# Embedded surface invariants via the Broda-Petit construction

I.J. Lee and D.N. Yetter

(Date: March 2023)

##  1\. Introduction

In the late 1990’s B. Broda described a surgical approach to Crane-Yetter
theory via Kirby calculus [8], which he subsequently sought to improve in [2]
by using only “bosonic” labels on the 2-handle attaching curves, while using
all the irreducible representations of $U_{q}(sl_{2})$ for $q$ a root of unity
on the 1-handle attaching curves. The a priori more refined “dichromatic”
invariant, nonetheless, reduced in the case of simply-connected 4-manifolds to
an encoding of signature and Euler characteristic, as did Crane-Yetter theory.

The construction was, however, felt to be of sufficient interest that it was
generalized by J. Petit [12] and Bärenz and Barrett [1] to give what they
called “dichromatic” invariants of smooth 4-manifolds. The details of the two
constructions are slightly different: Petit uses a nested pair of ribbon
fusion categories in imitation of Broda’s use of the “bosonic” representations
as a subcategory of $Rep(U_{q}(sl_{2}))$, while Bärenz and Barrett use a
functor from a spherical category into a ribbon category, as initial
“coloring” data.

The present paper will take Petit’s construction as a starting point. We will
then use a tangle-theoretic rendering of the work of Hughes, Kim and Miller
[6, 7], which gives a Kirby-calculus-type presentation of a 4-manifold with an
embedded surface as a framed link diagram with “dotted” components as in Kirby
[8], together with a “banded unknot”, to modify the construction to be
sensitive to the presence of a specified embedded surface in the 4-manifold,
or more precisely, a specified surface embedded in a 4-dimensional
(0,1,2)-handlebody. We will not consider the immersed case Hughes et al. deal
with in [7], but suggest that the reader will, as we did, find the clearer
diagrammatics for the embedded case given in that paper helpful.

Throughout we adopt the convention that whenever a diagram of objects and
arrows is used syntactically as a statement, the assertion being made is that
the diagram commutes. Moreover, all 4-manifolds considered are connected.

##  2\. The Broda-Petit construction

In this section we review Petit’s [12] generalization of Broda’s [2]
construction, beginning with the necessary notions from category theory to
describe the initial coloring data.

######  Definition 2.1.

A monoidal category $\mathcal{C}$ is a category $\mathcal{C}$ equipped with a
functor $\otimes:{\mathcal{C}}\times{\mathcal{C}}\rightarrow{\mathcal{C}}$ and
an object $I$, together with natural isomorphisms
$\alpha:\otimes(\otimes\times
1_{\mathcal{C}})\Rightarrow\otimes(1_{\mathcal{C}}\times\otimes)$,
$\rho:\otimes I\Rightarrow 1_{\mathcal{C}}$, and $\lambda:I\otimes\Rightarrow
1_{\mathcal{C}}$, satisfying the pentagon, triangle, and bigon
($\rho_{I}=\lambda_{I}$) coherence conditions (cf. [9]). A tensor category
over $K$, for $K$ some field $K$, is a monoidal abelian category linear over
$K$, with all $-\otimes X$ and $X\otimes-$ exact.

######  Definition 2.2.

An object $X$ in an abelian category over $K$ is simple if its only subobjects
are $0$ and $X$, in which case $hom(X,X)$ is a field extension of $K$. An
abelian category is semi-simple if every object in it isomorphic to a direct
sum of simple objects. A multifusion category is a semi-simple tensor category
with finitely many isomorphism classes of simple objects. A fusion category is
a multifusion category in which, moreover the monoidal identity $I$ satisfies
${\rm hom}(I,I)\cong K$ (and is thus a fortiori simple).

We will be concerned with monoidal categories with additional structure.

######  Definition 2.3.

A braided monoidal category is a monoidal category equipped with a monoidal
natural isomorphism $\sigma:\otimes\Rightarrow\otimes(tw)$, where
$tw:{\mathcal{C}}\times{\mathcal{C}}\rightarrow{\mathcal{C}}\times{\mathcal{C}}$
is the “twist functor” ($tw(f,g)=(g,f)$). And satisfying

$(A\otimes B)\otimes C$$A\otimes(B\otimes C)$$(B\otimes C)\otimes A$$(B\otimes
A)\otimes C$$B\otimes(A\otimes C)$$B\otimes(C\otimes A)$$\alpha$$\sigma^{\pm
1}$$\alpha$$\sigma^{\pm 1}\otimes C$$\alpha$$B\otimes\sigma^{\pm 1}$

Here the same choice of $+1$ or $-1$ exponents must be made in all three
instances.

A braided monoidal category is a symmetric monoidal category if the components
of $\sigma$ satisfy $\sigma_{B,A}(\sigma_{A,B})=1_{A\otimes B}$ for all
objects $A$ and $B$.

As an aside, many authors write out two hexagons involving $\sigma$. One is
the one given above, the other is parenthesized differently and is equivalent
to that above with $\sigma^{-1}$.

Braided monoidal categories are, of course, generalizations of the more
classical notion of a symmetric monoidal category in which
$\sigma_{B,A}=\Sigma_{A,B}^{-1}$. It will be important for our purposes to
distinguish those object for which this equation holds, even when it does not
in general:

######  Definition 2.4.

An object $X$ in a braided monoidal category
${\mathcal{C}},\otimes,I,\alpha,\rho,\lambda,\sigma$ is transparent if

$\forall Y\in Ob({\mathcal{C}})\;\;\sigma_{X,Y}=\sigma_{Y,X}^{-1}$

The full subcategory of transparent objects is called the Müger center of the
category. In a braided fusion category ${\mathcal{C}}$, if $S_{\mathcal{C}}$
is a set of representatives of the isomorphisms of simple objects, the denote
the subset of $S_{\mathcal{C}}$ consisting of transparent objects by
$T_{\mathcal{C}}$

Because we will be working in a context which there is a distinguished
subcategory ${\mathcal{B}}\subset{\mathcal{C}}$ of a larger braided category,
we will also need

######  Definition 2.5.

Given a subcategory ${\mathcal{B}}\subset{\mathcal{C}}$ of a braided category
${\mathcal{C}}$ an object $X$ of ${\mathcal{C}}$ if
${\mathcal{B}}$-transparent if

$\forall Y\in Ob({\mathcal{B}})\;\;\sigma_{X,Y}=\sigma_{Y,X}^{-1}$

######  Definition 2.6.

A right dual to an object $X$ in a monoidal category $\mathcal{C}$ is an
object $X^{\ast}$ equipped with maps $\epsilon:X\otimes X^{\ast}\rightarrow I$
and $\eta:I\rightarrow X^{\ast}\otimes X$ such that the compositions

$X\stackrel{{\scriptstyle\rho^{-1}}}{{\rightarrow}}X\otimes
I\stackrel{{\scriptstyle X\otimes\eta}}{{\rightarrow}}X\otimes(X^{\ast}\otimes
X)\stackrel{{\scriptstyle\alpha^{-1}}}{{\rightarrow}}(X\otimes
X^{\ast})\otimes X\stackrel{{\scriptstyle\epsilon\otimes
X}}{{\rightarrow}}I\otimes X\stackrel{{\scriptstyle\lambda}}{{\rightarrow}}X$

and

$X^{\ast}\stackrel{{\scriptstyle\lambda^{-1}}}{{\rightarrow}}I\otimes
X^{\ast}\stackrel{{\scriptstyle\eta\otimes
X^{\ast}}}{{\rightarrow}}(X^{\ast}\otimes X)\otimes
X^{\ast}\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}X^{\ast}\otimes(X\otimes
X^{\ast})\stackrel{{\scriptstyle
X^{\ast}\otimes\epsilon}}{{\rightarrow}}X^{\ast}\otimes
I\stackrel{{\scriptstyle\rho}}{{\rightarrow}}X^{\ast}$

are both identity maps.

A left dual ${}^{\ast}X$ is defined similarly, but with the objects placed on
opposite sides of the monoidal product.

This type of duality is an abstraction from the sort of duality which exists
in categories of finite dimensional vector spaces. It is not hard to show that
the canonical isomorphism from the second dual of a vector space to the space
generalizes to give canonical isomorphisms $k:^{\ast}(X^{\ast})\rightarrow X$
and $\kappa:(^{\ast}X)^{\ast}\rightarrow X$. In general, however, there are
not necessarily even any maps (or non-zero maps in the linear context) from
$X^{\ast\ast}$ or ${}^{\ast\ast}X$ to X. (cf. [5]). In cases where every
object admits a right (resp. left) dual, it is easy to show that a choice of
right (resp. left) dual for every object extends to a contravariant functor,
whose application to maps will be denoted $f^{\ast}$ (resp. ${}^{\ast}f$), and
that the canonical maps noted above become natural isomorphisms between the
compositions of these functors and the identity functor. Likewise, it is easy
to show that $(A\otimes B)^{\ast}$ is canonically isomorphic to
$B^{\ast}\otimes A^{\ast}$ and similarly for left duals.

######  Definition 2.7.

A pivotal category is a monoidal category with right duals together with a
natural isomorphism $a_{X}:X\rightarrow X^{\ast\ast}$.

It is easy to see that a pivotal structure induces the structure of a left
dual on each right dual.

######  Definition 2.8.

Let $X$ be an object in a pivotal category $\mathcal{C}$. Left and right
traces $tr_{L}$, $tr_{R}:\coprod_{X\in
Ob{\mathcal{C}}}\mathcal{C}(X,X)\to\mathcal{C}(I,I)$ are be defined as
follows:

$tr_{R}(f)=\epsilon_{X}\,(a_{X}^{-1}f\otimes 1)\,\eta_{X^{*}}$
$tr_{L}(f)=\epsilon_{X^{*}}\,(1\otimes fa_{X})\,\eta_{X}$

######  Definition 2.9.

A pivotal category $\mathcal{C}$ is spherical if it satisfies $tr_{L}(f)$ =
$tr_{R}(f)$ for all endomorphisms $f$. In the context of spherical categories,
we can thus suppress the distinction between left and right traces, and simply
write $tr(f)$.

In any spherical category, the (quantum) dimension of an object $X$ is the
endomorphism of the monoidal identity object given by $dim(X)=tr(1_{X})$, the
trace of its identity map. In the case of a spherical fusion category over
$K$, it may be regarded as an element of the field $K$.

The dimension of a spherical fusion category ${\mathcal{C}}$ is

$\Delta_{\mathcal{C}}:=\sum_{X\in S_{\mathcal{C}}}dim(X)^{2}$

where $S_{\mathcal{C}}$ is a set containing one object from each isomorphism
class of simple objects in ${\mathcal{C}}$.

The following proposition is easy to establish:

######  Proposition 2.10.

In any fusion category, the family of maps ${\mathsf{d}}_{X}:X\rightarrow X$
which multiplies each direct summand of $X$ by its quantum dimension is
independent of the choices decompositions into direct summands and forms a
natural transformation.

The components of ${\mathsf{d}}$ at the objects $B_{\mathcal{C}}:=\oplus_{X\in
S({\mathcal{C}})}X$ will be used in our description of Petit’s invariants. It
is easy to see that $tr({\mathsf{d}}_{B_{\mathcal{C}}}=\Delta_{\mathcal{C}}$.

In the case of a braided monoidal category every right dual is also a left
dual, in general in a non-canonical way (cf. [5]). In symmetric monoidal
categories, we return to the familiar: right duals are canonically left duals.
For non-symmetric braided monoidal categories, the structure will be canonical
only in the presence of additional structure. As we will be concerned only
with the case of braided categories in which all objects admit such two-sided
duals, we make

######  Definition 2.11.

A braided monoidal category $\mathcal{C}$ is ribbon (or tortile) if all
objects admit right duals, and it is equipped with a natural transformation
$\theta:1_{\mathcal{C}}\Rightarrow 1_{\mathcal{C}}$, which satisfies

$\theta_{A\otimes B}=\sigma_{B,A}(\sigma_{A,B}(\theta_{A}\otimes\theta_{B}))$

and

$\theta_{A^{\ast}}=\theta_{A}^{\ast}$

Ribbon fusion categories are also called premodular categories.

Any ribbon category is spherical when equipped with left duals whose structure
maps are obtained by pre- (resp. post-)composing $epsilon$ (resp. $\eta$) by a
component of $\sigma$ (resp. $\sigma^{-1}$). For details of the canonicity of
the correspondence between right and left duals in the ribbon or tortile case
see [16].

The importance of ribbon categories for knot theory and 3- and 4-manifold
topology is provided by an extremely beautiful special case of the coherence
theorem of Shum which characterized ribbon categories freely generated by a
small category [15]:

######  Theorem 2.12.

The ribbon category freely generated by a single object $X$ is monoidally
equivalent to the category of framed tangles $\mathcal{F}T$.

The relationship between framed links and surgical presentations of
4-manifolds provided by the Kirby calculus [8] then provides the starting
point for the construction of 4-manifold invariants given by Broda [2] and its
generalization by Petit [12]. Shum’s full coherence theorem also is the
justification for the now commonplace use of knot-theoretic features in
string-diagram calculations for maps in ribbon categories, which we will use
without further comment.

Before recalling the construction of Petit [12] in sufficient detail for our
purposes, we prove a purely knot-theoretic lemma, which we feel provides a
wholely satisfactory explanation of why functorial knot invariants can be
modified by introducing numerical coefficients (of quantum dimensions) on
direct summands of the object to which a strand in the knot diagram is mapped
has been done in [14, 2, 12]:

######  Theorem 2.13.

Suppose $L_{0}$ and $L_{1}$ are two equivalent links in ${\mathbb{R}}^{3}$,
and suppose, moreover, that each component of $L_{0}$ and $L_{1}$ has a unique
global minimum. Then there is an ambient isotopy $H$ which preserves the
component-wise global minima throughout the isotopy in the sense that if
$S_{t}$ denotes the set of component-wise global minima on $L_{t}=H(L_{0},t)$,
then $S_{t}=H(S_{0},t)$ for all $t\in[0,1]$.

###### Proof.

Begin by modifying each of $L_{0}$ and $L_{1}$ using an isotopy by
$\Delta$-moves, in the sense of Reidemeister [13], to move the component-wise
global minima below the plane $z=-1$ and leave the rest of each link lying
between $z=0$ and $z=1$, except for edges incident with the global minima.
Plainly, this can be done without moving any points below the minima of their
respective components.

It thus suffices to see that there is an isotopy between the modified links
$L_{0}^{\prime}$ and $L_{1}^{\prime}$ with the desired properties. Mark each
of the global minima with a $\wedge$ on the curve. Now view the links in
projection onto the $xy$-plane along a vector arbitrarily close to vertical
for which the projection is a knot diagram with only transverse double points.

Now, a sequence of Reidemeister moves and isotopies of the projection (all
corresponding to $\Delta$-moves in ${\mathbb{R}}^{3}$) will turn the one
diagram into the other, including sending the marked minima to each other.

The problem is that some of these moves may correspond to moves in
${\mathbb{R}}^{3}$ which lower points below the global minimum of the
component on which they lie.

Specifically, instances of the Reidemeister II (resp., III) move in which an
arc (resp., one or two arcs incident with a crossing) passes under another arc
containing a minimum, and isotopies of the projection in which an
undercrossing arc moves from one side of a minimum to the other correspond to
portions of an isotopy which may violate the preservation of global minima.
When the arc passed under belongs to a different component from the minimum,
this may not violate the condition. If it does, it is easily remedied: lower
the minimum of the underpassing component well below the minimum being passed
under before doing the move.

In the case where the undercrossing arc belongs to the same component, we need
to do more work: with the minimum set well below undercrossing arc, the move
on the diagram corresponds to a sequence of $\Delta$-moves, most of which
simply lower the offending arc without changing the diagram, one of which
passes it below what should have been the global minimum, and the rest of
which bring the moved arc back up into the region where the rest of the link
is to lie at the end of the isotopy.

The “before” and “after” positions of the arc changed by the move form a loop
which we may assume lies between $z=0$ and $z=1$. Lower the global min of the
component involved below all of the other global mins by an isotopy fixing the
link above $z=0$. (Doing this by $\Delta$-moves without violating the global
minimum preservation may require briefly introducing a vertical segment
projecting to the same point, but this is resolved immediately by another
$\Delta$-move.)

Now, complete the disk across which the errant move was made to a polyhedral
sphere, intersecting the initial and final links only in the segments changed
by the move, and such that the complementary disk forming the sphere lies
entirely above the just-lowered minimum. (The new faces will slope down and
slightly away from the boundary of the old disk to a level between the other
minima and the lowered one on the component being modified, then level off to
form a “floor” extending beyond the link in all horizontal directions, “walls”
extending above the link, and a “ceiling” completing the polyhedral sphere
above the link.) The move can be replaced with a sequence of $\Delta$-moves
across the complementary disk of the sphere.

Thus, no moves which fail to preserve the global minima are necessary and the
theorem holds. ∎

A convenient way to visualize the last step in the proof is to imagine the
component on which the offending move is to be made as a person holding a
jump-rope (resp. a pair of crossed jump-ropes for the Reidemeister III case),
and wanting to get it (resp. them) from in front of their feet to behind. The
offending Reidemeister move involves jumping, to bring the rope(s) under the
feet, but the same position is obtained by swinging the ropes over one’s head
and bringing them down behind the feet.

As an aside, the theorem very much depends on the availability of isotopies
which raise and lower the component-wise global minima. If the minima were
fixed, one could assume they are fixed on a “sphere at infinity” and one would
be effectively dealing with isotopy classes of string links, which it is well-
known are are not equivalent to isotopy classes of links.

######  Corollary 2.14.

Any ambient isotopy invariant of (framed) links equipped with a distinguished
base-point on each component gives rise to an ambient isotopy invariant of
links by choosing the global minimum with respect to some coordinate direction
to be the base-point. Moreover, the same is true for isotopy invariants of
colored (framed) links (that is links in which the components are divided into
disjoint subsets designated by different colors).

Note that the choice of base-points or coordinates is really irrelevant: given
base-points, one could first drag them down in the coordinate direction to
make them the global minima, or, given minima with respect to a different
coordinate direction, one could first rotate a the link so that the “downward”
directions align; then use the procedure of the proof to isotope one to the
other preserving the minima.

This lemma then gives a satisfactory explanation of how to introduce in a
functorial way the coefficients on which the constructions of 3-manifold
invariants of Reshetikhin and Turaev [14] and 4-manifold invariants of Broda
[2] and Petit [12] depend: the base point, regarded as an endomorphism of the
object whose identity arrow is denoted as a descending arc, is mapped to the
map we have denoted ${\mathsf{d}}$ which multiplies the direct summands of the
coloring object by their quantum dimensions.

We now redescribe the construction of Petit’s dichromatic invariants of
4-manifolds, making use of the foregoing, beginning by recalling the usual
notions from the Kirby calculus [8].

######  Definition 2.15.

A Kirby diagram is a blackboard framed diagram of an oriented framed link $L$,
with a distinguished subset of its components forming a 0-framed unlink,
$L_{0}$.

Traditionally the components of $L_{0}$ are indicated by placing a large dot
somewhere on the component.

From Kirby [8] recall that such a diagram describes a 4-dimensional
0-1-2-handlebody, by regarding it as lying in the bounding $S^{3}$ of a 4-ball
$B^{4}$ (the 0-handle), regarding the components of $L_{0}$ as locations from
which 2-handles are to be removed from the interior of $B^{4}$, thereby
effectively attaching a 1-handle which would be cancelled by the replacement
of the removed 2-handle, and the components of $L\setminus L_{0}$ as giving
instructions for the attaching of 2-handles. Moreover, if the boundary of the
resulting handlebody is diffeomorphic to the boundary of a 0-1-handlebody,
there will be a unique way to complete the handlebody with 3-handles and a
4-handle to give a closed 4-manifold, and all closed connected smooth
4-manifolds arise in this way.

Also, as shown in [8], two such diagrams describe diffeomorphic handlebodies
(closed 4-manifolds, when the boundary is appropriate) if and only if they are
related by a sequence of (1) isotopies (which, using the blackboard framing
convention are encoded by the framed Reidemeister moves, the usual first
Reidemeister move being replaced with a move which does not remove the curl,
but moves it to the other side of the strand, keeping the same crossing), (2)
handle slide moves of one component over another, in which the component “slid
over” the other is replaced with its band sum with a parallel copy of the the
component over which it was slid, pushed off in the direction of the framing,
with slides of components in $L_{0}$ over components not in $L_{0}$ being
forbidden, as they do not correspond to a geometric handle-slide, and (3)
stablilization moves: addition or removal of a 0-framed component linked with
a dotted component, or (for closed 4-manifolds) addition or removal of a
0-framed component, in either case, lying in a disk in the plane of the
diagram which is disjoint from the rest of the diagram.

######  Definition 2.16.

A based Kirby diagram is a Kirby diagram with a base point chosen on each
component, lying on a strand oriented downward.

In a category of tangles, a marked point on a strand may be taken to be an
endomorphism of the object whose identity map an unmarked strand represents.
Doing this allows us to make the usual surgical constructions of 3- and
4-manifold invariants via the Kirby calculus functorial.

Consider the free ribbon category ${\mathcal{K}}$ generated by a category with
two objects, ${\bf 1}$ and ${\bf 2}$, each of which have a copy of
$({\mathbb{N}},+,0)$ as their set of endomorphisms, these being the only
arrows in the generating category. An easy application of Shum’s [15] full
coherence theorem shows that this is equivalent to the category of framed
tangles with marked points (with no limit on how many marked points occur or
where they occur).

Based Kirby diagrams can be regarded as forming a particular sub-monoid (under
separated union) of the commutative monoid of endomorphisms of the monoidal
identity ${\mathcal{K}}$ – namely that sub-monoid whose elements, when written
as string diagrams, have exactly one marked point on each connected component,
and with connected components colored ${\bf 1}$ forming a 0-framed unlink.

The effects of Petit’s “Kirby colors” are then obtained by applying the
functor induced by freeness, mapping ${\bf 1}$ (resp. ${\bf 2}$) to the object
$B_{\mathcal{C}}:=\oplus_{X\in S({\mathcal{C}})}X$, (resp.
$B_{\mathcal{B}}:=\oplus_{X\in S({\mathcal{B}})}X$), and the marked point
($1\in{\mathbb{N}}$ as the generating endomorphism of the object), to
${\mathsf{d}}_{B_{\mathcal{C}}}$ (resp. ${\mathsf{d}}_{B_{\mathcal{B}}}$),
where $S({\mathcal{C}})$ (resp. $S({\mathcal{B}})$) is a set of representative
objects for the isomorphism classes of simple objects in ${\mathcal{C}}$
(resp. ${\mathcal{B}}$).

We call this functor “the Kirby functor”, denote it by $\big{<}-\big{>}$,
observe that as $\mathcal{C}$ is assumed to be a fusion category, its values
on based Kirby diagrams are scalar multiples of the identity arrow on the
monoidal identity, with which scalars we identify the images of based Kirby
diagrams under the functor.

By Shum’s coherence theorem, the values of the Kirby functor are isotopy
invariant. For invariance under handle-slides Petit [12] relies on results of
Brugières [3], but a clearer proof of the necessary invariance properties can
be found by applying Lemma 3.3 of Bärenz and Barrett [1] to the identity
functor on ${\mathcal{C}}$ and the inclusion functor of ${\mathcal{B}}$ into
${\mathcal{C}}$, which makes obvious why the sliding property holds, even if
the link-component slid over is knotted, non-trivally framed or linked with
other components.

We will also have call to use the generalization of the Lickorish encirclement
property given by Brugières [3]:

######  Lemma 2.17.

Let $\mathcal{C}$ be a ribbon fusion category with
$\Delta_{\mathcal{C}}\in\mathbb{k}^{*}$. For every object $X$ of
$\mathcal{C}$, we have:

$\displaystyle($ ϵ_B_C^* ⊗1_X )( 1 ⊗d_B_C⊗1_X) (1_B_C^*
⊗σ_B_C,Xσ_X,B_C)(η_B_C⊗1_X)
$\displaystyle=\left\\{\begin{array}[]{cl}\Delta_{\mathcal{C}}\,\,1_{X}&\textrm{if
$X$ is a transparent object},\\\ 0&\textrm{otherwise}.\end{array}\,\right.$

for every simple object $X$.

The map in Lemma 2.17 is given in string diagram notation by

Restated in our favored terminology, the main result of Petit [12] regarding
4-manifolds is then:

######  Theorem 2.18.

Let $\mathcal{B}\subset\mathcal{C}$ be ribbon fusion categories such that
their dimensions and the quantity

$\Delta_{\mathcal{C},\mathcal{B}}^{\prime\prime}=\sum_{X\in
S_{\mathcal{B}}\cap T_{\mathcal{C}}}\textrm{dim}(X)^{2}$

are invertible, $W$ a closed, compact, connected, and oriented 4-manifold, and
$L$ a based Kirby diagram which represents $W$, then the following scalar is
independent of the choice of the based Kirby diagram, and thus an invariant of
the 4-manifold $W$,

$I_{+}(W)=\frac{\big{<}L\big{>}}{(\Delta_{\mathcal{B}})^{b_{0}(L)}\,(\Delta_{\mathcal{C}}\,\Delta_{\mathcal{C},\mathcal{B}}^{\prime\prime})^{b_{+}(L)}},$

where $\displaystyle\Delta_{\mathcal{C},\mathcal{B}}^{\prime\prime}=\sum_{X\in
S_{\mathcal{B}}\cap T_{\mathcal{C}}}\textrm{dim}(X)^{2}$, and
$b_{\epsilon}(L)$ be the number eigenvalues of the linking matrix of $L$ with
the sign $\epsilon\in\\{0,-,+\\}.$

To prove that $I_{+}$ is an invariant of closed and oriented 4-manifolds, we
need to prove that $I_{+}$ is invariant under the Kirby moves. By construction
of Kirby functor, we know that $\big{<}L\big{>}$ is invariant under isotopies
of framed link and handle slides. In addition, the linking matrix of $L_{0}$
does not change under isotopies. Therefore, $I_{+}$ is invariant under
isotopies.

Separated union with a 0-framed component plainly increases the number
multiplicity of zero as an eigenvalue of the linking matrix by 1 and
multiplies $\big{<}L\big{>}$ by $\Delta_{\mathcal{B}}$, while separated union
with a Hopf link with one component 0-framed and one dotted component
increases the number of positive eigenvalues by one and, applying Lemma 2.17
to evaluate the functor on the Hopf link, multiplies $\big{<}L\big{>}$ by
$\Delta_{\mathcal{C}}\,\Delta_{\mathcal{C},\mathcal{B}}^{\prime\prime}$, so
plainly $I_{+}(W)$ is invariant under the stabilization moves.

As an aside we note that the powers on the normalization factors in the
denominator could have been chosen differently. In particular, instead of
$b_{+}(L)$, the rank of the linking matrix or the number of dotted components
could have been used, while $b_{0}(L)$ could be replace with the number of
2-handle attaching curves minus the number of dotted components. In fact, in
the present work in which after introducing surfaces, we obtain only
invariants of surfaces in (0,1,2)-handlebodies, the first factor in the
denominator could be omitted entirely. Different normalizations may be
preferable in various contexts, but we will not examine the question here and
simply use Petit’s choice.

##  3\. Generalized Broda-Petit invariants for 4-manifold/surface pairs

In [6] Hughes at al. described an extension of the Kirby calculus which
provided an encoding of smooth 4-manifolds with an embedded surface. This
encoding is provided by a more complicated diagrammatics similar to that of
Kirby [8]: besides the framed link encoding the 2-handle attachments and the
dotted 0-framed unlink encoding the 1-handles as locations to bore out a
2-handle, their diagrams have another (0-framed, since we are using blackboard
framings) unknot, between whose components there are bands.

This “banded unknot” encodes a surface by regarding the unlink as lying in a
3-sphere just inside the bounding 3-sphere of the 0-handle of the 4-manifold,
thought of as the standard 4-ball, and having disks hanging inward toward the
origin, as the 0-handles of the surface, the bands should be thought of as
rising slightly outward, and are the 1-handles of the surface. The 2-handles
are then disks attached to the circles formed by the closure of the symmetric
difference between union of the band boundaries and the unlink. The
requirements that these disks exist is a condition on the banded unlink, which
in particular implies that its components have linking number zero with those
of the dotted unlink encoding the 1-handles.

Moreover, Hughes et al. show that two presentations in this form of
4-manifolds with embedded surfaces give diffeomorphic pairs if and only if
they are related by a sequence of moves of the following types:

$\bullet$ isotopy:

sequences of framed Reidemeister moves;

$\bullet$ handle slides:

encoded as replacing the sliding component with its band-sum with the
component slide over, with the same restrictions as in the Kirby calculus [8];

$\bullet$ Kirby stablization:

separated union with a zero-framed 2-handle attaching curve, or with a Hopf
link formed of a 0-framed 2-handle attaching curve and a dotted unknot, or
undoing such a move;

$\bullet$ cap:

$\bullet$ cup:

$\bullet$ slides of surface:

encoded as band sum moves of a band over any component of the Kirby diagram,
or of an unlink component over a dotted component of the Kirby diagram

$\bullet$ band slide:

$\bullet$ band swim:

$\bullet$ band/2-handle swim:

$\includegraphics[width=108.405pt]{6.pdf}.$

In the moves shown, rather than described, above, the black arcs are part of
the unknot encoding the surface, the thick red arcs are bands in the surface
encoding, and the green arc is part of a 2-handle attaching curve.

The reader may note that we have drawn the last three moves in a slightly
different position than Hughes et al. do in [7]. Doing this makes their
translation into an algebraic structure in a ribbon category more obvious.

The key to translating the moves into algebraic form is the band slide move:
the geometric justification for it includes an intermediate step in which the
upper foot of the lower band in our diagram above in attached to the middle of
the upper band. Readers familiar with the development of algebraic approaches
to topological field theory will recognize that if the band (when running
vertically) is thought of as the identity arrow on an object in a ribbon
category, that object must have the structure of a symmetric Frobenius algebra
with the multiplication (or comultiplication, depending on the exact geometry)
given by the intermediate step in the band slide move (cf. [10]):

######  Definition 3.1.

In any monoidal category, a Frobenius algebra (object) is an object $F$,
equipped with an associative, unital multiplication $\mu:F\otimes F\rightarrow
F$ (which we will write as the null infix when applied to elements of
underlying sets or vector spaces, when the fusion category admits a faithful
underlying functor), with unit $e:I\rightarrow F$ and a coassociative,
counital comultiplication $\Delta:F\rightarrow F\otimes F$, with counit map
$\epsilon:F\rightarrow I$, and moreover satisfying

$(1\otimes\mu)(\Delta\otimes 1)=\Delta\mu=(\mu\otimes 1)(1\otimes\Delta)$

where we have suppressed mention of the associator by MacLane’s coherence
theorem.

And, more than that, the state of the band slide move shown on the right,
together with this intermediate state requires that the object whose identity
arrow is a vertical part of the unlink which is part of the surface encoding
must be a module over this Frobenius algebra. Here a little care is needed:
should it be a left module, a right module or a bimodule? To have a
satisfactory answer, we will require that our embedded surfaces be oriented.
This is accomplished by orienting the unlink encoding the surface, and
requiring that this orientation extend compatibly to the curve bounding the
upper disks in the description given above. The surface can then be given the
orientation such that the orientation on the 0-handles has the given
orientation on the unlink as boundary orientation.

Thus, if we perform isotopies so that bands have no half-twists, a band with
feet on vertical sections of the unknot must run from an upward oriented
section to a downward oriented section. Moreover, since the unknot has no
thickness, by performing an isotopy that lifts part of the band over (or
pushes part of the band under) an unknot component, we may assume that the
band alwasys approaches downward (resp. upward) oriented parts of the unknot
from the left (resp. right).

From this and the band slide move with its intermediate step included, we see
that a band foot, with the band approaching the unknot from above (resp.
below), is a string diagram representation of an action (resp. coaction) of a
Frobenius algebra on a module, this being a left module if the bit of unknot
is oriented downward, and a right module if it is oriented upward:

######  Definition 3.2.

Let $M$ be a left module over $F$ with a left action,

$\mu:F\otimes M\rightarrow M\,\,\,\textrm{by}\,\,\,a\otimes m\mapsto am$

satisfying

  * •

$a(bm)=(ab)m$

  * •

$(1_{F})m=m$

Right modules are defined similarly.

We can depict it using red lines as follows for $F$, and black for $M$; for
$a,b\in F,m\in M,$

$a$$m$$am$$\mu:F\otimes M\rightarrow M\,\,\,\,\,\textrm{by}\,\,\,\,\,a\otimes
m\mapsto am$

$a$$b$$m$$abm$$=$ |  $a$$b$$m$$abm$
---|---

$(1_{F})m$$=$ |  $m$
---|---

The corresponding equations for right modules have string diagram
representations which are mirror images of these in a vertical line of
reflection.

A module $M$ over $F$ always admits a comodule structure on the same side, the
coaction being give by precomposing the action by the monoidal product of the
copairing with the identity map on the module. With this coaction the
equations in string diagrams below always hold:

$=$ | $=$ |
---|---|---

When it is recalled that duality extends to a contravariant functor and that
Frobenius algebra objects are canonically self-dual, for $F$ a Frobenius
algebra object and $M$ any left module, we always have the following maps:

$\mu:F\otimes F\rightarrow F$ |  $\Delta:F\rightarrow F\otimes F$
---|---
$m:F\otimes M\rightarrow M$ |  $\delta:M\rightarrow F\otimes M$
$\delta^{*}=m:M^{*}\otimes F\rightarrow M^{*}$ |  $m^{*}=\delta:M^{*}\rightarrow M^{*}\otimes F$.

Notice that we have drawn the red arcs representing the identity maps on the
Frobenius algebra object with greater thickness than the black arcs
representing the module, in keeping with our argument that formally the bands
of Hughes et al. [6, 7] (when running vertically) should be (identity arrows
on) a Frobenius algebra object in a ribbon category.

The foregoing discussion, together with the arguments from Petit [12] has
almost established our main theorem. It remains only to discuss the
categorical conditions which give invariance under the cup, cap, surface slide
and swimming moves, which we explicitly state as part of the theorem:

######  Theorem 3.3.

If ${\mathcal{C}}\supset{\mathcal{B}}$ is nested pair of ribbon fusion
categories, such that $\Delta_{\mathcal{C}}$, $\Delta_{\mathcal{B}}$ and
$\Delta_{{\mathcal{C}},{\mathcal{B}}}^{\prime\prime}$ are non-zero, and
$(F,\mu,e,\Delta,\epsilon)$ is a symmetric Frobenius algebra object in
$\mathcal{B}$, $(M,m)$ a left $F$-module in $\mathcal{C}$, with induced
coaction $\delta:=1_{F}\otimes m(\Delta(e))\otimes 1_{M}$, moreover satisfying

  * cap

there exists a scalar $k$ such that

$m(\delta)=k\cdot 1_{M}$

  * cup

there exits a scalar $\kappa$ such that

$1_{M}\otimes ev_{M}(m\otimes m\otimes 1_{M^{*}}(1_{M}\otimes
coev_{M}))=\kappa\cdot 1_{M}$

  * swim

the image of

$(m\otimes m)(1\otimes\Delta(1_{F})\otimes 1):M^{*}\otimes M\rightarrow
M^{*}\otimes M$

is ${\mathcal{B}}$-transparent.

Then the evaluation resulting from coloring a based Kirby diagram with banded
unlink by

  * •

coloring every two-handle attaching curve with $B_{\mathcal{B}}$, and
inserting the map ${\mathsf{d}}_{B_{\mathcal{B}}}$ at the base point ,

  * •

coloring every one-handle attaching curve with $B_{\mathcal{C}}$, and
inserting the map ${\mathsf{d}}_{B_{\mathcal{C}}}$ at the base point ,

  * •

coloring each band with $F$

  * •

coloring each unlink component with $M$ on the downward sections and $M^{*}$
on the upward sections

and interpreting the meetings of bands and unlink components as the action,
when normalized by dividing by
$\Delta_{\mathcal{B}}^{b_{0}}(\Delta_{\mathcal{C}}\,\Delta_{\mathcal{C},\mathcal{B}}^{\prime\prime})^{b_{+}}k^{s}\kappa^{\omega}$,
where $b_{0}$ is the multiplicity of 0 as an eigenvalue in the linking matrix
of the framed link describing the 1- and 2-handles, $b_{+}$ is the number of
positive eigenvalues in the linking matrix of the framed link describing the
1- and 2-handles, $s$ is the number of self-bands (bands with both feet on the
same components of the unknot) and $\omega$ is the number of other-bands
(bands with feet on two different components of the unknot), is an invariant
of the surface-(0,1,2)-handlebody pair presented by the diagram.

###### Proof.

Invariance under the band slide move was discussed in our explanation of why a
Frobenius algebra object and a module over it are the appropriate coloring
data for the bands and unlink components.

Invariance under handle slide moves follows from the lemma of Bärenz and
Barrett [1] discussed previous section, which extends easily to give the
surface slide moves when the band is an object in $\mathcal{B}$.

The conditions cap and cup, together with the specified normalizations plainly
render the normalized evaluation invariant under the cap and cup moves.

Finally, invariance under the band and 2-handle swimming moves follows from
the $\mathcal{B}$-transparency of the image of the “action in the middle of
the copairing”, as the map given by left-hand side of the band swim and
band/2-handle swim moves, regarded as string diagrams are unchanged by
inserting a projection from $M^{*}\otimes M$ onto the image, followed by the
inclusion of the image just below where the legs of the copairing act on the
dual module and module. ∎

Notice that we have not claimed that the normalized evaluation is an invariant
of the surface-closed-4-manifold pair presented by the diagram, though for
consistency with Petit [12] we have retained the normalization factor which
makes the quantity invariant under the removal of a 2-handle cancellable by
the addition of a 3-handle.

This is because the evaluation is not invariant under the removal of 3-handle-
cancellable 2-handles from which some part of the surface cannot be removed by
ambient isotopy. We illustrate this lack of invariance in one of our examples
in the next section.

##  4\. Examples from group theoretic fusion categories

Having established our main theorem, it is incumbent upon us to supply at
least some class of examples of the necessary algebraic data. In this section
will discuss one such class of examples and do some sample calculations of the
resulting invariants. There are some unsatisfactory aspects to the class of
initial data we explicitly describe, which we will discuss in our concluding
section on directions for future research.

We begin with a standard result from the theory of fusion categories:

######  Proposition 4.1.

For any finite abelian group $G$, together with a bicharacter $\beta:G\times
G\rightarrow\mathbf{k}^{\times}$, and a 3-cocycle $a:G\times G\times
G\rightarrow\mathbf{k}^{\times}$ category $\mathbf{Vec}_{G}$ of $G$-graded
vector spaces over $\mathbf{k}$ is a ribbon category, with duals of
homogeneous objects given by the dual vector space in the inverse degree,
associator given by multiplication by $a$ on triples of homogeneous objects,
braiding given on homogeneous objects of degrees $g$ and $\gamma$ by
$\beta(g,\gamma)\cdot tw$, where $tw$ is the usual symmetry for tensor product
of vector spaces, and the ribbon twist given on objects homogeneous of degree
$g$ by multiplication by $\beta(g,g)$.

We thus have a large class of categories from which Petit’s construction [12]
gives rise to invariants of smooth 4-manifolds:
$\mathbf{Vec}_{H}\subset\mathbf{Vec}_{G}$ for any group-subgroup pair
$G\supset H$, with associator given by a 3-cocycle on $G$ and braiding and
ribbon structure induced by a bicharacter on $G$. Moreover, as all of the
simple objects are invertible, their quantum dimensions are all 1, so that
$\Delta_{\mathbf{Vec}_{\Gamma}}=|\Gamma|$ for $\Gamma=H,G$. Also because of
this, in this section we omit any mention of basepoints, as for these examples
${\mathsf{d}}$ is the identity natural transformation.

We wish to give examples of suitable Frobenius algebras in some of these and
modules over them which satisfy the hypotheses of the main theorem, and thus
give rise to invariants of 4-manifolds equipped with an embedded surface.

######  Definition 4.2.

A graded matrix algebra $M(n,\Gamma)$ in ${\mathbb{V}ec}_{A}$, for $A$ an
abelian group is the object

$\bigoplus_{i,j=1}^{n}\mathbf{k}_{g_{ij}}$

for $\Gamma=[g_{ij}]_{n\times n}$ a matrix of elements of $A$ satisfying

$\forall i,j,k\;g_{ij}+g_{jk}=g_{ij},$

equipped with matrix multiplication in the obvious sense as multiplication,
unit the identity matrix $I$ (with the 1’s on the diagonal all necessarily
being of degree $0$).

It is a Frobenius algebra when equipped with the functional

$tr=\sum_{i=1}^{n}p_{ii}:M(n,\Gamma)\rightarrow I=\mathbf{k}_{0}$

which induces a pairing $(X,Y)\mapsto tr(XY)$, thereby inducing a copairing
and the Frobenius comultiplication $\Delta$ (in terms of which the copairing
is $\Delta(I)$), as happens when the classical Frobenius algebra structure on
a matrix algebra is rewritten as an example of the categorical notion.

To construct the desired example of a category-subcategory pair suitable for
the Broda-Petit construction with a Frobenius algebra $F$ in the subcategory
${\mathcal{B}}$ and module over $M$ in the ambient category ${\mathcal{C}}$
for which none $F$, $M$ and $M^{*}\otimes M$ is either transparent or
$\mathcal{B}$-transparent, but the image of

$(m\otimes m)1\otimes\Delta(1_{F})\otimes 1):M^{*}\otimes M\rightarrow
M^{*}\otimes M$

is $\mathcal{B}$-transparent (and actually, transparent) we consider an
abelian group $G$ and a proper subgroup $H$ which contains an element $c\neq
0$

For any fixed $g\in G\setminus H$, the object
$X_{g}=\mathbf{k}_{c+g}\oplus\mathbf{k}_{g}$, regarded as column vectors with
lower entry of degree $g$, is then left module over $M(2,\Gamma_{c})$, where

$\Gamma_{c}:=\left[\begin{array}[]{cc}1&c\\\ -c&1\end{array}\right].$

Its dual object is $X_{g}^{*}=\mathbf{k}_{-c-g}\oplus\mathbf{k}_{-g}$,
regarded as row vectors, and is a right module over $M(2,\Gamma_{c})$.

By construction $F$ lies in $\mathbf{Vec}_{H}$ and $X_{g}$ does not. If $c$
and $g$ are chosen so that objects homogeneous of degrees $c$ and at least one
of $g$ or $c+g$ are not transparent, our non-transparency desideratum is
satisfied. To see that we have invariance under the swimming moves we proceed
as follows:

An easy calculation shows that

$\Delta(I)=\left[\begin{array}[]{cc}1&0\\\
0&0\end{array}\right]\otimes\left[\begin{array}[]{cc}1&0\\\
0&0\end{array}\right]+\left[\begin{array}[]{cc}0&1\\\
0&0\end{array}\right]\otimes\left[\begin{array}[]{cc}0&0\\\
1&0\end{array}\right]$

$+\left[\begin{array}[]{cc}0&0\\\
1&0\end{array}\right]\otimes\left[\begin{array}[]{cc}0&1\\\
0&0\end{array}\right]+\left[\begin{array}[]{cc}0&0\\\
0&1\end{array}\right]\otimes\left[\begin{array}[]{cc}0&0\\\
0&1\end{array}\right]$

in which we have suppressed writing the degrees. Recall the diagonal entries
are of degree $0$, that in the upper right is of degree $c$ and that in the
lower left is of degree $-c$.

We can thus calculate the image of this element acting on the left on
$X_{g}^{*}$ and on the right on $X_{g}$, the object which must be transparent
for the swimming moves to hold by calculating its action on generic elements:

$\left[\begin{array}[]{cc}x_{-c-g}&y_{-g}\end{array}\right]\Bigg{(}\left[\begin{array}[]{cc}1&0\\\
0&0\end{array}\right]\otimes\left[\begin{array}[]{cc}1&0\\\
0&0\end{array}\right]+\left[\begin{array}[]{cc}0&1\\\
0&0\end{array}\right]\otimes\left[\begin{array}[]{cc}0&0\\\
1&0\end{array}\right]$

$+\left[\begin{array}[]{cc}0&0\\\
1&0\end{array}\right]\otimes\left[\begin{array}[]{cc}0&1\\\
0&0\end{array}\right]+\left[\begin{array}[]{cc}0&0\\\
0&1\end{array}\right]\otimes\left[\begin{array}[]{cc}0&0\\\
0&1\end{array}\right]\Bigg{)}\left[\begin{array}[]{c}z_{c+g}\\\
w_{g}\end{array}\right]$

$=\left[\begin{array}[]{cc}x_{-c-g}&0\end{array}\right]\otimes\left[\begin{array}[]{c}z_{c+g}\\\
0\end{array}\right]+\left[\begin{array}[]{cc}0&x_{-g}\end{array}\right]\otimes\left[\begin{array}[]{c}0\\\
z_{g}\end{array}\right]+\left[\begin{array}[]{cc}y_{-c-g}&0\end{array}\right]\otimes\left[\begin{array}[]{c}w_{c+g}\\\
0\end{array}\right]$
$+\left[\begin{array}[]{cc}0&y_{-g}\end{array}\right]\otimes\left[\begin{array}[]{c}0\\\
w_{g}\end{array}\right]$

each summand of which is an element homogenous of degree $0$, thus showing the
desired image is transparent, and thus a fortiori $\mathcal{B}$-transparent as
required.

Similar calculations show that in this instance, $k$ and $\kappa$ are both 2.

A minimal example illustrating the desired features is obtained by letting
$G=({\mathbb{Z}}/\ell m,+)$, and $H$ be the copy of $({\mathbb{Z}}/\ell,+)$
generated by $m$. This group has an obvious bicharacter
$\beta(a,b)=\zeta^{ab}$ where $\zeta$ is a primitive $\ell m^{th}$-root of
unity, thereby creating a braiding on ${\mathcal{C}}={\bf Vec}_{G}$. We use
the trivial 3-cocycle to induce the (strict) associator. Taking
${\mathcal{B}}={\bf Vec}_{H}$, choosing $c\in\\{m,2m,3m,\ldots(\ell-1)m\\}$
and $g\in{\mathbb{Z}}/\ell m\setminus\\{0,m,2m,\ldots(\ell-1)m\\}$ and using
$F=M(2,\gamma_{c})$ and $M=X_{g}$ give us our desiderata.

Calculating we find that $\Delta_{\mathcal{B}}=\ell$,
$\Delta_{\mathcal{C}}\Delta_{{\mathcal{B}},{\mathcal{C}}}^{\prime\prime}=m^{2}\ell$

We now give some sample calculations of our invariants for this sort of
initial data, concluding each with actual numerical values in the special case
$\ell=3,m=2,c=2,\zeta=\frac{1}{2}+\frac{\sqrt{3}}{2}i$ and $g=1$

The 4-sphere has invariant 1 (being given by an empty Kirby diagram), while
the 4-sphere with 2-unlink of $n$ components (represented by the bandless
unknot of $n$ components) has invariant $2^{n}$ (2 being the quantum dimension
of the module $M$ used to color the unlink representing the surface). A
4-sphere with a trivally embedded torus, whose banded unknot presentation
$L_{\rm torus}$ is given below

has invariant $\frac{1}{2}$. A tedious calculation shows that $\big{<}L_{\rm
torus}\big{>}$ is 2, so dividing by $k^{2}$ for the two self-bands gives
$\frac{1}{2}$.

Calculating the invariant for the simplest knotted 2-sphere in $S^{4}$, the
spun trefoil, which has banded unknot presentation given below,

gives 1, distinguishing it from the unknotted 2-sphere, whose invariant was 2.

We can calculate the invariant for ${\mathbb{C}}{\mathbb{P}}^{2}$, presented
as an unknot with a single maximum, single minimum and a negative crossing as
follows:

$\displaystyle 1$ $\displaystyle\mapsto$
$\displaystyle\sum_{i=0}^{\ell-1}1_{-im}\otimes 1_{im}$ $\displaystyle\mapsto$
$\displaystyle\sum_{i=0}^{\ell-1}\zeta^{-i^{2}m^{2}}1_{-im}\otimes 1_{im}$
$\displaystyle\mapsto$ $\displaystyle\sum_{i=0}^{\ell-1}\zeta^{-i^{2}m^{2}}$

Taking the particular case of
$\ell=3,m=2,\zeta=\frac{1}{2}+\frac{\sqrt{3}}{2}i$ this gives
$1+\zeta^{-4}+\zeta^{-16}=1+2\zeta^{2}=\sqrt{3}i$. Note, that as the only
eigenvalue of the linking matrix in this case is $-1$ the normalization factor
is 1.

It is easy to see that the value of $\big{<}-\big{>}$ on the diagram with the
crossing reversed to present $\overline{{\mathbb{C}}{\mathbb{P}}}^{2}$ is the
complex conjugate, and thus with Petit’s normalization that
$I_{+}(\overline{{\mathbb{C}}{\mathbb{P}}}^{2})=\frac{1}{3}\sum_{i=0}^{\ell-1}\zeta^{-i^{2}m^{2}}$,
which in our chosen particular case gives $-\frac{\sqrt{3}}{3}i$.

Now, consider a sphere embedded in ${\mathbb{C}}{\mathbb{P}}^{2}$ (or rather,
the (0,2)-handlebody completion of which by adding a 4-handle gives
${\mathbb{C}}{\mathbb{P}}^{2}$), which represents the homology class $nE$
where E is the exceptional divisor, that is the class of the sphere at
infinity, and $n\in{\mathbb{Z}}$. In the banded-unknot presentation of Hughes
et. al [6] this is given by a single (bandless) unlink component linking the
(-1)-framed 2-handle attaching curve of the simplest Kirby diagram for
${\mathbb{C}}{\mathbb{P}}^{2}$ with linking number $n$.

Calculating as above gives

$\displaystyle 1$ $\displaystyle\mapsto$
$\displaystyle\left(\sum_{i=0}^{\ell-1}1_{-im}\otimes
1_{im}\right)\otimes\left(\left[\begin{array}[]{c}1_{c+g}\\\
0\end{array}\right]\otimes\left[\begin{array}[]{cc}1_{-c-g}&0\end{array}\right]+\left[\begin{array}[]{c}0\\\
1_{g}\end{array}\right]\otimes\left[\begin{array}[]{cc}0&1_{-g}\end{array}\right]\right)$
$\displaystyle=$ $\displaystyle\left(\sum_{i=0}^{\ell-1}1_{-im}\otimes
1_{im}\otimes\left[\begin{array}[]{c}1_{c+g}\\\
0\end{array}\right]\otimes\left[\begin{array}[]{cc}1_{-c-g}&0\end{array}\right]\right)+$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\ \left(\sum_{i=0}^{\ell-1}1_{-im}\otimes
1_{im}\otimes\left[\begin{array}[]{c}0\\\
1_{g}\end{array}\right]\otimes\left[\begin{array}[]{cc}0&1_{-g}\end{array}\right]\right)$
$\displaystyle\mapsto$
$\displaystyle\left(\sum_{i=0}^{\ell-1}\zeta^{-i^{2}m^{2}+2ink(c+g)}1_{-im}\otimes
1_{im}\otimes\left[\begin{array}[]{c}1_{c+g}\\\
0\end{array}\right]\otimes\left[\begin{array}[]{cc}1_{-c-g}&0\end{array}\right]\right)+$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\
\left(\sum_{i=0}^{\ell-1}\zeta^{-i^{2}m^{2}+2inkg}1_{-im}\otimes
1_{im}\otimes\left[\begin{array}[]{c}0\\\
1_{g}\end{array}\right]\otimes\left[\begin{array}[]{cc}0&1_{-g}\end{array}\right]\right)$
$\displaystyle\mapsto$
$\displaystyle\sum_{i=0}^{\ell-1}\zeta^{-i^{2}m^{2}+2ink(c+g)}+\zeta^{-i^{2}m^{2}+2inkg}$

Thus the value is a sum of two generalized quadratic Gauss sums. In general,
if $c$ and $g$ are chosen correctly, the generalized Gauss sums will pick out
the multiples of the exceptional divisor which are 0 modulo $\ell$.

Taking the particular case of
$\ell=3,m=2,c=2,\zeta=\frac{1}{2}+\frac{\sqrt{3}}{2}i$ and $g=1$ gives
$\frac{3}{2}+\frac{\sqrt{3}}{2}i$ if $n\neq 0$ mod 3, and $2\sqrt{3}i$ if
$n=0$ mod 3. Note in this case, since $b_{0}$, $b_{+}$, $s$ and $\omega$ are
all zero the normalizing factors in the denominator are all 1.

The example of the sphere $S^{2}\times\\{0\\}\subset S^{2}\times{\mathbb{C}}$
illustrates the lack of invariance under cancellation of a 2-handle which
necessarily intersects all elements of the surface’s ambient isotopy class in
a (0,1,2)-handlebody. The open 4-manifold is a handlebody with a single
0-handle and a single 2-handle attached on a 0-framed unknot. The
surface-4-manifold pair is presented by a Hopf link one component of which is
the 2-handle attaching curve, and the other component of which is the unlink
in a (band-free) banded unlink presentation of the sphere.

Evaluating in our general example gives

$\frac{1}{\ell}\sum_{j=0}^{\ell-1}\zeta^{-2(c+g)mj}+\zeta^{-2gmj},$

which in our specific example with
$\ell=3,m=2,c=2,\zeta=\frac{1}{2}+\frac{\sqrt{3}}{2}i$ and $g=1$ gives 3 for
the evaluation of the diagram and 1 for the invariant. (The first summands in
each term of the indexed summation are each 1, while the second summands add
to 0).

As completing the handle-body to a closed 4-manifold by adjoining a 3-handle
cancelling the 2-handle and a 4-handle to give the 4-sphere results in a
trivially embedded 2-sphere, were our scalar also invariant under this move,
the result should be the same as for an unlink consisting of a 0-framed
unknotted 2-handle attaching curve and an unknot giving the(band-free) banded
unlink presentation of the sphere, which as we noted before evaluates to 2 for
any instance of our explicitly given initial data.

##  5\. Directions for future research

Several obvious directions for future research arise from the present paper.

Most basic is the calculation of the invariants explicitly presented here for
more substantial examples.

Also, beyond the scope of this paper, but worth immediate investigation is
whether there is a way of normalizing our invariants, using only the
combinatorial data of a Kirby diagram with banded unlink, so that they become
invariant under 2-3-handle cancellations in which the 2-handle intersects a
part of the surface that cannot be isotoped out of the handle? If this can be
done, the result would be give an invariant of smooth surface-
closed-4-manifold pairs. If such a renormalization is available, it would then
be worthwhile to find further renormalization that is insensitive to the
addition of trivally embedded handles to the surface, thus giving rise to a
functional on the second homology of the 4-manifold, which would be an
invariant of the 4-manifold itself.

Others involve seeking better initial data.

In particular, are there examples in which the image of the copairing acting
between the dual module and the module is not transparent, but only
${\mathcal{B}}$-transparent? If the image of the “copairing acting in the
middle” is transparent in the ambient category $\mathcal{C}$, rather than only
$\mathcal{B}$-transparent, as in the examples we have explicitly constructed
herein, the invariant will also be unchanged by the ungeometric “band swim”
moves of 1-handle attaching curves and components of the surface-encoding
unlink. This is disappointing, as invariance under geometrically nonsensical
moves generally weakens the topological significance of an invariant. Initial
data in which the image is only ${\mathcal{B}}$-transparent, but not
transparent, would not be invariant under these non-geometric moves and thus
would be expected to yield more sensitive topological invariants.

Of course, as always for any new “quantum” topological invariant, there is the
vexing question of giving a sensible geometric interpretation of the
information it provides.

Finally, our fully functorial approach to “Kirby coloring” using an
endomorphism, rather than a formal linear combination of simple objects, leads
to the question of whether there are objects in other ribbon categories,
possibly even non-semisimple ribbon categories, which have suitable
endomorphisms and actions by other objects so that the analogue of Lemma 3.3
of Bärenz and Barrett [1] gives rise to handle-slide invariance.

## References

  * [1] Bärenz, M. and Barrett, J., “Dichromatic state-sum models for four-manifolds from pivotal functors,” Comm. Math. Phys., 360, (2018) 663–714, doi.org/10.1007/s00220-017-3012-9 .
  * [2] Broda, B., “A gauge-field approach to 3- and 4-manifold invariants,” Symplectic Singularities and Geometry of Gauge Fields, Banach Center Publications, Vol. 39, Polish Academy of Sciences, Warsaw (1997) 201-209.
  * [3] Brugières, A., “Catégories prémodulaires, modularisations et invariants de variétés de dimension 3,” Math. Ann. 316 (2000) 215-236.
  * [4] Etingof, P., Gelaki, S., Nikshych,D., and Ostrik, V., Tensor categories, Mathematical Surveys and Monographs,Vol. 205, American Mathematical Soc. (2016).
  * [5] Freyd, P., and Yetter, D.N., “Coherence Theorems via Knot Theory”, J. Pure and App. Alg., 78 (1992) 47-76.
  * [6] Hughes, M., Kim, S. and Miller, M., “Isotopies of surfaces in 4-manifolds via banded unlink diagrams,” Geometry & Topology 24 (2020) 1519–1569 DOI: 10.2140/gt.2020.24.1519. also arXiv:1804.09169.v4.
  * [7] Hughes, M., Kim, S. and Miller, M., “Band diagrams of immersed surfaces in 4-manifolds,” arXiv:2109.12794.v1.
  * [8] Kirby, R., “The calculus of framed links in $S^{3}$,” Invent. Math. 45 (1978) 35-56.
  * [9] Mac Lane, S., Categories for the Working Mathematician (2nd ed.) Springer (1978).
  * [10] Lauda, A., “Frobenius algebras and ambidextrous adjunctions,” Theory and Applications of Categories 16 (2006) 84-122.
  * [11] Nikshych, D., “Classifying braidings on fusion categories,” arXiv:1801.06125v2 (2018).
  * [12] Petit, J., “The dichromatic invariants of smooth 4-manifolds,” Global Journal of Pure and Applied Mathematics 4(3) (2008) 1–16.
  * [13] Reidemeister, K., Knottentheorie, Springer-Verlag, Berlin (1932).
  * [14] Reshetikhin, N. and Turaev, V.G., “Invariants of 3-manifolds via link polynomials and quantum groups,” Invent. Math. 103 (1991) 547-597.
  * [15] Shum, M.-C., “Tortile tensor categories,” J. Pure and App. Alg. 93 (1) (1994) 57-110.
  * [16] Yetter, D.N., “Framed Tangles and a Theorem of Deligne on Braided Deformations of Tannakian Categories”, in Deformation Theory and Quantum Groups with Applications to Mathematical Physics (M. Gerstenhaber and J.D. Stasheff, eds.) AMS Contemp. Math. vol. 134 (1992) 325-350.