for any $S_0$, $S_0' \in \Surf(K)$ and $t \in [0,2]$.
\end{prop}

\begin{proof}
Choose basepoints $w_i$ and $z_i$ on $K_i$ for $i \in \{0,1\}$,
and a decoration $\cA$ on $S$ such that $S_{\ws}$ is a strip
containing $w_0$ and $w_1$.
Using the functoriality of the link cobordism maps,
it sends $\ve{t}_{S_0, \zs}^\infty(1)$ to $\ve{t}_{S_1, \zs}^\infty(1)$
and $\ve{t}_{S_0', \zs}^\infty(1)$ to $\ve{t}_{S_1', \zs}^\infty(1)$. Furthermore, by \cite[Theorem~1.4]{ZemAbsoluteGradings}, the map $F_{I \times \Sphere^3, (S, \cA)}$
increases the Alexander grading by $g(S)$. From these two facts, all claims can be proven quickly.
\end{proof}

A concordance $\cC$ from $K_0$ to $K_1$ is called \emph{invertible}
if there is a concordance $\cC'$ from $K_1$ to $K_0$ such that $\cC' \circ \cC$
is the identity cobordism from $K_0$ to itself; see Sumners~\cite{Sumners}.

\begin{cor}
Suppose that $K_0$ and $K_1$ are knots in $\Sphere^3$ and $S_0$, $S_0' \in \Surf(K)$.
If $\cC$ is an invertible concordance from $K_0$ to $K_1$,
let $S_1$ and $S_1'$ denote the surfaces in  $\Surf(K_1)$
obtained by stacking $S_0$ or $S_0'$ and $\cC$, respectively. Then
\[
\omega(S_0, S_0') = \omega(S_1, S_1')
\]
for $\omega \in \{\tau, \Upsilon\}$.
If $g(S_0) = g(S_0')$, then the same equality holds for $\omega \in  \{V_k,\kappa\}$, provided $k\ge g$.
\end{cor}

\begin{proof}
Let $\cC'$ be a left inverse of $\cC$.
We first apply Proposition~\ref{prop:concordancemonotonicity}
to the surfaces $S_0$, $S_0'$, and to the concordance $\cC$,
to obtain that $\omega(S_0, S_0') \ge \omega(S_1, S_1')$.
Note that we recover $S_0$ if we stack $S_1$ and $\cC'$,
and $S_0'$ if we stack $S_1'$ and $\cC'$.
Hence, if we apply Proposition~\ref{prop:concordancemonotonicity}
to the surfaces $S_1$, $S_1'$, and to the concordance $\cC'$,
we obtain that $\omega(S_1, S_1') \ge \omega(S_0, S_0')$.
\end{proof}

Like $\mu_{\st}$ and $\mu_{\Sing}$, our secondary invariants satisfy
the following ultrametric inequality:

\begin{prop}\label{prop:ultra}
If $K$ is a knot in $\Sphere^3$ and $S_1$, $S_2$,  $S_3 \in \Surf(K)$, then
\[
\omega(S_1, S_3) \le \max\{\, \omega(S_1, S_2), \omega(S_2, S_3) \,\}
\]
for $\omega\in \{\tau,\Upsilon\}$, where the inequality is to be
taken pointwise when $\omega = \Upsilon$.
If $g(S_1) = g(S_2) = g(S_3)$, then the
inequality holds with $\omega\in \{V_k,\kappa\}$, as well.
\end{prop}

\begin{proof}
All of the invariants are described in terms of when two distinguished
elements become equal in homology, after multiplying by some power of $V$, $\hat{U}$, or $v$.
\end{proof}

\begin{lem}
Let $K$ be a knot in $\Sphere^3$, and suppose $S_1$, $S_2$, $S_3 \in \Surf(K)$.
If we endow $\N \times \bar{\Z}^{\le 0}$ with the lexicographic ordering,
then the map $\tau^+$ satisfies the ultrametric inequality
\[
\tau^+(S_1, S_3) \le \max\{ \tau^+(S_1, S_2), \tau^+(S_2, S_3) \}.
\]
\end{lem}

\begin{proof}
For $i$, $j \in \{1,2,3\}$, let us write $\tau_{ij} = \tau(S_i, S_j)$ and $\tau'_{ij} = \tau'(S_i, S_j)$,
and let
\[
\ve{t}_i := V^{-g(S_i)} \cdot \ve{t}_{S_i, \zs}^-(1).
\]
Without loss of generality, we can suppose that $\tau_{12} \le \tau_{23}$.
By Proposition~\ref{prop:ultra}, we have $\tau_{13} \le \max \{\tau_{12}, \tau_{23}\} = \tau_{23}$.
As we have endowed $\N \times \bar{\Z}^{\le 0}$ with the lexicographic ordering,
it suffices to consider the case when $\tau_{13} = \tau_{23}$, which we denote by $\tau$,
as otherwise $\tau^+(S_1, S_3) < \tau^+(S_2, S_3)$.

Choose basepoints $w$ and $z$ on $K$, and write $\bK = (K, w, z)$.
First, assume that $\tau_{12} < \tau$. By the definition of $\tau_{12}$, there
exists $x \in C(\bK, \I{\tau_{12}})$ such that $\d x = \ve{t}_1 - \ve{t}_2$.
If we write $m := -\tau'_{23}$, by definition, there is an $y \in C(\bK, \L{m}{\tau})$ such that
$\d y = \ve{t}_2 - \ve{t}_3$. Since $\tau_{12} < \tau$ and $\d$ respects the Alexander filtration,
there is an inclusion of complexes
\[
\iota \colon C(\bK, \I{\tau_{12}}) \to C(\bK, \L{m}{\tau}).
\]
It follows that $\iota(x) + y \in C(\bK, \L{m}{\tau})$ satisfies
$\d(\iota(x) + y) = \ve{t}_1 - \ve{t}_3$. As $\tau_{13} = \tau$,
it follows that $-\tau'_{13} \ge m$, hence $\tau'_{13} \le \tau'_{23}$
and $\tau^+(S_1, S_3) \le \tau^+(S_2, S_3)$.

Now suppose that $\tau_{12} = \tau$. If we write $m' := -\tau'_{12}$ and $m := -\tau'_{23}$,
then there exist $x \in C(\bK, \L{m'}{\tau})$ and $y \in C(\bK, \L{m}{\tau})$ such that
$\d x = \ve{t}_1 - \ve{t}_2$ and $\d y = \ve{t}_2 - \ve{t}_3$.
Without loss of generality, we can assume that $m' \ge m$. Then there is a natural projection
\[
\pi \colon C(\bK, \L{m'}{\tau}) \to C(\bK, \L{m}{\tau})
\]
that is a chain map, and preserves the elements $\ve{t}_i$ for $i \in \{1,2,3\}$.
Since $x + \pi(y) \in C(\L{m}{\tau})$
and $\d(x + \pi(y)) = \ve{t}_1 - \ve{t}_3$, we have $-\tau'_{13} \ge m$,
hence $\tau'_{13} \le \tau'_{23}$ and $\tau^+(S_1, S_3) \le \tau^+(S_2, S_3)$.
\end{proof}

\section{Link Floer homology and the stabilization distance}\label{sec:stab}

In this section we prove our main technical results about stabilizations and the link Floer TQFT,
and show that our invariants $\tau$ and $V_k$ give lower bounds on $\mu_{\st}$,
while $\kappa_0$ and $\cI$ give lower bounds on $g_{\dest}$.

\subsection{Algebraic reduction and knotification functors}

Suppose $\bU$ is a doubly-based unknot in a 3-manifold $Y$, and $D$ is a choice of
Seifert disk for $\bU$. There is a simple algebraic relationship between the
filtered knot invariant $\cCFL^-(Y,\bU,\frs)$ and the 3-manifold invariant
$\CF^-(Y,p,\frs)$, where $p$ denotes a basepoint in $Y$.

There are two algebraic reduction operations that reduce $\cCFL^-$ to $\CF^-$.
These correspond to setting one of $U$ or $V$ equal to 1, and
identifying the remaining variable with a third variable, $\hat{U}$. Algebraically,
given a multi-based link $\bL = (L, \ws, \zs)$, we define the $V = 1$ and $U = 1$ reductions by the formulas
\begin{equation}
\begin{split}
\Red_{V=1}(\cCFL^-(Y,\bL,\frs)) &:= \cCFL^-(Y,\bL,\frs) \otimes_{\bF_2[U,V]} \bF_2[\hat{U},U,V]/(\hat{U}-U,V-1)\\
\Red_{U=1}(\cCFL^-(Y,\bL,\frs)) &:= \cCFL^-(Y,\bL,\frs) \otimes_{\bF_2[U,V]} \bF_2[\hat{U},U,V]/(\hat{U}-V,U-1).
\label{eq:tensorproductreduction}
\end{split}
\end{equation}
There are canonical isomorphism of $\bF_2[\hat{U}]$-modules
\[
\begin{split}
\Red_{V=1}(\cCFL^-(Y,\bL,\frs)) &\iso \CF^-(Y,\ws,\frs),\\
\Red_{U=1}(\cCFL^-(Y,\bL,\frs)) &\iso \CF^-(Y,\zs,\frs-\PD[L]).
\end{split}
\]
The $V = 1$ and $U = 1$  reduction operations are functorial: If $(W,\cF)$ is a
decorated link cobordism, then the map $F_{W,\cF,\frs}$ determines two maps
\[
\begin{split}
\Red_{V=1}(F_{W,\cF,\frs}) &= F_{W,\cF,\frs} \otimes_{\bF_2[U,V]} \id_{\bF_2[\hat{U},U,V]/(\hat{U}-U,V-1)} \text{ and } \\
\Red_{U=1}(F_{W,\cF,\frs}) &= F_{W,\cF,\frs} \otimes_{\bF_2[U,V]} \id_{\bF_2[\hat{U},U,V]/(\hat{U}-V,U-1)}
\end{split}
\]
on $\CF^-$.

There is a functor in the opposite direction, which we call
\emph{knotification}. Algebraically, if $p \in Y$ is a basepoint (or, more
generally, a collection of basepoints), the knotification functor is defined as
\[
\Kn(\CF^-(Y,p,\frs)) := \CF^-(Y,p,\frs) \otimes_{\bF_2[\hat{U}]} \bF_2[U,V],
\]
where $\hat{U}$ acts on $\bF_2[U,V]$ as the product $UV$. If
\[
h \colon \CF^-(Y_1,p_1,\frs_1) \to \CF^-(Y_2,p_2,\frs_2)
\]
is an $\bF_2[\hat{U}]$-equivariant map, then there is an induced map
\[
\Kn(h) := h \otimes_{\bF_2[\hat{U}]} \id_{\bF_2[U,V]} \colon
\Kn(\CF^-(Y_1, p_1,\frs_1)) \to \Kn((\CF^-(Y, p_2, \frs_2)),
\]
which is $\Z \oplus \Z$-filtered and $\bF_2[U,V]$-equivariant.
Furthermore, if $h' \colon \CF^-(Y_2,p_2,\frs_2) \to \CF^-(Y_3,p_3,\frs_3) $ is $\bF_2[\hat{U}]$-equivariant,
\begin{equation}\label{eqn:commute}
\Kn(h' \circ h) = \Kn(h') \circ \Kn(h).
\end{equation}
We define the boundary map on $\Kn(\CF^-(Y,p,\frs))$ as $\Kn(\d_{\CF^-(Y,p)})$;
we have $\Kn(\d_{\CF^-(Y,p)})^2 = 0$ by equation~\eqref{eqn:commute}.

If $F \colon \CF^-(Y_1,p_1,\frs_1) \to \CF^-(Y_2,p_2,\frs_2)$ is a chain map,
then $\Kn(F)$ is also a chain map by equation~\eqref{eqn:commute}, and by
the definition of the boundary map on the knotification as $\Kn(\d_{\CF^-(Y,p)})$.
Equation~\eqref{eqn:commute} further implies that knotification is
also functorial on the level of $\bF_2[\hat{U}]$-equivariant chain homotopy
types of maps: If
\[
F \text{, } G \text{, } H \colon \CF^-(Y_1,p_1,\frs_1) \to \CF^-(Y_2,p_2,\frs_2)
\]
are $\bF_2[\hat{U}]$-equivariant and $F + G = \d \circ H + H \circ \d$, then
\[
\Kn(F) + \Kn(G) = \d \circ \Kn(H) + \Kn(H) \circ \d.
\]

There is a canonical chain isomorphism
\begin{equation}
\Kn(\CF^-(Y,p,\frs)) \iso \cCFL^-(Y,\bU(p),\frs),
\label{eq:knotificationisomorphism}
\end{equation}
where $\bU(p)$ is the doubly-based unknot corresponding to $p$. The
isomorphism from equation~\eqref{eq:knotificationisomorphism} is obtained by
taking a Heegaard diagram $(\Sigma,\as,\bs,p)$ for $(Y,p)$, and replacing $p$
with a pair of basepoints $w$ and $z$ that are immediately adjacent to each
other. On such a diagram, any holomorphic disk counted by the differential
passes over $w$ and $z$ an equal number of times. Hence
\begin{equation}
\d_{\cCFL^-(Y,\bU(p))} = \Kn(\d_{\CF^-(Y,p)}).
\label{eq:knotificationdifferential}
\end{equation}

In order to compute the link cobordism maps for closed surfaces, it is
natural to consider the compositions $\Kn \circ \Red_{U=1}$ and $\Kn \circ
\Red_{V=1}$. Obviously, these two compositions are not the identity for
an arbitrary link $\bL$. Less obviously, even if we restrict to unknots, there
is in general no canonical isomorphism between $(\Kn \circ
\Red_{V=1})(\cCFL^-(Y,\bU,\frs))$ and $\cCFL^-(Y,\bU,\frs)$. Topologically,
this is due to the fact that $\cCFL^-(Y,\bU,\frs)$ is defined independently
of a choice of Seifert surface, but when we reduce then knotify, we are
implicitly constructing a Seifert disk for $\bU$.

If we fix a Seifert disk $D$ of $\bU$, then there is a canonical isomorphism
between $\cCFL^-(Y,\bU,\frs)$ and $(\Kn \circ
\Red_{V=1})(\cCFL^-(Y,\bU,\frs))$, which indeed depends on $D$. The
isomorphism is obtained by restricting to Heegaard diagrams
$(\Sigma,\as,\bs,w,z)$ of $(Y,\bU)$ where $D$ intersects $\Sigma$ in an arc,
connecting $w$ and $z$, and disjoint from $\as \cup \bs$. We can also
assume that $p$ is a point along $D \cap \Sigma$. The complex
$\CF^-(\Sigma,\as,\bs,p,\frs)$ is generated by monomials $\hat{U}^i \cdot
\ve{x}$, with $i \ge 0$. The complex $\cCFL^-(\Sigma,\as,\bs,w,z,\frs)$ is
generated by monomials of the form $U^i V^j \cdot \ve{x}$ with $i$, $j \ge 0$.
The canonical isomorphism between $\Kn(\CF^-(Y,p,\frs))$ and
$\CF^-(Y,\bU(p),\frs)$ is then obtained by sending $\xs \otimes 1$ to $\xs$,
extending equivariantly over $\bF_2[U,V]$.

Given an $\bF[\hat{U}]$-equivariant map
\[
F \colon \bF_2[\hat{U}] \to \CF^-(\Sigma,\as,\bs,p),
\]
the knotified map
\[
\Kn(F) \colon \bF_2[U,V] \to \cCFL^-(\Sigma,\as,\bs,w,z)
\]
can be described concretely. If $F(1)$ contains the summand $\hat{U}^i \cdot \xs$,
then the map $\Kn(F)(1)$ contains the summand $U^i V^i \cdot \xs$.

Even when we equip $\bU$ with a Seifert disk $D$, there is another way in
which the composition $\Kn \circ \Red_{V=1}$ fails to be the identity: on
morphisms. If $F \colon \bF_2[U,V] \to \cCFL^-(\Sigma,\as,\bs,w,z,\frs)$ is a
map that increases Alexander grading by $k$, and $F(1)$ contains the summand
$U^i V^j \cdot \xs$, then $(\Kn \circ \Red_{V=1})(F)(1)$ contains the summand
$U^i V^i \cdot \xs$, which may not be the same as $F(1)$.

The failure of $\Kn \circ \Red_{V=1}$ to be the identity can be described in terms of Alexander
gradings. A choice of Seifert disk $D$ induces an Alexander grading $A_D$ on
the complex $\cCFL^-(\Sigma,\as,\bs,w,z)$ by \cite{ZemAbsoluteGradings}*{Theorem~2.13}.
When $D$ intersects $\Sigma$ in an arc connecting $w$ and $z$ disjoint from $\as \cup \bs$,
the Alexander grading is simply
\begin{equation}
A_D(U^i V^j \cdot \xs) = j - i.\label{eq:Alexandergradingformulaunknot}
\end{equation}
Hence, if $F \colon \bF_2[U,V] \to \cCFL^-(\Sigma,\as,\bs,w,z,\frs)$ is an
Alexander graded map, we can recover $F$ from the knotification of its
reductions by correcting the powers of the $V$ variables in the map
$(\Kn \circ \Red_{V=1})(F)$, or by correcting the powers of the $U$ variables
in the map $(\Kn \circ \Red_{U=1})(F)$. We state this precisely in the following lemma:

\begin{lem}\label{lem:relknotifiedredandoriginal}
Suppose $\bU$ is a doubly-based unknot in $Y$, with a chosen Seifert disk $D$. If
\[
F \colon \bF_2[U,V] \to \cCFL^-(Y,\bU,\frs)
\]
is an $\bF_2[U,V]$-equivariant map of $A_D$-grading $k$, then the following hold:
\begin{enumerate}
\item If $k \ge 0$, then
\[
V^k\cdot (\Kn\circ \Red_{V=1})(F) \simeq F \text{ and } (\Kn \circ \Red_{U=1})(F) \simeq U^k \cdot F.
\]
\item If $k\le 0$, then
\[
(\Kn \circ \Red_{V=1})(F) \simeq V^{-k} \cdot F \text{ and } U^{-k} \cdot (\Kn \circ \Red_{U=1})(F) \simeq F.
\]
\end{enumerate}
\end{lem}

\begin{proof}
Let us consider the case when $k \ge 0$. In this case, both $F$ and
$V^k \cdot (\Kn\circ \Red_{V=1})(F)$ are well-defined up to chain homotopy, so it is
sufficient to show the claim for a particular representative of the maps.
Upon examination, $(\Kn \circ \Red_{V=1})(F)$ and $F$ agree up to a power of
$V$, and the discrepancy is exactly the Alexander grading change, by
equation~\eqref{eq:Alexandergradingformulaunknot}. The other relations are
proven similarly.
\end{proof}

\subsection{Stabilizations and link Floer homology}\label{sec:stabandHFL}

In this section, we prove our main computational results about stabilizations
and link Floer homology.
Before we state our computational results, we recall that the link cobordism
maps admit extensions
\[
F_{W,\cF,\frs} \colon \Lambda^*(H_1(W)/\Tors) \otimes \cCFL^-(Y_1,\bL_1,\frs|_{Y_1}) \to
\cCFL^-(Y_2,\bL_2,\frs|_{Y_2})
\]
that incorporate the action of $\Lambda^* (H_1(W)/\Tors)$, similar to the
cobordism maps of Ozsv\'ath and Szab\'o~\cite{OSTriangles}; see
\cite{ZemAbsoluteGradings}*{Section~12.2} for a description.

If $F$ and $G$ are two maps from $\bF_2[U,V]$ to
$\cCFL^-(Y,\bL,\frs)$, we say that $F \simeq G$ \emph{modulo the action of $H_1(Y)$}, and write
\[
F \simeq G \mod H_1(Y),
\]
if there are classes $\gamma_1, \dots, \gamma_k \in H_1(Y)$, as well as
filtered, $\F_2[U,V]$-equivariant maps $J_1, \dots, J_k$ from $\bF_2[U,V]$ to
$\cCFL^-(Y,\bL,\frs)$, such that
\[
F + G \simeq \sum_{i=1}^k A_{\gamma_i} \circ J_i.
\]

Next, we note that, analogously to
equation~\eqref{eq:knotificationdifferential}, the $H_1(Y)/\Tors$ actions
on $\cCFL^-$ and $\CF^-$ commute with the knotification functor, in the sense that
\begin{equation}
A_\gamma \simeq \Kn(A_\gamma),
\label{eq:kncommutesH1action}
\end{equation}
since both sides can be computed by counting the same holomorphic disks on a
knot diagram $(\Sigma,\as,\bs,w,z)$, where $w$ is immediately adjacent to $z$, and
the multiplicities of such a disk at $w$ and $z$ agree.

\begin{lem}\label{lem:closedsurfacecomputation}
Suppose that $(W,\cF) \colon \emptyset \to (Y,\bU)$ is a decorated link
cobordism from the empty set to a doubly-based unknot $\bU$ in $Y$, equipped
with a Seifert disk $D$, and let $\frs \in \Spin^c(W)$.
Write $\cF = (S,\cA)$, and suppose that $H_1(Y) \to H_1(W)$ is a surjection.
\begin{enumerate}
\item\label{it:claim1} Suppose $\cA$ has a single component (necessarily a non-closed arc),
and write
\[
h(S\cup D,\frs) := \frac{\langle\, c_1(\frs), [S \cup D] \,\rangle - [S \cup D] \cdot [S \cup D]}{2}.
\]
If $h(S \cup D,\frs) + g(S_{\zs}) - g(S_{\ws}) \ge 0$, then
\[
F_{W,\cF,\frs} \simeq  U^{g(S_{\ws})} V^{g(S_{\zs}) + h(S \cup D, \frs)} \cdot \Kn(F_{W,\frs}) \mod H_1(Y).
\]
If $h(S \cup D,\frs) + g(S_{\zs}) - g(S_{\ws}) \le 0$, then
\[
F_{W,\cF,\frs} \simeq U^{g(S_{\ws}) - h(S \cup D, \frs)} V^{g(S_{\zs})} \cdot \Kn(F_{W,\frs - \PD[S]}) \mod H_1(Y).
\]
\item\label{it:claim2} If $\cA$ has a closed component $\gamma$, then
\[
F_{W,\cF,\frs} \simeq 0 \mod H_1(Y).
\]
\end{enumerate}
\end{lem}

\begin{proof}
The proof is a modification of \cite{ZemAbsoluteGradings}*{Proposition~9.7}.
Consider first Claim~\eqref{it:claim1}, in  the case when $h(S \cup
D,\frs) + g(S_{\zs}) - g(S_{\ws}) \ge 0$. Note that the latter quantity is the
Alexander grading change of the map $F_{W,\cF,\frs}$.  The reduction
$\Red_{V=1}(F_{W,\cF,\frs})$ is computed explicitly in
\cite{ZemCFLTQFT}*{Theorem~C}, and depends only on $W$, $\frs$, and the
embedding of the subsurface $S_{\ws}$ in $W$. When $S_{\ws}$ is a connected
surface with a single boundary component, according to
\cite{ZemAbsoluteGradings}*{Lemma~9.6}, the $V = 1$ reduction satisfies
\begin{equation}
\Red_{V=1}(F_{W,\cF,\frs}) \simeq F_{W,\frs}(\xi_{\ws} \otimes -), \label{eq:V=1reductioncomputation1}
\end{equation}
where $\xi_{\ws} \in \Lambda^*(H_1(W)/\Tors) \otimes \bF_2[\hat{U}]$ is an element
equal to $\hat{U}^{g(\Sigma_{\ws})}$ modulo $H_1(W)$. Since $H_1(Y)$
surjects onto $H_1(W)$, we can commute any summand of $\xi_{\ws}$ involving
the $H_1$-action past $F_{W,\frs}$ to obtain the relation
\[
\Red_{V=1}(F_{W,\cF,\frs}) \simeq \hat{U}^{g(S_{\ws})} \cdot F_{W,\frs} \mod H_1(Y).
\]
Applying the knotification functor, and using equation~\eqref{eq:kncommutesH1action}, we see that
\begin{equation}
(\Kn \circ \Red_{V=1})(F_{W,\cF,\frs}) \simeq \hat{U}^{g(S_{\ws})} \cdot \Kn(F_{W,\frs}) \mod H_1(Y).
\label{eq:relknotifiedtooriginal}
\end{equation}
Combining equation~\eqref{eq:relknotifiedtooriginal} and Lemma~\ref{lem:relknotifiedredandoriginal}, we conclude that
\begin{equation}
F_{W,\cF,\frs} \simeq  V^{h(S\cup D,\frs) + g(S_{\zs}) - g(S_{\ws})} \hat{U}^{g(S_{\ws})} \cdot \Kn(F_{W,\frs}) \mod H_1(Y). \label{eq:almostclaim1}
\end{equation}
Claim~\eqref{it:claim1} in this case follows by rearranging equation~\eqref{eq:almostclaim1}
using the fact that $\hat{U} = UV$.

The argument fails when the Alexander grading $h(S \cup
D,\frs) + g(S_{\zs}) - g(S_{\ws})$ is negative, since multiplication by
$V^{h(S \cup D,\frs) + g(S_{\zs}) - g(S_{\ws})}$ is not a filtered map. Instead we
must consider the $U = 1$ reduction of $F_{W,\cF,\frs}$. According to
\cite{ZemAbsoluteGradings}*{Lemma~9.6}, the $U = 1$ reduction satisfies
\[
\Red_{U=1}(F_{W,\cF,\frs}) \simeq F_{W,\frs - \PD[S]}(\xi_{\zs}\otimes -),
\]
for an element $\xi_{\zs} \in \Lambda^*(H_1(W)/\Tors) \otimes \bF_2[\hat{U}]$
equal to $\hat{U}^{g(S_{\zs})}$ modulo $H_1(W)$. Using this fact, the formula
\[
F_{W,\cF,\frs} \simeq U^{g(S_{\ws}) - h(S\cup D,\frs)} V^{g(S_{\zs})} \cdot \Kn(F_{W,\frs-\PD[S]}) \mod H_1(Y)
\]
can be established by the same strategy as before.

Next, we consider Claim~\eqref{it:claim2}, where $\cA$ contains a closed component. As in
the proof of Claim~\eqref{it:claim2}, the key will be to analyze the knotifications of
$\Red_{V=1}(F_{W,\cF,\frs})$ and $\Red_{U=1}(F_{W,\cF,\frs})$.
Let $\Delta$ denote the quantity
\[
\Delta := h(S \cup D,\frs) + \frac{\chi(S_{\ws}) - \chi(S_{\zs})}{2},
\]
which we note is the Alexander grading of the map $F_{W,\cF,\frs}$.

We first consider the case when $\Delta\ge 0$.
Let us write $C_{\ws,0}$ and $C_{\zs,0}$ for the components of $S_{\ws}$ and
$S_{\zs}$ that intersect $\d S$. We will reduce to the case when
$\d C_{\ws,0}$ contains a closed component disjoint from $\d S$. If $C_{\ws,0} = S_{\ws}$,
then $\d C_{\ws,0}$ trivially contains a closed component
disjoint from $\d S$. If $C_{\ws,0}$ is not the only component of $S_{\ws}$,
then, since $S$ is connected, we can find a properly embedded path
$\gamma_{\zs} \colon I \to C_{\zs,0}$, with both endpoints on $\cA$, such that
$\gamma_{\zs}(0) \in C_{\ws,0}$, and $\gamma_{\zs}(1)$ is a point in the
boundary of another component, $C_{\ws,1}$ of $S_{\ws}$. There are four cases we consider:
\begin{enumerate}
\item\label{it:1} $C_{\ws,1}$ is planar, and $|\d C_{\ws,1}| = 1$.
\item\label{it:2} $C_{\ws,1}$ is planar, and $|\d C_{\ws,1}| = 2$.
\item\label{it:3} $C_{\ws,1}$ is planar, and $|\d C_{\ws,1}| > 2$.
\item\label{it:4}  $g(C_{\ws,1})>0$.
\end{enumerate}

In Case~\eqref{it:1}, the surface $C_{\ws,1}$ is topologically a disk, which is
necessarily disjoint from $\d S$, since $|\cA \cap \d S| = 2$. We claim that the
map $F_{W,\cF,\frs} \simeq 0$. Indeed, the cobordism map
$F_{W,\cF,\frs}$ can be factored through the composition of a
quasi-stabilization, followed by a quasi-destabilization, and such a
composition clearly vanishes.

We next consider Case~\eqref{it:2}, when $C_{\ws,1}$ is an annulus, which is
disjoint from $\d S$. In this case, we also have $F_{W,\cF,\frs} \simeq 0$. To
see this, pick a properly embedded path $\gamma_{\ws} \colon I \to C_{\ws,1}$
that connects the two boundary components of $C_{\ws,1}$, and such that
$\gamma_{\ws}(0) = \gamma_{\zs}(1)$; see the top of Figure~\ref{fig::12}.
We concatenate $\gamma_{\zs}$ and $\gamma_{\ws}$ to get a path $\gamma$. A neighborhood of $\gamma$ is the
domain of a bypass. Let $\cA'$ and $\cA''$ denote the other two dividing sets
in the bypass triple; see the bottom row of Figure~\ref{fig::12}.
The bypass relation \cite{ZemConnectedSums}*{Lemma~1.4} implies that
\begin{equation}
F_{W,(S,\cA),\frs} \simeq F_{W,(S,\cA'),\frs} + F_{W,(S,\cA''),\frs}.
\label{eq:bypassforannulus}
\end{equation}
The key observation is that $\cA'$ and $\cA''$ are actually isotopic, so
equation~\eqref{eq:bypassforannulus} implies that $F_{W,(S,\cA),\frs} \simeq 0$.
The isotopy between $\cA'$ and $\cA''$ is shown in Figure~\ref{fig::12}.

\begin{figure}[ht!]
\centering
\input{fig12.pdf_tex}
\caption{When $C_{\ws,1}$ is an annulus, the two other dividing sets in a bypass are isotopic.}
\label{fig::12}
\end{figure}

We now consider Cases~\eqref{it:3} and~\eqref{it:4}, when $C_{\ws,1}$ is
planar and $|\d C_{\ws,1}| > 2$, or when $g(C_{\ws,1}) > 0$, respectively. In
both cases, we let $\gamma_{\ws} \colon I \to C_{\ws,1}$ be a properly
embedded curve which is non-separating and satisfies
$\gamma_{\zs}(1) = \gamma_{\ws}(0)$; see the top of Figure~\ref{fig::11}.
If $g(C_{\ws,1}) > 0$, we require that both
ends of $\gamma_{\ws}$ are on the same component of $\d C_{\ws,1}$. We let
$\gamma$ denote the concatenation of $\gamma_{\ws}$ and $\gamma_{\zs}$. As in
Case~\eqref{it:2}, we consider the bypass triple obtained by taking a regular
neighborhood of the image of $\gamma$.  Let $\cA'$ and $\cA''$ denote the
other two dividing sets in the bypass triple, shown on the bottom of Figure~\ref{fig::11}.

Let $C_{\ws,0}'$ and $C_{\ws,0}''$ denote the type-$\ws$ subregions of
$S \setminus \cA'$ and $S \setminus \cA''$ that intersect $\d S$.
In Cases~\eqref{it:3} and~\eqref{it:4}, it is easy to check that
$\d C_{\ws,0}'$ and $\d C_{\ws,0}''$ both contain a closed curve disjoint from $\d S$,
so it is sufficient to show the main claim for each of $F_{W,(S,\cA'),\frs}$ and
$F_{W,(S,\cA''),\frs}$ separately. Case~\eqref{it:4} is
illustrated in Figure~\ref{fig::11}.

\begin{figure}[ht!]
\centering
\input{fig11.pdf_tex}
\caption{The bypass relation in Case~\eqref{it:4}, when $g(C_{\ws,1}) > 0$. On the
bottom row, the domain of the bypass is shown. The dotted lines outside of
the domain of the bypass indicate the configuration of the dividing set
outside the bypass region.}
\label{fig::11}
\end{figure}

We now proceed to show that if $\cA$ is a dividing set on $S$ such that
$\d C_{\ws,0}$ contains a closed component $\gamma$ disjoint from $\d S$, then
\begin{equation}
\Red_{V=1}(F_{W,\cF,\frs}) = A_\gamma \circ G
\label{eq:closedloopimplieshomologyaction}
\end{equation}
for some map $G$, where $A_\gamma$ denotes the homology action of the curve
$\gamma$. To establish equation~\eqref{eq:closedloopimplieshomologyaction},
we must recall some additional facts about the functor
$\Red_{V=1}(F_{W,\cF,\frs})$. According to \cite{ZemCFLTQFT}*{Theorem~C}, the
chain homotopy type of the map $\Red_{V=1}(F_{W,\cF,\frs})$ depends only on
$W$, the embedded surface $S_{\ws}$ (which is not properly embedded) and
$\frs\in \Spin^c(W)$. To describe the reduction in more detail, we recall
that a \emph{ribbon 1-skeleton} of $S_{\ws}$ is a choice of embedded graph
$\Gamma_{\ws} \subset S_{\ws}$ such that $\Gamma_{\ws}\cap \d S_{\ws} = \{w\}$,
and $S_{\ws}$ is a regular neighborhood of $\Gamma_{\ws}$ in $S$; see
\cite{ZemCFLTQFT}*{Definition~14.5}.

There is a simple way to construct the ribbon 1-skeleton of the subsurface
$S_{\ws}$. One starts with a collection of arcs $a \subset S_{\ws}$ such that
$a \cap \d S_{\ws} = \{w\}$, and such that each component of $S_{\ws}$ contains
exactly one arc. One then takes inward translates $C_1, \dots, C_n$ of all but
one boundary component of each component of $S_{\ws}$, and connects them to