# Generalized Mazur patterns and immersed Heegaard Floer homology

Jay Patwardhan<EMAIL_ADDRESS>Rutgers University New
Brunswick, New Brunswick, NJ, USA  and  Zheheng Xiao<EMAIL_ADDRESS>Columbia University, New York, NY, USA

###### Abstract.

Generalizing prior work of Levine, we give infinitely many examples of pattern
knots $P$ such that $P(K)$ is not slice in any rational homology $4$-ball, for
any companion knot $K$. To show this, we establish a closed formula for the
concordance invariants $\tau$ and $\epsilon$ of a family of satellite knots
obtained from generalized Mazur patterns. Our main computational tool is the
immersed curve technique from bordered Heegaard Floer homology arising from
the work of Chen-Hanselman.

JP was partially supported by NSF CAREER Grant DMS-2019396.

ZX was partially supported by NSF CAREER Grant DMS-2019396.

###### Contents

  1. 1 Introduction
  2. 2 Knot Floer homology
    1. 2.1 The knot Floer complex and concordance invariants
    2. 2.2 Bordered Heegaard Floer homology
    3. 2.3 Obtaining type $A$ structures from Heegaard diagrams
    4. 2.4 Obtaining type $D$ structures from $CFK^{-}(K)$
  3. 3 Immersed Heegaard Floer homology
    1. 3.1 From type $D$ structures to immersed curves
    2. 3.2 Pairing diagrams
    3. 3.3 Computation of $\tau(P(K))$ and $\epsilon(P(K))$ from the pairing diagram
  4. 4 Generalized Mazur patterns
  5. 5 Computation of $\tau(Q_{m,n}(K))$ and $\epsilon(Q_{m,n}(K))$
  6. A Computation of $\tau(Q_{m,n}(K))$ using bordered Heegaard Floer homology
    1. A.1 Bordered strategy to determine $\tau(P(K))$
    2. A.2 Examples
    3. A.3 Computation of $\tau(Q_{m,n}(K))$

##  1\. Introduction

Two knots in $S^{3}$ are said to be smoothly concordant if they co-bound a
smoothly embedded annulus in $S^{3}\times I$. The set of concordance classes
of knots form a group $\mathcal{C}$, with addition given by connect sum, and
identity given by the concordance class of the unknot. In particular, knots in
the concordance class of the unknot are called smoothly slice. The classical
study of knot concordance looks to classify which knots in $S^{3}$ are
smoothly slice in $D^{4}$.

One may extend the notion of sliceness to knots to more general $3$-manifolds.
A knot $K$ in the boundary of a smooth $4$-manifold $M$ is said to be smoothly
slice if it bounds a smoothly embedded disk in $M$. There are also weaker
notions of concordance that we consider: two knots are exotically concordant
if they co-bound a smoothly embedded annulus in a smooth $4$-manifold
homeomorphic to $S^{3}\times I$ but possibly with an exotic smooth structure,
and (for a ring $R$) $R$-homology concordant if they co-bound a smoothly
embedded annulus in a smooth manifold with the same $R$-homology of
$S^{3}\times I$. Then, a knot $K\subset S^{3}$ is exotically slice or
$R$-homology slice if it is exotically or $R$-homology concordant to the
unknot, respectively; this is the same as saying that $K$ bounds an embedded
disk in a contractible $4$-manifold (which is homeomorphic to $D^{4}$ by
Freedman [Fre82]) or an $R$-homology $4$-ball, respectively. In particular,
$\mathbb{Q}$-homology concordance is stronger than exotic concordance.

For any two knots $K\subset S^{3}$ and $P\subset S^{1}\times D^{2}$, let
$P(K)$ denote the satellite knot of $K$ with pattern $P$. If $K$ is concordant
to $K^{\prime}$, then $P(K)$ is concordant to $P(K^{\prime})$, so we may
regard $P$ as an operator $P:\mathcal{C}\rightarrow\mathcal{C}$, called the
satellite operator. Note that this map is generally not a homomorphism. The
satellite operator is well-studied in literature; see for example [Hed07,
Hom14a, Lev16]. Moreover, we define the winding number $w(P)$ to be the number
of signed intersections of a meridional disk with $P$.

Problem 1.45 in Kirby’s problem list [Kir97], attributed to Akbulut, asks
whether there exists a winding number $\pm 1$ satellite operator $P$ for which
$P(K)$ is never exotically slice in any rational homology $4$-ball with
boundary $S^{3}$. Levine answered a stronger version of this in the
affirmative:

######  Theorem 1.1.

[Lev16, Theorem 1.2] There exists a pattern knot $P\subset S^{1}\times D^{2}$
with winding number $1$ such that for any knot $K\subset S^{3}$, $P(K)$ is not
slice in any rational homology $4$-ball.

Levine’s strategy was to find a pattern $Q$ which induces a non-surjective
satellite operator on the rational homology concordance classes, so that there
exists a knot $L$ which is not concordant to $Q(K)$ for all $K\subset S^{3}$.
Then, $P=Q\\#-L\subset S^{1}\times D^{2}$ satisfies the conclusion of the
Theorem. In particular, he chose $Q$ to be the Mazur pattern, shown in Figure
1.

Figure 1. The Mazur pattern $Q_{2,1}$ embedded in the solid torus
$V=S^{1}\times D^{2}$.

The goal of this paper is to expand on this result to obtain an infinite
family of pattern knots with winding number $\pm 1$ whose satellites are never
slice in any rational homology $4$-ball. To do this, we consider a
generalization of the Mazur pattern $Q$.

######  Definition 1.2.

Starting with a point in the solid torus, wind $m$ times around the torus,
then turn around and wind $n$ times. Join the top endpoint of the arc to the
bottom endpoint by crossing under the first $n$ times, and over the next $m$
times, resulting in a clasp where the pattern turns around. The resulting
pattern knot is a generalized Mazur pattern $Q_{m,n}$, and a picture of this
is shown in Figure 2.

Figure 2. The generalized Mazur pattern $Q_{m,n}$

A pattern knot $P$ is a $(1,1)$-pattern if it admits a genus one doubly-
pointed Heegaard diagram, and is an unknot pattern if $P(U)\simeq U$ in
$S^{3}$, where $U$ is the unknot [Che19, CH23]. Generalized Mazur patterns
$Q_{m,n}$ are part of the class of $(1,1)$-unknot patterns, and have winding
number $\pm(m-n)$ depending on the orientation. We show that the pattern
remains the same if we switch $m$ and $n$.

######  Proposition 1.3.

Inside the solid torus $S^{1}\times D^{2}$, $Q_{m,n}$ is isotopic to
$Q_{n,m}$.

In the early 2000s, Ozsváth and Szabó [OS04b, OS04c] introduced Heegaard Floer
homology, a collection of invariants of three-manifolds and knots and links
inside them. The knot version, also independently introduced by Rasmussen
[Ras03], associates to every knot $K\subset S^{3}$ a
$\mathbb{Z}\oplus\mathbb{Z}$-filtered, free $\mathbb{F}[U,U^{-1}]$-complex
$CFK^{\infty}(K)$, called the knot Floer complex of $K$. Knot Floer homology
has several nice properties; for example, it categorifies the Alexander
polynomial [OS04b], detects the knot genus [OS04a], and detects fiberedness
[Ghi08, Ni07]. There are a variety of concordance invariants arising from
$CFK^{\infty}(K)$; in this paper, we are interested in the integer-valued
$\tau$-invariant [OS03] and the $\\{-1,0,1\\}$-valued $\epsilon$-invariant
[Hom14b], which are maps with domain the set of rational homology concordance
classes.

In [Lev16], Levine computed $\tau(Q(K))$ and $\epsilon(Q(K))$ for satellites
of knots along the Mazur pattern.

######  Theorem 1.4.

[Lev16, Theorem 1.6] If $Q$ is the Mazur pattern, then for any knot $K\subset
S^{3}$, we have

$\tau(Q(K))=\begin{cases}\tau(K)&\text{ if }\tau(K)\leq 0\text{ and
}\epsilon(K)\in\\{0,1\\}\\\ \tau(K)+1&\text{ if }\tau(K)>0\text{ or
}\epsilon(K)=-1.\end{cases}$

and

$\epsilon(Q(K))=\begin{cases}0&\text{ if }\tau(K)=\epsilon(K)=0\\\ 1&\text{
otherwise }\end{cases}$

In this paper, we generalize this result to the generalized Mazur patterns
$Q_{m,n}(K)$:

######  Theorem 1.5.

Let $Q_{m,n}$ be the generalized Mazur pattern embedded in the solid torus
$V$. If $m\neq n$, then for any knot $K\subset S^{3}$, we have

(1.1) $\tau(Q_{m,n}(K))=\begin{cases}|m-n|\tau(K)&\text{if }\tau(K)\leq
0\text{ and }\epsilon(K)\in\\{0,1\\},\\\ |m-n|\tau(K)+|m-n|&\text{if
}\tau(K)<0\text{ and }\epsilon(K)=-1,\\\ |m-n|\tau(K)+\min(m,n)&\text{if
}\tau(K)>0\text{ and }\epsilon(K)=1,\\\ |m-n|\tau(K)+\max(m,n)-1&\text{if
}\tau(K)\geq 0\text{ and }\epsilon(K)=-1.\end{cases}$

In the case where $m=n$, we have

(1.2) $\tau(Q_{m,m}(K))=\begin{cases}0&\text{if }\tau(K)<0,\\\ m-1&\text{if
}\tau(K)=0,\\\ m&\text{if }\tau(K)>0.\end{cases}$

Also,

(1.3) $\epsilon(Q_{m,n}(K))=\begin{cases}0&\text{if }\tau(K)=\epsilon(K)=0,\\\
1&\text{otherwise}\end{cases}$

######  Remark 1.6.

To put this computation into context, we make the following remark about
$(1,1)$-patterns. Given a fixed pattern knot and a fixed companion knot, there
are many available tools in the literature that compute the $\tau$-invariant
of $P(K)$, stemming from the bordered Heegaard Floer homology invariants.
However, it is challenging to find a closed formula for $\tau(P(K))$ as we
vary the companion knot. The formula above recovers the results in the
literature for satellites along the Whitehead double $Q_{1,1}$ [Hed07, Theorem
1.4] and the Mazur pattern $Q_{2,1}$ [Lev16, Theorem 1.6]. Partial results for
this formula have also been computed in [Ray15, Proposition 3.5], using
certain conditions on the Thurston Bennet number.111In Definition 1.2, we
require $m$ and $n$ to be positive because if $m$ (equivalently $n$) is $0$,
then $Q_{0,p}\simeq Q_{p,0}$ is the $(p,1)$-cable. Hom has computed this case
in [Hom14a], and the result does not follow the pattern of Theorem 1.5. This
is expected, since the methods we used to obtain the results above rely on the
parameters $m$ and $n$ being strictly positive.

Similar to [Lev16], we use techniques in bordered Heegaard Floer homology, due
to Lipshitz, Ozsváth and Thurston [LOT18], which is well-adapted to study
three-manifolds with parametrized boundary. In particular, we use the immersed
curve interpretation of the bordered pairing theorem developed by Hanselman
[Han23]. The proof strategy for Theorem 1.5 is inspired by [CH23, Section 6],
which recovers Levine’s computation using immersed curve techniques.

We are now ready to construct our infinite family of winding number $\pm 1$
pattern knots which induce non-surjective satellite operators on the rational
concordance groups. Since $\epsilon(Q_{m,m-1}(K))\neq-1$ for all $K\subset
S^{3}$, we may pick a knot $L\subset S^{3}$ such that $\epsilon(L)=-1$, such
as the left handed trefoil. Then, $P_{m,m-1}=Q_{m,m-1}\\#-L\subset S^{1}\times
D^{2}$ satisfies the conclusion of Theorem 1.1. Thus, we have shown the
following.

######  Corollary 1.7.

For $m>1$, the infinite family of winding number 1 pattern knots $P_{m,m-1}$
have the property that $P_{m,m-1}(K)$ is not smoothly slice in any rational
homology $4$-ball for any knot $K\subset S^{3}$.

The computation of Theorem 1.5 also allows us to recover the pattern knot
genus of generalized Mazur patterns $Q_{m,n}$. Recall that for a pattern $P$
with winding number $w(P)$, a relative Seifert surface for $P$ is a surface
$\widehat{\Sigma}$ in $S^{1}\times D^{2}$ such that the interior of
$\widehat{\Sigma}$ is disjoint from $P$, and the boundary of
$\widehat{\Sigma}$ consists of $P$ together with $w(P)$ coherently oriented
longitudes. The genus of a pattern $g(P)$ inside the solid torus is defined as
the minimal genus of a relative Seifert surface for $P$. For a satellite knot
$P(K)$ with nontrivial companion $K$, a classical result of Schubert [Sch53]
shows the relation between the three-genus of the satellite knot $g(P(K))$ and
$w(P)$, $g(P)$, and $g(K)$:

(1.4) $g(P(K))=|w(P)|g(K)+g(P).$

We know $w(Q_{m,n})$, so the value of $g(Q_{m,n})$ is determined by
$g(Q_{m,n}(K))$ and $g(K)$ for some companion knot $K$ of our choice. Using
the fact that knot Floer homology detects knot genus, we obtain a formula for
the genus of $Q_{m,n}$.

######  Proposition 1.8.

The patterns $Q_{m,n}$ have genus $g(Q_{m,n})=\min(m,n)$.

We end with a discussion on a related problem. While Theorem 1.5 allows us to
compute the $\tau$ and $\epsilon$-invariant of generalized Mazur patterns, one
might hope that there is a general formula for $\tau(P(K))$ and
$\epsilon(P(K))$ for any $(1,1)$-unknot pattern $P$. Moreover, we know that
$(1,1)$-unknot patterns can be parameterized by a pair of integers $(r,s)$, so
we expect that $\tau(P(K))$ and $\epsilon(P(K))$ can also be obtained with
respect to $r$ and $s$ [Che19].

######  Question 1.9.

Is there a closed formula for the $\tau$-invariant of satellite knots with
$(1,1)$-unknot patterns?

Organization. In Sections 2 and 3, we review some background from bordered and
immersed Heegaard Floer homology and introduce the concordance invariants
$\tau$ and $\epsilon$. In particular, we focus on obtaining immersed curves
for knot complements, methods to recover Alexander gradings, and the relevant
pairing theorems which give a strategy for computing $\tau(Q_{m,n}(K))$ and
$\epsilon(Q_{m,n}(K))$. In Section 4, we describe the generalized Mazur
patterns and some properties, including a constructive procedure to recover
their bordered Heegaard diagrams, and their 2-bridge link representation.
Here, we recover Proposition 1.3. In Section 5, we find both the $\tau$ and
$\epsilon$-invariant for $Q_{m,n}$ using techniques from immersed Heegaard
Floer homology, proving Theorem 1.5. We also use the pairing diagrams to
recover Proposition 1.8. Lastly, Appendix 5 recovers the computation for the
$\tau$-invariant using only the ordinary bordered theory. Throughout the
paper, the coefficients of Floer homology groups are taken in
$\mathbb{F}=\mathbb{Z}/2\mathbb{Z}$.

Acknowledgements. The authors are grateful to Kristen Hendricks and Abhishek
Mallick for their guidance, to Robert Lipshitz, Jonathan Hanselman, Wenzhao
Chen, and Adam Levine for helpful discussions, and to the Summer 2023 DIMACS
REU for providing the opportunity for their research.

##  2\. Knot Floer homology

In this section, we will briefly review those features of most relevance. We
assume that the reader is familiar with Heegaard Floer homology. For an
introductory overview, see [Man14] and [OS06]. In Sections 2.1 to 2.3, we
closely follow the notation from [Hom14a, Hom20]. In Section 2.4, we follow
the notation from [Lev16].

###  2.1. The knot Floer complex and concordance invariants

To each $K\subset S^{3}$ we associate a $\mathbb{Z}\oplus\mathbb{Z}$-filtered
(freely finitely generated) $\mathbb{F}[U,U^{-1}]$ chain complex
$CFK^{\infty}(K)$, called the full knot Floer complex of $K$[OS03]. This
complex is well-defined up to chain homotopy equivalence. It admits two
$\mathbb{Z}$-gradings called the Alexander and Maslov (or homological)
gradings, denoted respectively by the maps $A,M:I\to\mathbb{Z}$, where $I$ is
a finite set of points specified by $K$.

Equivalently, $CFK^{\infty}$ can be seen as an $\mathbb{F}$-vector space,
freely generated by elements of the form

$[x;i,j],\textrm{ where }x\in\mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta},\
(i,j)\in\mathbb{Z}\oplus\mathbb{Z},\ \textrm{and }j-i=A(x).$

The triple $[x;i,j]$ corresponds to the generator $U^{-i}x$, and the
$(i,j)$-filtration level is given by
$\mathcal{F}_{i,j}=\\{[x,i^{\prime},j^{\prime}]\in CFK^{\infty}:i^{\prime}\leq
i,j^{\prime}\leq j\\}$. The differential $\partial$ decreases the Maslov
grading by 1, respects the Alexander filtration, and is $U$-equivariant, that
is,

  * •

$M(\partial x)=M(x)-1$

  * •

$A(\partial x)\leq A(x)$

  * •

$\partial(U^{n}x)=U^{n}\partial x$

Moreover, multiplication by $U$ decreases each filtration level by 1, lowers
the Maslov grading by 2, and the Alexander grading by 1. In other words,

  * •

$U[x;i,j]=[x;i-1,j-1]$

  * •

$M(U[x;i,j])=M([x;i-1,j-1])=M([x;i,j])-2$

  * •

$A(U[x;i,j])=A([x;i-1,j-1])=A([x;i,j])-1$

The Maslov and Alexander gradings for $[x;i,j]$ are then given by

$M([x;i,j])=M(x)+2i,\ \ \ A([x;i,j])=j-i.$

Graphically, we can represent $CFK^{\infty}$ on a plane by drawing the element
$[x;i,j]$ at $(i,j)$ and the differentials as arrows that point (non-strictly)
downwards and to the left, as seen in Figure 3. Multiplication by $U$
decreases the Alexander grading of a generator by 1. The $j$ coordinate is the
generator’s Alexander grading, and the $i$ coordinate is the negative of its
$U$ power. The Maslov grading is not represented in this picture.

######  Example 2.1.

The full knot Floer complex for the right-handed trefoil $T_{2,3}$ is depicted
in Figure 3. As a $\mathbb{F}[U,U^{-1}]$-vector space, it has three generators
$a,b,c$, with differentials

$\partial a=Ub+c,\ \ \partial b=\partial c=0.$

The Alexander gradings of $a,b,c$ are given by their $j$-coordinates, which
are $1,0,-1$, respectively. The Maslov gradings of $a,b,c$ are $-1,0,-2$,
respectively. The homology of this complex is generated by $[b]=[U^{-1}c]$
over $\mathbb{F}[U,U^{-1}]$.

Figure 3. The full knot Floer complex for the right-handed trefoil.

We can obtain various flavors of the knot Floer complex by taking different
subcomplexes of $CFK^{\infty}(K)$. For a set
$S\subset\mathbb{Z}\oplus\mathbb{Z}$, let $C\\{S\\}$ be the set of elements in
$CFK^{\infty}(K)$ whose $(i,j)$-coordinates are in $S$. We then define

$CFK^{-}(K)=C\\{i\leq 0\\}$

to be the $\mathbb{F}[U]$-module whose elements have non-positive
$i$-coordinates, and whose differential is the induced differential. This
complex has a natural $\mathbb{Z}$-filtration, induced by the Alexander
filtration of $CFK^{\infty}(K)$. Denote the associated graded of $CFK^{-}(K)$
by $gCFK^{-}(K)$, and let the homology of the associated graded be

$HFK^{-}=H_{*}(gCFK^{-}(K)).$

Similarly, we can also take the $\mathbb{Z}$-filtered chain complex

$\widehat{CFK}(K)=C\\{i=0\\}$

with the induced differential, and denote the homology of the associated
graded of $\widehat{CFK}(K)$ by

$\widehat{HFK}(K)=H_{*}(g\widehat{CFK}).$

This is commonly referred to as the knot Floer homology of $K$. While the
homology of $\widehat{CFK}(K)$ is always $\mathbb{F}$, the homology of
$g\widehat{CFK}(K)$ is more interesting. As a bigraded vector space, the knot
Floer homology decomposes as

$\widehat{HFK}(K)=\bigoplus_{i,j}\widehat{HFK}_{i}(K,j),$

where $i$ and $j$ indicate the Maslov and Alexander grading, respectively. It
also satisfies symmetry under orientation reversal [OS04b, Section 3.5]:

$\widehat{HFK}_{i}(K,j)=\widehat{HFK}_{i-2j}(K,j).$

Furthermore, the knot Floer homology categorifies the Alexander polynomial in
the following sense. Its graded Euler characteristic is the Alexander
polynomial [OS04b, Equation $(1)$]:

$\Delta_{K}(t)=\sum_{i,j}(-1)^{i}\dim\widehat{HFK}_{j}(K,s)t^{s}.$

While the Alexander polynomial bounds the Seifert genus of $K$ from below, the
knot Floer homology detects the genus [OS04a] by

$g(K)=\max\\{s|\ \widehat{HFK}(K,s)\neq 0\\}.$

And whereas the Alexander polynomial obstructs fiberedness, knot Floer
homology detects it [Ghi08, Ni07]:

$K\textrm{ is fibered }\Longleftrightarrow\widehat{HFK}(K,g(K))=\mathbb{F}.$

Note that the full knot Floer complex is defined over the base ring
$\mathbb{F}[U,U^{-1}]$. We may define analogously a bigraded chain complex
over the base ring $\mathcal{R}=\mathbb{F}[U,V]/UV$, denoted by
$CFK_{\mathcal{R}}(K)$. To obtain $CFK_{\mathcal{R}}(K)$, we decorate each
vertical arrow in $CFK^{\infty}(K)$ with $V^{n}$, where $n$ is its vertical
length, then set all the $UV$ arrows to 0. In particular, $CFK^{\infty}(K)$
and $CFK_{\mathcal{R}}(K)$ have the same set of original generators and
differentials, except that for $CFK^{\infty}(K)$ we extend them linearly over
$\mathbb{F}[U,U^{-1}]$, whereas for $CFK_{\mathcal{R}}(K)$ we extend them
linearly over $\mathcal{R}$. We may also take subcomplexes of
$CFK_{\mathcal{R}}(K)$ to obtain different flavors of the knot Floer chain
complex over $\mathcal{R}$, such as $\widehat{CFK}_{\mathcal{R}}(K)$. We will
make use of these chain complexes over $\mathcal{R}$ in Section 5.

Next, we introduce some concordance invariants arising from the knot Floer
complex. Recall that $\widehat{CF}$ is an invariant of 3-manifolds, that
$\widehat{HF}$ is the homology of $\widehat{CF}$, and that in particular
$\widehat{HF}(S^{3})=\mathbb{F}$ (see [OS04a, OS04c]). For a knot $K\subset
S^{3}$, the Alexander filtration on $\widehat{CFK}(K)$ induces a spectral
sequence that converges to $\widehat{HF}(S^{3})$. The invariant $\tau(K)$ is
defined to be the Alexander grading of the unique cycle that survives to the
$E_{\infty}$ page. By symmetry of the full knot Floer complex, we have
equivalently

$\tau(K)=-\max\\{s\mid U^{n}\cdot HFK^{-}(K,s)\neq 0\textrm{ for all }n\geq
0\\}.$

In other words, $\tau(K)$ is minus the Alexander grading of the non-vanishing
generator for $\mathbb{F}[U]$ in $HFK^{-}(S^{3},K)$. This is different from
but equivalent to the original definition of $\tau(K)$ in [OS03]; for a proof
of the equivalence, see [OST08, Lemma A.2].

Define the horizontal complex

$C^{\mathrm{horz}}=C\\{j=0\\}$

to be the subquotient complex of $CFK^{\infty}(K)$, for which the elements
have zero $j$-coordinates, and the differential $\partial^{\mathrm{horz}}$ is
the induced differential. The horizontal complex has a $\mathbb{Z}$-filtration
induced by $\mathit{CFK}^{\infty}(K)$. Similarly, we define the vertical
complex

$C^{\mathrm{vert}}=C\\{i=0\\},$

to be the subquotient complex along the $i$-axis, which we previously called
$\widehat{CFK}(K)$, with induced differential $\partial^{\mathrm{vert}}$. Note
that the vertical complex and the horizontal complex are homotopy equivalent.

For any $\mathbb{Z}\oplus\mathbb{Z}$-filtered chain complex $(C,\partial)$, we
say that $\\{x_{i}\\}$ is a filtered basis for $(C,\partial)$ if for all pairs
$(a,b)$, the set

$\big{\\{}x_{i}\mid x_{i}\in C\\{i\leq a,j\leq b\\}\big{\\}}$

is a basis for $C\\{i\leq a,j\leq b\\}$. Let $\\{\eta_{i}\\}$ be a filtered
basis over $\mathbb{F}[U]$ for a reduced complex $CFK^{-}(K)$, that is, the
arrows in $CFK^{-}(K)$ point strictly downwards or to the left (or both). We
say that $\\{\eta_{i}\\}$ is horizontally simplified if exactly one of the
following situations holds for every $\eta_{i}$:

  1. (1)

$\eta_{i}\in\operatorname{im}(\partial^{\mathrm{horz}})$, and there exists a
unique $\eta_{i-1}$ such that $\partial^{\mathrm{horz}}\eta_{i-1}=\eta_{i}$.

  2. (2)

$\eta_{i}\not\in\operatorname{im}(\partial^{\mathrm{horz}})$, but
$\eta_{i}\in\ker(\partial^{\mathrm{horz}})$

  3. (3)

$\eta_{i}\not\in\ker(\partial^{\mathrm{horz}})$, and
$\partial^{\mathrm{horz}}\eta_{i}=\eta_{i+1}$.

Recall that $H_{*}(C^{\mathrm{horz}})\cong
H_{*}(\widehat{CFK(K)})\cong\mathbb{F}$, i.e. the horizontal complex is
generated by one distinguished basis element upon taking homology. After
reordering, we call this distinguished element $\eta_{0}$.

A vertically simplified basis $\\{\xi_{j}\\}$ is defined similarly, replacing
$\partial^{\mathrm{horz}}$ by $\partial^{\mathrm{vert}}$. Let $\xi_{0}$ be the
distinguished non-vanishing basis element that generates
$H_{*}(C^{\mathrm{vert}})$ after reordering the basis. Note that if we have a
horizontally simplified basis $\\{\eta_{i}\\}$, then $\tau(K)=-A(\eta_{0}).$
If we have a vertically simplified basis $\\{\xi_{j}\\}$, then
$\tau(K)=A(\xi_{0}).$

Hom [Hom14a, Lemma 3.2 and 3.3] showed that $CFK^{-}(K)$ is always homotopy
equivalent to a reduced chain complex $\mathcal{C}$, which admits a
horizontally simplified basis $\\{\eta_{i}\\}$ such that some $\eta_{i}$ is
the distinguished element of a vertically simplified basis. We then consider
the position of $\eta_{i}$ in the horizontal complex. It can have three
situations, as enumerated above. To each situation we assign the numbers $1$,
$0$ and $-1$, respectively; see Figure 4.

$\star$$\eta_{i}$ (a) $\epsilon(K)=1$

$\eta_{i}$ (b) $\epsilon(K)=0$

$\eta_{i}$$\star$ (c) $\epsilon(K)=-1$

Figure 4. Three positions of $\eta_{i}$ in the horizontal complex.

This assignment is well-defined up to concordance, giving us the
$\epsilon$-invariant,

$\epsilon:\mathcal{C}\to\\{-1,0,1\\}$

which satisfies the following properties [Hom14a]:

  * •

$\epsilon(K)=-\epsilon(\bar{K})$.

  * •

If $K$ is alternating, then $\epsilon(K)=\mathrm{sgn}(\tau(K)).$

  * •

If $\epsilon(K)=\epsilon(K^{\prime})$, then
$\epsilon(K\\#K^{\prime})=\epsilon(K)=\epsilon(K^{\prime})$. If
$\epsilon(K)=0$, then $\epsilon(K\\#K^{\prime})=\epsilon(K^{\prime}).$

  * •

If $\epsilon(K)=0$, then $\tau(K)=0$.

By symmetry, $CFK^{-}(K)$ also admits a vertically simplified basis
$\\{\xi_{j}\\}$ such that some $\xi_{j}$ is the distinguished element of a
horizontally simplified basis, and we can similarly obtain the
$\epsilon$-invariant by looking at vertical position of $\xi_{j}$.

######  Example 2.2.

In the full knot Floer complex of the right-handed trefoil shown in Figure 3,
$\\{a,b,c\\}$ is a vertically simplified basis, with $b$ being the
distinguished basis element. So we have $\tau(T_{2,3})=A(b)=1$. Looking at the
horizontal position of $b$, we get $\epsilon(T_{2,3})=1$.

Even though we defined the $\tau$ and $\epsilon$ invariants using the $CFK$
complexes, they can be computed in exactly the same way using the
$CFK_{\mathcal{R}}$ complexes. This is because $CFK^{\infty}(K)$ and
$CFK_{\mathcal{R}}(K)$ have the same set of generators, and almost the same
set of differentials on those generators, except that on
$CFK_{\mathcal{R}}(K)$ we decorate the vertical arrows with the extra $V$
variable. In particular, $CFK^{\infty}(K)$ and $CFK_{\mathcal{R}}K$ have the
same vertical and horizontal complexes. So if we have a horizontally
simplified basis $\\{\eta_{i}\\}$ on $CFK_{\mathcal{R}}(K)$, then
$\tau(K)=-A(\eta_{0})=A(\xi_{0})$ and $\epsilon(K)$ is determined by the
vertical position of $\eta_{0}$. This is the strategy we will employ to
determine the $\epsilon$-invariant in Section 5.

The $\tau$-invariant and $\epsilon$-invariant often admit a nice decomposition
under the satellite operation. For instance, Hom [Hom14a, Theorem 1] showed
that for the $(p,q)$-cable of a knot $K$, denoted $K_{p,q}$, we have

$\displaystyle\tau(K_{p,q})$
$\displaystyle=\begin{cases}p\tau(K)+\frac{(p-1)(q-1)}{2}&\text{ if
}\epsilon(K)=1,\text{ or }\epsilon(K)=0\text{ and }q>0,\\\
p\tau(K)+\frac{(p-1)(q+1)}{2}&\text{ if }\epsilon(K)=-1,\text{ or
}\epsilon(K)=0\text{ and }q<0.\\\ \end{cases}$
$\displaystyle\epsilon(K_{p,q})$
$\displaystyle=\begin{cases}\epsilon(K)&\text{ if }\epsilon(K)\neq 0,\\\
-1&\text{ if }q<-1\textrm{ and }\epsilon(K)=0,\\\ 0&\text{ if }|q|=1\textrm{
and }\epsilon(K)=0,\\\ 1&\text{ if }q>1\textrm{ and
}\epsilon(K)=0.\end{cases}$

Shortly thereafter, Levine gave a formula for the $\tau$ and $\epsilon$
invariants of satellite knots with Mazur pattern $Q$, as stated in Theorem
1.4:

$\displaystyle\tau(Q(K))=\begin{cases}\tau(K)&\text{ if }\tau(K)\leq 0\text{
and }\epsilon(K)\in\\{0,1\\},\\\ \tau(K)+1&\text{ if }\tau(K)>0\text{ or
}\epsilon(K)=-1.\end{cases}$
$\displaystyle\epsilon(Q(K))=\begin{cases}0&\text{ if
}\tau(K)=\epsilon(K)=0,\\\ 1&\text{ otherwise.}\end{cases}$

The main goal of this paper is to find an analogous result for the generalized
Mazur patterns $Q_{m,n}$, that is, calculate $\tau(Q_{m,n}(K))$ and
$\epsilon(Q_{m,n}(K))$.

###  2.2. Bordered Heegaard Floer homology

The bordered version of Heegaard Floer homology extends the theory to
manifolds with boundary. In this section, we give a brief tour of the tools we
will use in bordered Heegaard Floer homology; a more comprehensive discussion
is given in [LOT18]. For most of this section, we follow the notation in
[Hom14a].

First, we define the algebraic structures in bordered Heegaard Floer homology,
such as $A_{\infty}$-modules and type $D$ structures. Let $\mathcal{A}$ be a
unital differential graded algebra over $\mathbb{F}$, equipped with a
subalgebra of idempotents $\mathcal{I}\subset\mathcal{A}$ generated by an
orthogonal basis that sum up to 1. Let $M$ be a (right) differential graded
module over $\mathcal{A}$. Denote $M[n]$ as the module defined by
$M[n]_{d}=M_{d-n}$. We say that $M$ is a (right unital)
$\mathcal{A}_{\infty}$-module if the family of right $\mathcal{I}$-actions

$m_{i}:M\otimes\mathcal{A}^{i-1}\to M[2-i],\ \ \ i\geq 1$

satisfies the $A_{\infty}$ relations

$\displaystyle 0$
$\displaystyle=\sum_{i=0}^{n}m_{n-i+1}\left(m_{i+1}\left(x\otimes
a_{1}\otimes\cdots\otimes a_{i}\right)\otimes a_{i+1}\otimes\cdots\otimes
a_{n}\right)$ $\displaystyle+\sum_{i=1}^{n}m_{n+1}\left(x\otimes
a_{1}\otimes\cdots\otimes a_{i-1}\otimes d\left(a_{i}\right)\otimes
a_{i+1}\otimes\cdots\otimes a_{n}\right)$
$\displaystyle+\sum_{i=1}^{n-1}m_{n}\left(x\otimes a_{1}\otimes\cdots\otimes
a_{i-1}\otimes a_{i}a_{i+1}\otimes a_{i+2}\otimes\cdots\otimes a_{n}\right)$

and the unital conditions

$\displaystyle m_{2}(x,1)$ $\displaystyle=x,$ $\displaystyle
m_{i}(x,\cdots,1,\cdots,)$ $\displaystyle=0,\ \ \ i>2.$

A type D structure over $\mathcal{A}$ is a $\mathbb{F}$-vector space $N$, with
left $\mathcal{I}$-action satisfying

$N=\bigoplus_{i=1}^{n}\iota_{i}N,$

and a map

$\delta_{1}:N\to\mathcal{A}\otimes_{\mathcal{I}}N,$

satisfying the type $D$ relations

$(\mu\otimes\operatorname{Id}_{N})\circ(\operatorname{Id}_{\mathcal{A}}\otimes\delta_{1})\circ\delta_{1}+(d\otimes\operatorname{Id}_{N})\circ\delta_{1}=0,$

where $\mu:\mathcal{A}\times\mathcal{A}\to\mathcal{A}$ denotes the
multiplication map on $\mathcal{A}$. Inductively, we can define maps on $N$

$\delta_{k}:N\to\mathcal{A}^{\otimes_{k}}\otimes_{\mathcal{I}}N$

by setting

$\displaystyle\delta_{0}$ $\displaystyle=\operatorname{Id}_{N}$
$\displaystyle\delta_{i}$
$\displaystyle=\operatorname{Id}_{\mathcal{A}}^{\otimes_{i-1}}\otimes\delta_{1}\circ\delta_{i-1}.$

Given an $A_{\infty}$-module $M$ and a type $D$ structure $N$, we define the
box tensor product $M\boxtimes N$ to be the $\mathbb{F}$-vector space
$M\otimes N$, equipped with the differential map

$\partial^{\boxtimes}(x\otimes
y)=\sum_{k=0}^{\infty}(m_{k+1}\otimes\operatorname{Id}_{N})(x\otimes\delta_{k}(y)).$

Let $Y$ be a compact, oriented $3$-manifold with connected boundary $\partial
Y=F$. To the surface $F$, we associate a differential graded algebra
$\mathcal{A}(F)$. To the $3$-manifold $Y$, we associate two invariants:
$\widehat{CFD}(Y)$, which is a type $D$ structure, and $\widehat{CFA}(Y)$,
which is a right $\mathcal{A}_{\infty}$-module over $\mathcal{A}(F)$. To a
knot $K_{1}$ in $Y_{1}$, we may associate either $\widehat{CFA}(Y_{1},K_{1})$,
which is $\mathbb{Z}$-filtered $\mathcal{A}_{\infty}$-module, or
$CFA^{-}(Y_{1},K_{1})$, which is an $\mathcal{A}_{\infty}$-module over
$\mathcal{A}(F)$ with ground ring $\mathbb{F}[U]$. The pairing theorem [LOT18,
Theorem 1.3 and 11.19] states that gluing 3-manifolds along their boundaries
corresponds to taking the box tensor of their invariants. More concretely, let
$Y_{1}$ and $Y_{2}$ be compact oriented 3-manifolds with boundary, along with
an orientation-reversing diffeomorphism $f$ from $\partial Y_{1}$ to $\partial
Y_{2}$. Let $Y=Y_{1}\cup_{f}Y_{2}$ be the 3-manifold obtained by gluing
$Y_{1}$ and $Y_{2}$. Then there exists a homotopy equivalence

$\widehat{CF}(Y)\simeq\widehat{CFA}(Y_{1})\boxtimes\widehat{CFD}(Y_{2}).$

Moreover, if we have a knot $K_{1}\subset Y_{1}$ whose image $K\subset Y$
under identification is null-homologous, then we get a homotopy equivalence of
$\mathbb{F}[U]$-modules

$gCFK^{-}(Y,K)\simeq CFA^{-}(Y_{1},K_{1})\boxtimes\widehat{CFD}(Y_{2}),$

where $gCFK^{-}(Y,K)$ denotes the associated graded of $CFK^{-}(Y,K)$. We will
discuss these invariants in more detail in Sections 2.3 and 2.4.

######  Remark 2.3.

There is an alternate version of the pairing theorem, which gives a homotopy
of $\mathbb{Z}$-filtered chain complexes

$\widehat{CFK}(Y,K)\simeq\widehat{CFA}(Y_{1},K_{1})\boxtimes\widehat{CFD}(Y_{2}),$

where $\widehat{CFA}$ is the hat version of $CFA^{-}$. For more detail, see
[LOT18].

In this paper, we are concerned with satellite knots, so we restrict our
discussion to the case where $F$ is the torus $T^{2}$. We let $Y_{1}$ be the
solid torus $V$ equipped with a pattern knot $P$, and $Y_{2}$ be the bordered
manifold $X_{K}=S^{3}\backslash N^{\mathrm{o}}(K)$ with the bordered structure
given by the $0$-framing (where $N^{\mathrm{o}}(K)$ denotes a regular
neighborhood of $K$). Upon gluing $Y_{1}$ and $Y_{2}$, the knot $K\subset Y$
becomes the satellite knot $P(K)$ inside of $S^{3}$. The pairing theorem then
tells us that

(2.1) $gCFK^{-}(P(K))\simeq CFA^{-}(V,P)\boxtimes\widehat{CFD}(X_{K}).$

Setting $P$ to be the generalized Mazur patterns, we see that computing
$gCFK^{-}(Q_{m,n}(K))$ reduces to finding the $A_{\infty}$-module for the
generalized Mazur patterns and the type $D$ structure for $X_{K}$. We discuss
how one may obtain these two components in Sections 2.3 and 2.4.

Next, we describe the graded algebra $\mathcal{A}(F)$ in the case where
$F=T^{2}$; in the general case where $g(F)\geq 2$, $\mathcal{A}(F)$ is a
differential graded algebra, as discussed in [LOT18]. Recall that one may
specify a torus via handle decomposition: a disk $D^{2}$ with two $1$-handles
attached, such that the boundary is connected and can be capped off with a
disk. We represent this information by a pointed matched circle
$(Z,z,\\{a_{1},a_{3}\\},\\{a_{2},a_{4}\\})$, where $Z$ is an oriented circle,
$z\in Z$ is a fixed basepoint, and $\\{a_{1},a_{3}\\},\\{a_{2},a_{4}\\}$ are
two pairs of points on $Z$ disjoint from $z$ (see Figure 5). We may recover
the torus from the pointed matched circle; see [LOT14, Construction 1.2].

Figure 5. Left: the pointed matched circle for the surface $T^{2}$. Right: the
same pointed matched circle cut open at $z.$

Given a torus $T^{2}$ parametrized by a pointed matched circle, the graded
algebra $\mathcal{A}(T^{2})$ is generated over $\mathbb{F}$ by two idempotents
$\iota_{0}$ and $\iota_{1}$ (where $\iota_{0}+\iota_{1}=1$) and the six “Reeb”
elements $\rho_{1}$, $\rho_{2}$, $\rho_{3}$, $\rho_{12}$, $\rho_{23}$,
$\rho_{123}$, satisfying the comptability conditions

$\begin{array}[]{ccc}\rho_{1}=\iota_{1}\rho_{1}=\rho_{1}\iota_{2}&\rho_{2}=\iota_{2}\rho_{2}=\rho_{2}\iota_{1}&\rho_{3}=\iota_{1}\rho_{3}=\rho_{3}\iota_{2}\\\
\rho_{12}=\iota_{1}\rho_{12}=\rho_{12}\iota_{1}&\rho_{23}=\iota_{2}\rho_{23}=\rho_{23}\iota_{2}&\rho_{123}=\iota_{1}\rho_{123}=\rho_{123}\iota_{2}\end{array}$

and the nonzero products

(2.2)
$\begin{array}[]{ccc}\rho_{1}\rho_{2}=\rho_{12},&\rho_{2}\rho_{3}=\rho_{23},&\rho_{1}\rho_{2}\rho_{3}=\rho_{12}\rho_{3}=\rho_{1}\rho_{23}=\rho_{123}.\end{array}$

Schematically, the algebra elements are shown in Figure 6.

Figure 6. The idempotents and the “Reeb” elements in $\mathcal{A}(T^{2})$

For a three manifold $Y$ with $\partial Y=T^{2}$, a doubly pointed bordered
Heegaard diagram for $Y$ is a tuple
$\mathcal{H}=(\bar{\Sigma},\alpha^{c},\alpha^{a},\beta,z,w)$ such that

  * •

$\bar{\Sigma}$ is a compact oriented surface of genus $g$ with a single
boundary component.

  * •

There is a $(g-1)$-tuple of pairwise disjoint circles
$\alpha^{c}=(\alpha_{1}^{c},\cdots,\alpha_{g}^{c})$ in
$\bar{\Sigma}^{\mathrm{o}}$, the interior of $\bar{\Sigma}$.

  * •

There is a pair of disjoint arcs $\alpha^{a}=(\alpha_{1}^{a},\alpha_{2}^{a})$
in $\bar{\Sigma}\backslash\alpha^{c}$ with endpoints on
$\partial\bar{\Sigma}$.

  * •

There is a $g$-tuple of pairwise disjoint circles
$\beta=(\beta_{1},\cdots,\beta_{g})$ in $\bar{\Sigma}^{\mathrm{o}}$.

  * •

$z$ and $w$ are basepoints in
$\bar{\Sigma}\backslash(\alpha^{a}\cup\alpha^{c}\cup\beta)$, with
$z\in\partial\bar{\Sigma}$.

In addition, we require that the $\alpha$-curves and $\beta$-curves intersect
transversely, that the complement of the $\alpha$-curves and the complement of
the $\beta$-curves are connected, and that
$(\partial\bar{\Sigma},z,\partial\alpha_{1}^{a},\partial\alpha_{2}^{a})$
describes a pointed matched circle. Whenever the context is clear, we refer to
doubly pointed bordered Heegaard diagrams simply as Heegaard diagrams.

A Heegaard diagram specifies a knot embedded in $Y$ by connecting $w$ to $z$
in the complement of the $\beta$ curves and connecting $z$ to $w$ in the
complement of the $\alpha$ arcs. If the two sets of arcs cross, the second set
crosses under the first set. Moreover, every knot in a bordered three manifold
can be realized by a doubly pointed Heegaard diagram.

A pattern knot embedded in the solid torus $V=S^{1}\times D^{2}$ is called a
$(1,1)$-pattern knot if it admits a Heegaard diagram with genus 1 [Che19]. All
the pattern knots in this paper are $(1,1)$-pattern knots, including the Mazur
pattern whose bordered Heegaard diagram was shown in Figure 8 [Lev16]. The arc
$\alpha^{a}_{1}$ represents a meridian
$\mu_{V}=\\{\textrm{pt}\\}\times\partial D^{2}$, and the arc $\alpha^{a}_{2}$
represents a longitude $\lambda_{V}=S^{1}\times\\{\textrm{pt}\\}$. Note that
there are no $\alpha$-arcs in the interior of our Heegaard diagrams because
their genus is 1.

Figure 7. Pseudoholomorphic disks bounded by an $\alpha$-arc (red) and a
$\beta$-arc (blue). The $CFA^{-}(V,P)$ relations corresponing to these disks
are $m_{1}(a)=Ub$ (left) and $m_{2}(a,\rho_{1})=b$ (right) Figure 8. Genus 1
Heegaard diagram for the Mazur pattern $Q$ (equivalently $Q_{2,1}$)

###  2.3. Obtaining type $A$ structures from Heegaard diagrams

Recall that the pairing theorem states that

$gCFK^{-}(Y,K)\simeq CFA^{-}(Y_{1},K_{1})\boxtimes\widehat{CFD}(Y_{2}).$

In this subsection, we outline how one may obtain $CFA^{-}(V,P)$ from a
Heegaard diagram $\mathcal{H}$ associated with $P$, and in the next
subsection, we discuss how one determines the type $D$ structure. We may
assume $P$ to be a $(1,1)$-pattern knot, since all pattern knots are $(1,1)$
in this paper. As such, the $\mathbb{F}[U]$-vector space $CFA^{-}(V,P)$ is
generated by the set of intersection points of $\alpha$-arcs and $\beta$-arcs,
denoted $\mathfrak{S}(\mathcal{H})$. The right $\mathcal{I}$-action is given
by

$\displaystyle x\cdot\iota_{0}$ $\displaystyle=\begin{cases}x&\textrm{if
}x\textrm{ lies on the arc }\alpha_{1}^{a},\\\
0&\textrm{otherwise,}\end{cases}$ $\displaystyle x\cdot\iota_{1}$
$\displaystyle=\begin{cases}x&\textrm{if }x\textrm{ lies on the arc
}\alpha_{2}^{a},\\\ 0&\textrm{otherwise.}\end{cases}$

We may obtain the $A_{\infty}$-structure on $CFA^{-}(V,P)$ by counting certain
pseudoholomorphic curves, giving us the family of $\mathcal{I}$-actions:

$m_{j+1}:CFA^{-}(V,P)\otimes\mathcal{A}^{\otimes j}\to CFA^{-}(V,P).$

Specifically, we extend the Heegaard diagram $\mathcal{H}$ to its universal
cover, and count the pseudoholomorphic disks $\phi$ that are bounded by an
$\alpha$-arc and a $\beta$-arc intersecting acutely at two points in
$\mathfrak{S}(\mathcal{H})$. Label the intersection points by $a$ and $b$,
such that if we start at $a$ and travel counterclockwise along the boundary,
we encounter the $\alpha$-arc first (see Figure 7). Note that the boundary of
$\phi$ may contain parts of the pointed matched circle, which we denote by
their corresponding algebra elements $\rho_{i_{1}},\cdots,\rho_{i_{j}}$
(concatenated when possible per equation (2.2)), in the order that we
encounter travelling counterclockwise from $a$. The relation in $CFA^{-}(V,P)$
corresponding to $\phi$ is

$m_{j+1}(a,\rho_{i_{1}},\cdots,\rho_{i_{j}})=U^{n_{w}(\phi)}b,$

where $n_{w}(\phi)$ is the number of times $w$ appears inside $\phi$. Note
that the region bounded by $\phi$ must be a disk, and in particular has
trivial fundamental group (this is true only in the genus one case).
Furthermore, $\phi$ may not intersect the top left corner of $\mathcal{H}$. In
Figure 7, the corresponding relations are $m_{1}(a)=Ub$, and $m_{2}(a)=b$,
respectively. An example of finding these disks and writing their
corresponding relations in $CFA^{-}$ is given in Example A.2.

###  2.4. Obtaining type $D$ structures from $CFK^{-}(K)$

The module $\widehat{CFD}(X_{K})$ may be algorithmically computed in terms of
a basis in $CFK^{-}(K)$. The algorithm in the following theorem is originally
due to [LOT18]; we state a slight enhancement of it, due to [Lev16].

Recall that $CFK^{-}(K)$ always admit a horizontally and a vertically
simplified basis up to chain homotopy. Let
$\big{\\{}\widetilde{\xi}_{0},\ldots,\widetilde{\xi}_{2n}\big{\\}}$ and
$\big{\\{}\widetilde{{\eta}}_{0},\ldots,\widetilde{{\eta}}_{2n}\big{\\}}$ be
its vertically and horizontally simplified basis, respectively. Then the type
$D$ structure $\widehat{CFD}(X_{K})$ may be described as follows.

######  Theorem 2.4.

[Lev16, Theorem 2.6] Let $K$ be a knot in $S^{3}$, and let
$\big{\\{}\widetilde{\xi}_{0},\ldots,\widetilde{\xi}_{2n}\big{\\}}$ be the
vertically and horizontally simplified basis described above. The type D
structure $\widehat{\mathrm{CFD}}\left(X_{K}\right)$ satisfies the following
properties:

  * •

The summand $\iota_{0}\cdot\widehat{\mathrm{CFD}}\left(X_{K}\right)$ has
dimension $2n+1$, with designated bases
$\left\\{\xi_{0},\ldots,\xi_{2n}\right\\}$ and
$\left\\{\eta_{0},\ldots,\eta_{2n}\right\\}$ related by

$\xi_{p}=\sum_{q=0}^{2n}a_{p,q}\eta_{q}\quad\text{ and
}\quad\eta_{p}=\sum_{q=0}^{2n}b_{p,q}\xi_{q}.$

These elements are all homogeneous with respect to the grading by relative
spinc structures.

  * •

The summand $\iota_{1}\cdot\widehat{\mathrm{CFD}}\left(X_{K}\right)$ has
dimension $\sum_{j=1}^{n}\left(k_{j}+l_{j}\right)+s$, where $s=$ $2|\tau(K)|$,
with basis

$\bigcup_{j=1}^{n}\left\\{\kappa_{1}^{j},\ldots,\kappa_{k_{j}}^{j}\right\\}\cup\bigcup_{j=1}^{n}\left\\{\lambda_{1}^{j},\ldots,\lambda_{l_{j}}^{j}\right\\}\cup\left\\{\mu_{1},\ldots,\mu_{s}\right\\}.$

  * •

For $j=1,\ldots,n$, corresponding to the vertical arrow
$\xi_{2j-1}\rightarrow\xi_{2j}$, there are coefficient maps

(2.3) $\xi_{2j}\stackrel{{\scriptstyle
D_{123}}}{{\longrightarrow}}\kappa_{1}^{j}\stackrel{{\scriptstyle
D_{23}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
D_{23}}}{{\longrightarrow}}\kappa_{k_{j}}^{j}\stackrel{{\scriptstyle
D_{1}}}{{\longleftarrow}}\xi_{2j-1}.$

  * •

For $j=1,\ldots,n$, corresponding to the horizontal arrow
${\eta}_{2j-1}\rightarrow{\eta}_{2j}$, there are coefficient maps

(2.4) $\eta_{2j-1}\stackrel{{\scriptstyle
D_{3}}}{{\longrightarrow}}\lambda_{1}^{j}\stackrel{{\scriptstyle
D_{23}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
D_{23}}}{{\longrightarrow}}\lambda_{l_{j}}^{j}\stackrel{{\scriptstyle
D_{2}}}{{\longrightarrow}}\eta_{2j}$

  * •

Depending on $\tau(K)$, there are additional coefficient maps

(2.5) $\begin{cases}\eta_{0}\stackrel{{\scriptstyle
D_{3}}}{{\longrightarrow}}\mu_{1}\stackrel{{\scriptstyle
D_{23}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
D_{23}}}{{\longrightarrow}}\mu_{s}\stackrel{{\scriptstyle
D_{1}}}{{\longleftarrow}}\xi_{0}&\tau(K)>0\\\ \xi_{0}\stackrel{{\scriptstyle
D_{12}}}{{\longrightarrow}}\eta_{0}&\tau(K)=0\\\
\xi_{0}\stackrel{{\scriptstyle
D_{123}}}{{\longrightarrow}}\mu_{1}\stackrel{{\scriptstyle
D_{23}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
D_{23}}}{{\longrightarrow}}\mu_{0}\stackrel{{\scriptstyle
D_{2}}}{{\longrightarrow}}\eta_{0}&\tau(K)<0\end{cases}$

We refer to the subspaces of $\widehat{CFD}(X_{K})$ spanned by the generators
in (2.3), (2.4), and (2.5) as the vertical chains, horizontal chains, and
unstable chain, respectively.

For example, running the algorithm for the right-handed trefoil $T_{2,3}$, we
obtain the type $D$ structure for its complement in $S^{3}$, as shown in
Figure 9. The basis for $\iota_{0}\cdot\widehat{CFD}(X_{T_{2,3}})$ is
$\\{\xi_{0},\xi_{1},\xi_{2}\\}$, and the basis for
$\iota_{1}\cdot\widehat{CFD}(X_{T_{2,3}})$ is
$\\{\kappa,\lambda,\mu_{1},\mu_{2}\\}$.

${\xi_{0}}$${\lambda}$${\xi_{1}}$${\mu_{1}}$${\mu_{2}}$${\kappa}$${\xi_{2}}$$\scriptstyle{\rho_{1}}$$\scriptstyle{\rho_{2}}$$\scriptstyle{\rho_{3}}$$\scriptstyle{\rho_{1}}$$\scriptstyle{\rho_{23}}$$\scriptstyle{\rho_{3}}$$\scriptstyle{\rho_{123}}$
Figure 9. Getting $\widehat{CFD}$ form $CFK^{-}$ for the right-handed trefoil
complement

In general, we may use Theorem 2.4 to determine the structure for
$\widehat{CFD}(X_{K})$, where $K$ is an arbitrary knot in $S^{3}$. The type D
structure when $\tau(K)>0$ is displayed in Figure 10, and the type D
structures when $\tau(K)\leq 0$ are determined similarly. In practice, we are
concerned with finding a particular summand of the box tensor
$CFA^{-}(V,P)\boxtimes\widehat{CFD}(X_{K})$. For this purpose, we will only
need to consider a neighborhood of the unstable chain in
$\widehat{CFD}(X_{K})$, which we call the unstable neighborhood.

${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\eta_{1}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta_{2}}$${\xi_{0}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\lambda^{1}_{l_{1}}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\cdots}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\lambda^{1}_{1}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\eta_{1}}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\lambda^{1}_{1}}$${\mu_{s}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\xi_{1}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\kappa^{1}_{k_{1}}}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vdots}$${\ddots}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\vdots}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\lambda^{1}_{l_{1}}}$${\mu_{1}}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\eta_{2}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\kappa^{1}_{1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\xi_{2}}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\kappa^{1}_{1}}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\cdots}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\kappa^{1}_{k_{1}}}$${\eta_{0}}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\xi_{1}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\xi_{2}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}D_{3}}$$\scriptstyle{D_{1}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D_{2}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D_{23}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D_{23}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D_{3}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}D_{23}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D_{1}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}D_{23}}$$\scriptstyle{D_{23}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D_{23}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}D_{2}}$$\scriptstyle{D_{23}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D_{23}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}D_{123}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}D_{23}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}D_{23}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}D_{1}}$$\scriptstyle{D_{3}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D_{123}}$
Figure 10. Neighborhood of the unstable chain of $\widehat{CFD}(X_{K})$ for
$\tau(K)>0$. When $\epsilon=-1$, the unstable neighborhood is the left and
bottom edges of the diagram square, colored in red. When $\epsilon=1$, the
unstable neighborhood is the left and bottom edges of the diagram square,
colored in blue.

##  3\. Immersed Heegaard Floer homology

We introduce the immersed curve interpretation of the pairing theorem in
[LOT18, Theorem 1.3]. In Section 3.1, we sketch out how one may convert a type
$D$ structure to an immersed curve in the torus $T^{2}$. In Section 3.2, we
introduce the notion of a pairing diagram and the method to determine
Alexander grading. This gives us an algorithmic strategy to compute the $\tau$
and $\epsilon$ invariant of the satellite $P(K)$, which is sketched out in
Section 3.3.

###  3.1. From type $D$ structures to immersed curves

Given a type D stucture $N=\widehat{CFD}(X_{K})$ over the torus algebra
$\mathcal{A}$, we can convert it into an immersed curve in the torus $T^{2}$
using an algorithm described in [HRW23, Sections 2.3 and 2.4].

The first step is to convert $N$ into an $\mathcal{A}$-decorated graph
$\Gamma$, which is a directed graph whose vertex is either $\bullet$ or
$\circ$, and whose edges are labelled with one element from
$\\{\emptyset,1,2,3,12,23,123\\}$, along with additional requirements
specified in [HRW23, Section 2.4]. The conversion is given as follows. For
each generator of $N$ which is either in $\iota_{0}$ or $\iota_{1}$, we put a
$\bullet$ or a $\circ$ in $\Gamma$, respectively. Suppose that there are two
vertices corresponding to generators $x$ and $y$ such that $\rho_{I}\otimes y$
is a summand of $\delta_{1}(x)$; in this case, we put a corresponding edge
labeled by $I$ from $x$ to $y$ in $\Gamma$. The higher differentials
$\delta_{k}$ in the type D structure then correspond to direct paths in
$\Gamma$. We say a decorated graph (and its associated type D structure) is
reduced if there are no edges labeled by $\emptyset$.

Next, we convert the $\mathcal{A}$-decorated graph $\Gamma$ into an immersed
train track. Let $T$ be a torus $\mathbb{R}^{2}/\mathbb{Z}^{2}$ punctured at
$z=(1-\epsilon,1-\epsilon)$, and let $\mu$ and $\lambda$ be the images of the
$x$ and $y$ axis, respectively. We embed the vertices of $\Gamma$ so that the
$\bullet$’s are distinct points in the interval
$\\{0\\}\times[\frac{1}{4},\frac{3}{4}]$ within $\lambda$, and the $\circ$’s
are distinct points in the interval $[\frac{1}{4},\frac{3}{4}]\times\\{0\\}$
within $\mu$. Then, we embed the edges of $\Gamma$ according to Figure 11.

Figure 11. Correspondence between the edges of the decorated graph $\Sigma$
and directed edges in the punctured torus $T$

This gives us an immersion from $\Sigma$ to $T$, which we denote as
$\alpha_{K}$. We further require that all intersections in the image of
$\alpha_{K}$ are transversal, and edges intersect in points that are away from
$\lambda$ and $\mu$. In this case, we call $\alpha_{K}$ the immersed train
track associated with $K$. Figure 12 shows the immersed train track associated
with the right-handed trefoil.

Figure 12. Immersed train track associated with the right-handed trefoil.

For an arbituary knot $K$, we know how to compute $\widehat{CFD}(X_{K})$ from
Proposition 2.4 (also see Figure 10). Given the algorithm converting
$\widehat{CFD}$ to immersed curves stated above, we may describe the shape of
$\alpha_{K}$ when lifted to the universal cover. First, we describe the
segment in $\alpha_{K}$ corresponding to the unstable chain, which we call the
unstable segment of $\alpha_{K}$.

######  Lemma 3.1.

[Bod23, Lemma 2.2] Suppose that $K$ is a knot in $S^{3}$ and that $\gamma_{0}$
is the unstable segment of $\alpha_{K}$ lifted to the universal cover.

  * •

If $\epsilon(K)=1$ and $\tau(K)\geq 0$, $\gamma_{0}$ slopes upwards for
$2\tau(K)$ rows and turns down at the top and up at the bottom.

  * •

If $\epsilon(K)=-1$ and $\tau(K)\geq 0$, $\gamma_{0}$ slopes upwards for
$2\tau(K)$ rows and turns up at the top and down at the bottom.

  * •

If $\epsilon(K)=1$ and $\tau(K)\leq 0$, $\gamma_{0}$ slopes downwards for
$2\tau(K)$ rows and turns down at the bottom and up at the top.

  * •

If $\epsilon(K)=-1$ and $\tau(K)\leq 0$, $\gamma_{0}$ slopes downwards for
$2\tau(K)$ rows and turns up at the bottom and down at the top.

  * •

If $\epsilon(K)=0$, then $\tau(K)=0$ and $\gamma_{0}$ is horizontal at height
$0$.

Figure 13. The unstable segment and its neighborhood in both directions in
$\alpha_{K}$. The points $\xi_{0}$ and $\eta_{0}$ are at heights $\tau(K)$ and
$-\tau(K)$ respectively when $\tau(K)\geq 0$, and the points $\xi_{0}$ and
$\eta_{0}$ are at heights $-\tau(K)$ and $\tau(K)$ respectively when
$\tau(K)\leq 0$. The length of the dotted neighborhood can be arbitrarily
large or small.

We may extend the results of Lemma 3.1 to describe a neighborhood of the
unstable segment using the correspondence between elements of
$\widehat{CFD}(X_{K})$ and their immersed curve representations shown in
Figure 11. This is displayed in Figure 13. For example, the top figures shows
the neighborhood of the unstable segment when $\tau(K)\geq 0$, corresponding
to the unstable neighborhood in $\widehat{CFD}(X_{K})$ appearing previously in
Figure 10. The bottom figure displays the neighborhood of the unstable segment
when $\tau(K)<0$.

###  3.2. Pairing diagrams

In this section, we introduce the notion of a pairing diagram for the
satellite $P(K)$. Let $\alpha_{K}$ be the immersed curve associated to $K$ as
previously described, and let $\beta_{P}$ be the $\beta$-curve in
$\mathcal{H}$, the genus-one Heegaard diagram of $P$. Following the notation
in [CH23], we denote the pairing diagram of $P(K)$ by
$\mathcal{H}(\alpha_{K})$. Intuitively, the pairing diagram is a way of laying
$\beta_{P}$ over $\alpha_{K}$ in the torus $T^{2}=[0,1]^{2}/\sim$.
Specifically, we divide $T^{2}$ into four quadrants, upon which we include
$\alpha_{K}$ into the first quadrant and $\beta_{P}$ (along with the $w,z$
basepoints) into the third quadrant. For the second and fourth quadrant,
extend both curves horizontally and vertically. See Figure 14(a) for an
example of a pairing diagram of the Mazur pattern $Q_{2,1}$ with the right-
handed trefoil as companion. The intersection points in the second quadrant
correspond to the generators in
$CFA^{-}(S^{3},P)\boxtimes\widehat{CFD}(S^{3},X_{K})$ coming from the
$\iota_{0}$ idempotent, and the intersection points in the fourth quadrant
correspond to those generators coming from the $\iota_{1}$ idempotent.

(a) Pairing diagram of $Q_{2,1}(T_{2,3})$

(b) Pairing diagram of $Q_{2,1}(T_{2,3})$ in the universal cover, after
isotopy

In practice, we usually draw the pairing diagram $\mathcal{H}(\alpha_{K}$ in
the universal cover $\pi:\mathbb{R}^{2}\to\mathbb{R}^{2}/\mathbb{Z}^{2}$, with
a single lift of $\alpha_{K}$ and a single lift of $\beta_{P}$, which we
denote by $\tilde{\alpha}$ and $\tilde{\beta}$, respectively. Intuitively, we
can imagine the universal cover $\mathbb{R}^{2}$ as a board with a lattice of
pegs nailed in at the $w$ and $z$ basepoints, and the curve $\tilde{\alpha}$
as a rubber band that winds around those pegs. This curve is called a peg-
board diagram. When $\tilde{\alpha}$ is pulled tight, we require that it
intersects $\tilde{\beta}$ transversally, and that every Whitney disk
connecting two intersection points must contain at least a $w$ or a $z$
basepoint. Figure 14(b) shows the pairing diagram of $Q_{2,1}(T_{2,3})$ in the
universal cover after isotopy, so that there are minimal intersection points
between $\tilde{\alpha}$ and $\tilde{\beta}$.

Given a pairing diagram, we may recover the Alexander grading of intersection
points representing generators of $gCFK^{-}(P(K))$, which we describe below.
First, we can recover the relative Alexander grading with the following Lemma.

######  Lemma 3.2.

[Che19, Lemma 4.1] Let $x,y$ be two intersection points of $\alpha$ and
$\beta$. Let $l$ be the section of the $\beta$-curve from $x$ to $y$, and let
$\delta_{w,z}$ be the straight arc from $w$ to $z$. Then

$A(y)-A(x)=l\cdot\delta_{w,z},$

where multiplication on the right counts the algebraic intersection between
$l$ and $\delta_{w,z}$.

To determine the absolute Alexander grading, note that the pairing diagram is
symmetric under hyperelliptic involution. In other words, the complex remains
the same if we rotate it by $\pi$ and exchange the $w$ and $z$ basepoints. The
intersection point fixed under this involution must necessarily have Alexander
grading 0. From this we may recover the absolute gradings of all intersection
points using Lemma 3.2. In Example 3.3, we use this method to compute the
absolute gradings of certain intersection points in the pairing diagram of
$Q_{2,1}(T_{2,3})$.

###  3.3. Computation of $\tau(P(K))$ and $\epsilon(P(K))$ from the pairing
diagram

From the pairing diagram $\mathcal{H}(\alpha_{K})$ for the satellite $P(K)$,
we may obtain $\tau(P(K))$ and $\epsilon(P(K))$ via a combinatorial
computation similar to [CH23], which we describe here. First, we construct a
bigraded chain complex $CFK_{\mathcal{R}}(\mathcal{H}(\alpha_{K}),\partial)$
over $\mathcal{R}$ by counting bigons combinatorially in the pairing diagram
$\mathcal{H}(K)$. Specifically, the generators of the complex are
intersections of the $\tilde{\alpha}$ and $\tilde{\beta}$, and the
differentials are given by bigons connecting two generators. Suppose there is
a bigon $\phi$ bounded by $\tilde{\alpha}$ and $\tilde{\beta}$ intersecting
acutely at $a$ and $b$, such that we encounter $\tilde{\alpha}$ first if we
travel counterclockwise from $a$. The differential in
$CFK_{\mathcal{R}}(\mathcal{H}(\alpha_{K}),\partial)$ corresponding to $\phi$
is

$\partial a=U^{n_{w}(\phi)}V^{n_{z}(\phi)}b,$

where $n_{w}(\phi)$, $n_{z}(\phi)$ are the number of $w$ and $z$ basepoints in
$\phi$, respectively. As we are working with $CFK_{\mathcal{R}}$, we recall
that $UV=0$. See [CH23, Figure 1] for an example of the complex
$CFK_{\mathcal{R}}$ of the $(2,1)$ cable paired with the right-handed trefoil
$T_{2,3}$.

Recall the pairing theorem [LOT18, Theorem 11.19] which says

$\displaystyle gCFK^{-}(Y,K)$ $\displaystyle\simeq
CFA^{-}(Y_{1},K_{1})\boxtimes\widehat{CFD}(Y_{2}).$

In [CH23], Chen and Hanselman reinterpreted the pairing theorem in terms of
the immersed curves in the case when $CFA^{-}(V,P)$ is paired with
$\widehat{CFD}(X_{K})$, thus obtaining $gCFK^{-1}(S^{3},P(K))$. (Here $P$ must
be a $(1,1)$-pattern knot, as is the case for generalized Mazur patterns). The
main theorem in [CH23] states that there is a bigraded homotopy equivalence of
chain complexes

$CFK_{\mathcal{R}}(\mathcal{H}(\alpha_{K}),\mathcal{\partial})\cong
CFK_{\mathcal{R}}(P(K)).$

In other words, the pairing of $CFA^{-}(V,P)$ with $\widehat{CFD}(X_{K})$ is
entirely captured in the $CFK_{\mathcal{R}}$ complex of the pairing diagram.
Hence, to recover the $\tau$ and $\epsilon$ invariants of $P(K)$, it suffices
to compute the relevant portion of
$CFK_{\mathcal{R}}(\mathcal{H}(\alpha_{K}),\mathcal{\partial})$.

First, we describe how to recover $\tau(P(K))$, using an algorithm described
in [Che19]. Recall that the Alexander filtration on $\widehat{CFK}(K)$ induces
a spectral sequence converging to $\widehat{HF}(S^{3})=\mathbb{F}$, and that
the $\tau$-invariant is the Alexander grading of the cycle that survives to
the $E_{\infty}$ page. Passing from one page to the next in the spectral
sequence amounts to eliminating the differentials that connect elements of
minimal Alexander filtration difference. (For a more detailed discussion of
why this is the case, see for example [Bod23, Che19].) This can be performed
in the pairing diagram: the differentials correspond to Whitney disks that
contain $z$ basepoints but not $w$ basepoints, and eliminating the
differential amounts to isotoping the $\beta$-curve over such Whitney disks
that connect intersection points with minimal filtration difference, as seen
in Figure 15. Each time we perform such an isotopy, we eliminate a pair of
intersection points, so that at the end of such isotopies, we only have one
intersection point left. This intersection point corresponds to the cycle that
survives to the $E_{\infty}$ page, hence its Alexander grading is the
$\tau$-invariant of the satellite knot.

Figure 15. Eliminating a pair of intersection points, $x$ and $y$ with minimal
filtration difference. The Whitney disks connecting $x$ and $y$ is highlighted
in pink. The small black arrow is the A-buoy placed along the $\beta$-curve.

In practice, we need to remember the filtration difference of the intersection
points, and we do so by placing A-buoys along the $\beta$-curve, which are
small arrows introduced in [Che19, Section 4]. See Figure 15 for the placement
of these A-buoys. Recall that from Lemma 3.2 that the relative Alexander
grading between intersection points $x$ and $y$ is given by the algebraic
intersection between the arc on the $\beta$-curve connecting $x$ and $y$ and
$\delta_{w,z}$. When we perform the isotopies described above, we may change
this algebraic intersection; to count the filtration difference of the
remaining intersection pooints, we must count the algebraic intersection of
the $\beta$-curve with both $\delta_{w,z}$ and the A-buoys.

Note that this algorithm corresponds to doing successive pages of a spectral
sequence, so we have to eliminate pairs of intersection points in order of
their original filtration difference. In other words, we need to eliminate all
disks connecting points of filtration difference one, then once none are left,
all disks connecting points of filtration distance two, and so on, making sure
to count the A-buoys.

To compute $\epsilon(P(K))$, we construct a subcomplex of
$CFK_{\mathcal{R}}(\mathcal{H}(\alpha_{K}))$ that contains the cycle which
survives to the $E_{\infty}$ page. This subcomplex must contain the
distinguished element of some horizontally simplified basis, whose vertical
situation indicates the value of $\epsilon(P(K))$. Note that in general, it is
difficult to locate a subcomplex in
$CFK_{\mathcal{R}}(\mathcal{H}(\alpha_{K}))$ that contains the distinguished
element. Doing isotopy across $z$ basepoints allow us to track down one
element in that subcomplex, hence recovering the entire subcomplex becomes
computationally easy.

Then, we perform isotopies to simplify the subcomplex (i.e. reduce the number
of elements as much as possible) while ensuring that it is reduced.
Differentials obtained from crossing over $z$ are called vertical
differentials, and are labelled with powers of $V$ corresponding to the
multiplicity of $z$, and differentials obtained from crossing over $w$ are
called horizontal differentials, labelled with powers of $U$ corresponding to
the multiplicity of $w$. In the horizontal complex consisting of the
horizontal differentials, we look for a cycle which survives in the homology
$\widehat{HF}(S^{3})$; this is the distinguished element whose vertical
situation gives us $\epsilon(P(K))$, and whose Alexander grading gives us
$\tau(P(K))$.

######  Example 3.3.

We compute $\tau(Q_{2,1}(T_{2,3}))$ using the algorithm described above. The
pairing diagram of the Mazur pattern and the right-handed trefoil is given in
Figure 16.

Figure 16. Pairing diagram of $Q_{2,1}(T_{2,3})$

First, we eliminate all pairs of intersection points of filtration difference
1 (see Figure 17). After isotoping the right side of the $\beta$-curve to
obtain Figure 18, there are no more differentials in the new complex, and the
only remaining intersection point is $x_{3}$. By the symmetry under elliptic
involution, we see that $A(x_{15})=0$. Then using Lemma 3.2, we have
$A(x_{3})=-2$, therefore $\tau(Q_{2,1}(T_{2,3}))=-A(x_{3})=2$. This matches up
with the bordered computation in [Lev16, Theorem 1.6]. To compute
$\epsilon(Q_{2,1}(T_{2,3}))$, we refer the reader to [CH23], where the
differentials in the $CFK_{\mathcal{R}}$ complex are given. Note that the
Mazur pattern in [CH23] has the opposite orientation, which amounts to
switching the $w$ and $z$ basepoint in the pairing diagram. The differentials
in $CFK_{R}$ are changed accordingly by switching the $U$ and $V$ powers. From
there, one may compute the homology and find that the generator containing
$x_{3}$ has an incoming vertical arrow. Therefore
$\epsilon(Q_{2,1}(T_{2,3}))=1$.

Figure 17. Eliminating pairs of intersection points of filtration difference
1. The Whitney disks connecting those pairs are highlighted in pink. After the
isotopies, we get the dark green curve as our new $\beta$-curve.  Figure 18.
$x_{3}$ is the only intersection point remaining.

##  4\. Generalized Mazur patterns

In this section, we describe the generalized Mazur patterns $Q_{m,n}$ and some
of their properties, including representations of their associated 2-bridge
links. We begin by describing how to construct bordered Heegaard diagrams for
$Q_{m,n}$ pattern knots, which will be utilized in our computation of the
$\tau$ and $\epsilon$-invariant in Section 5.

######  Lemma 4.1.

We may obtain a Heegaard diagram for $Q_{m,n}$ starting from $Q_{1,1}$ by an
inductive algorithm on $m$ and $n$.

###### Proof.

The inductive procedure begins with a Heegaard diagram for $Q_{1,1}$, then
constructs $Q_{m,1}$, $Q_{m,2}$, and $Q_{m,n}$.

The Heegaard diagram for the $Q_{1,1}$ case is shown in Figure 19. A Heegaard
diagram has left and right vertical $\alpha_{1}$ arcs, top and bottom
$\alpha_{2}$ arcs, and a $\beta$ curve which crosses the $\alpha_{i}$ arcs,
breaking it into strands inside the diagram. To describe the algorithm, we
utilize the terminology of Figure 20, which label the strands of the $\beta$
curve based on where they are located in the diagram and how they intersect
the $\alpha$-arcs. We first describe the strands which intersect both the
$\alpha_{2}$ and $\alpha_{1}$ arcs: the strand $T/L$ intersects the top
$\alpha_{2}$ arc and the left $\alpha_{1}$ arc; $B/L$ intersects the bottom
$\alpha_{2}$ arc and the left $\alpha_{1}$ arc; $T/R$ intersects the top
$\alpha_{2}$ arc and the right $\alpha_{1}$ arc; $B/R$ intersects the bottom
$\alpha_{2}$ arc and the right $\alpha_{1}$ arc. Next, we describe the strands
which only intersect the $\alpha_{2}$ arcs: $V/M/L$ and $V/M/R$ intersect both
the top and bottom of the $\alpha_{2}$ arcs, and these are placed in between
the four aforementioned strands which intersect both $\alpha_{2}$ and
$\alpha_{1}$ arcs. The final label $L$ or $R$ determine whether they are to
the left or right of the final type of strand: $B/M/RNB$. This strand
intersects the bottom $\alpha_{2}$ arc twice, and takes the shape of a
rainbow. If we label the horizontal intersection points lying on the
$\alpha_{2}$ arcs in a sequential order, the $V/M/L$ arc passes from an
intersection point $x_{i}$ upwards to an intersection point $x_{j}$ with
$i<j$, while the $V/M/R$ arc passes from an intersection point $x_{i}$ upwards
to an intersection point $x_{j}$ with $i>j$.

We now describe the algorithm to construct $Q_{m,n}$ diagrams for any $m$ and
$n$. We add $\beta$ strands at each step, but the order of adding them does
not matter for the steps $Q_{1,1}$ to $Q_{m,1}$ and $Q_{m,2}$ to $Q_{m,n}$:
each strand type is independent of the others. However, the step $Q_{m,1}$ to
$Q_{m,2}$ requires a vital modification: the strands $V/M/L$ are introduced
into the diagrams, and they must be on the left of the rainbows $B/M/RNB$. For
this reason, the first step in this case which constructs a $B/M/RNB$ strand
from a $B/L$ strand must come first before adding any other strand.

  * •

$Q_{1,1}$ to $Q_{m,1}$: To construct $Q_{k+1,1}$ from $Q_{k,1}$, add the
following strands:

    * –

$2$ $T/L$

    * –

$1$ $T/R$

    * –

$1$ $B/R$

    * –

$1$ $V/M/R$

    * –

$1$ $B/M/RNB$

To see a precise example, we may obtain the Mazur pattern $Q_{2,1}$ in Figure
8 from the Whitehead double $Q_{1,1}$ in Figure 19. General $Q_{m,1}$ diagrams
are shown in Figure 22.

  * •

$Q_{m,1}$ to $Q_{m,2}$: add the following strands:

    * –

Take the strand of $B/L$ which lies on the left $\alpha_{1}$ arc (there should
be only one at this stage) and push it down through $\rho_{3}$ such that we
obtain a rainbow $B/M/RNB$. There should now be $m+1$ $B/M/RNB$ rainbows in
the diagram.

    * –

$1$ $T/L$

    * –

$1$ $V/M/R$

    * –

$2m$ $V/M/L$

A schematic for this is seen in Figure 21, which uses $m=1$ for simplicity,
and omits all parts of the diagram except for the one $B/M/RNB$ rainbow
present in the diagram for $Q_{1,1}$.

  * •

$Q_{m,2}$ to $Q_{m,n}$: To construct $Q_{m,k+1}$ from $Q_{m,k}$, add the
following strands:

    * –

$2m+1$ $V/M/L$

    * –

$1$ $V/M/R$

General $Q_{m,n}$ diagrams can be seen in Figure 23.

To verify this construction, one may perform isotopies on the curve obtained
from the Heegaard diagram. It is not immediately obvious that the curve is the
desired $Q_{m,n}$ pattern, but well-chosen isotopies will reduce the number of
self intersections until the same ones as described in Definition 1.2 remain.
This results in $1+2m$ intersection points down the vertical alpha curve and
$2m+2n+2nm-2$ intersection points along the horizontal alpha curve. To
circumvent the technical details of isotoping Heegaard diagrams, we defer a
formal proof that this construction is valid until the end of the section.

Figure 19. The Heegaard Diagram for $Q_{1,1}$ Figure 20. The terminology used
to describe the construction of $Q_{m,n}$ diagrams. The labels represent the
strands, not the regions. Figure 21. The Heegaard diagram transformation for
the $n=2$ case, utilizing $m=1$ for simplicity and avoiding all arcs except
for B/M/RNB Figure 22. Heegaard diagrams for $Q_{m,1}$ patterns Figure 23.
Heegaard diagrams for $Q_{m,n}$ patterns, when $n\geq 2$

∎

With these bordered Heegaard diagrams, we look to prove that the patterns
$Q_{m,n}$ and $Q_{n,m}$ are isotopic, fixing $m$ and $n$. We utilize an
approach developed in [Che19], which passes from Heegaard diagrams to their
associated $2$-bridge links, and extract the Schubert normal form $b(p,q)$ for
the $2$-bridge link.

We may describe a pattern obtained by a $2$-bridge link as such: remove a
regular neighborhood of one component of the $2$-bridge link, and this leaves
the other component as a pattern knot inside the solid torus. Theorem 1.6 in
[Che19] states that patterns obtained by $2$-bridge links are in bijective
correspondence with $(1,1)$-unknot patterns. This bijection gurantees that the
Schubert normal form for an $(1,1)$-unknot pattern is unique up to the
following equivalence condition: $b(p,q)$ and $b(p^{\prime},q^{\prime})$ are
isotopic if and only if $p=p^{\prime}$ and $q^{\prime}\equiv q^{\pm 1}$ (mod
$p$) [Sch53].

To pass from a Heegaard diagram to a $2$-bridge link, we follow the procedure
outlined in [Che19]: one can extract integers $r$ and $s$ associated to the
diagram, and use the following Theorem to find their Schubert normal form.

######  Theorem 4.2.

[Che19, Theorem 5.4] Let $P$ be a $(1,1)$ unknot pattern obtained by a genus-
one doubly pointed Heegaard diagram of parameter $(r,s)$. Then the link
consists of $P$ and the meridian of the solid torus is the 2-bridge link
$b(2|s|+4|r|,\epsilon(r)(2|r|-1))$, where $\epsilon(r)$ is the sign function
of r.

These integers $r$ and $s$ are described as the loops and strands of the
diagram. In particular, $r$ is the number of loops around $w$ and $z$, and $s$
is the number of strands of the $\beta$ curve which runs between the loops
around $w$ and $z$, separating them. An example of finding $r$ and $s$ for the
Mazur pattern is shown in Figure 24. Loops are colored in green, and strands
are colored in purple, so $(r,s)=(3,1)$.

Figure 24. Determining the number of loops and strands for the Mazur pattern
$Q_{2,1}$

The integers $r$ and $s$ for generalized Mazur patterns $Q_{m,n}$ are
$(m+1,-(2mn+n-m-2))$, where the minus comes from sign conventions. Thus, by
Theorem 4.2, the Schubert normal form of the $2$-bridge link associated to
$Q_{m,n}$ is $b(4mn+2n+2m,2m+1)$. For $Q_{m,n}$, we have $b(4mn+2n+2m,2n+1)$.
Since $(2m+1)(2n+1)\equiv 1$ (mod $4mn+2n+2m$), we may conclude the following.

######  Proposition 4.3.

$Q_{m,n}$ is isotopic to $Q_{n,m}$.

######  Remark 4.4.

Utilizing the bordered Heegaard diagrams for the $(p,1)$-cable patterns found
in [Hom14a], we obtain the parameters $(r,s)=(1,-(p-2))$, which implies that
$(p,1)$-cables correspond to the GMP notation $Q_{0,n}=Q_{m,0}$.

######  Remark 4.5.

In particular, the Heegaard diagrams specified by lemma 4.1 are for a specific
orientation of $Q_{m,n}$ corresponding to a winding number of $-(m-n)$.
Aligning orientations, Proposition 4.3 says that $Q_{m,n}$ is isotopic to
$rQ_{n,m}$. Throughout the paper, the use of these Heegaard diagrams to do
computation comes with the implicit detail that the winding number associated
to this pattern is $-(m-n)$.

With this proposition, it is possible to make statements about $Q_{m,n}$ for
$m\geq n$, and have them hold for all $Q_{m,n}$ patterns.

We now prove that the Heegaard diagrams from lemma 4.1 recover the generalized
Mazur patterns. To do this, we determine the braid decomposition for the
generalized Mazur patterns and their associated rational tangles. If the
fraction of the tangle associated to the braid decomposition matches the
fraction associated to the Schubert normal form coming from the Heegaard
diagrams, we conclude that the Heegaard diagrams are indeed valid.

Let $L$ be the 2-bridge link associated to $Q_{m,n}$. For simplicity, assume
that the bridge presentation consists entirely of whole twists, and that the
top and bottom closures consists of arcs which are neither overlapping nor
braided. To be precise, suppose $L$ is the plat closure of the 4-stranded
braid

(4.1)
$\sigma_{2}^{2a_{0}}\sigma_{1}^{2a_{1}}\cdots\sigma_{2}^{2a_{r-2}}\sigma_{1}^{2a_{r-1}}\sigma_{2}^{2a_{r}}$

for some nonzero integers $a_{0},\ldots,a_{r}$, where $r$ is even and the
braid is read from top to bottom. For example, the Mazur pattern has $r=2$ and
$(a_{0},a_{1},a_{2})=(2,-1,-1)$.

Figure 25. The two bridge link $L_{Q_{m,n}}$ along with a dotted arc joining
the two components. There are $2(m-1)$ blue components added, and $2(n-1)$ red
components added for a given $m$ and $n$.

######  Proposition 4.6.

The braid decomposition for $Q_{m,n}$ of the form in Equation 4.1 is given by
$r=2$ and $(a_{0},a_{1},a_{2})=(m,-1,-n)$. The 2-bridge link with a dotted arc
joining the two components is given in Figure 25.

###### Proof.

If we start with no red or blue arcs, then Figure 25 is the 2-bridge link for
the Whitehead double. Adding two blue arcs corresponds to adding one
longitudinal arc in the set of m strands, and this is the figure for the Mazur
pattern $Q_{2,1}$ (a figure for this can also be found in [Lev16]). Likewise,
adding two red arcs instead of two blue arcs corresponds to adding a
longitudinal arc in the set of n strands. The figure we recover corresponds to
$Q_{1,2}$, which by symmetry of the two bridge link we find is the same as
$Q_{2,1}$. From here, an induction of adding 2 arcs in the blue and red
components correspond to adding longitudinal strands in the generalized Mazur
patterns $Q_{m,n}$. ∎

Figure 26. A tangle diagram for generalized Mazur patterns, describing the
rational tangle $C(2m,1,2n)$

This construction may also be used to describe the Conway rational tangle
associated to the patterns $Q_{m,n}$. An isotopy of Figure 25 results in
Figure 26, from which we extract the rational tangle $C(2n,1,2m)$, obtained by
counting the number of crossings at each section of the tangle diagram.

Finally, we recall the correspondence between the Schubert normal form and
Conway’s rational tangles [Lic97]: $b(\alpha,\beta)$ is equivalent to the
rational tangle $C(a_{1},...,a_{n})$, where $a_{i}$ is the $i$th coefficient
in the continued fraction expansion

(4.2)
$\dfrac{\beta}{\alpha}=\dfrac{1}{a_{1}+\dfrac{1}{a_{2}+\cdots\dfrac{1}{a_{n-1}+\dfrac{1}{a_{n}}}}}$

Thus, $b(4mn+2n+2m,2m+1)=C(2n,1,2m)$, and we conclude that the Heegaard
diagrams for $Q_{m,n}$ are valid.

##  5\. Computation of $\tau(Q_{m,n}(K))$ and $\epsilon(Q_{m,n}(K))$

In this section, we compute the invariants $\tau(Q_{m,n}(K))$ and
$\epsilon(Q_{m,n}(K))$ for generalized Mazur patterns, giving a proof of
Theorem 1.5. This is the technical core of our paper. By Proposition 1.3, we
may reduce our computation to $Q_{m,n}$ with $m\geq n$.

######  Theorem 5.1.

Let $Q_{m,n}$ be the generalized Mazur pattern embedded in the solid torus
$V$. If $m\neq n$, then for any knot $K\subset S^{3}$, we have

(5.1) $\tau(Q_{m,n}(K))=\begin{cases}|m-n|\tau(K)&\text{if }\tau(K)\leq
0\text{ and }\epsilon(K)\in\\{0,1\\},\\\ |m-n|\tau(K)+|m-n|&\text{if
}\tau(K)<0\text{ and }\epsilon(K)=-1,\\\ |m-n|\tau(K)+\min(m,n)&\text{if
}\tau(K)>0\text{ and }\epsilon(K)=1,\\\ |m-n|\tau(K)+\max(m,n)-1&\text{if
}\tau(K)\geq 0\text{ and }\epsilon(K)=-1.\end{cases}$

In the case where $m=n$, we have

(5.2) $\tau(Q_{m,m}(K))=\begin{cases}0&\text{if }\tau(K)<0,\\\ m-1&\text{if
}\tau(K)=0,\\\ m&\text{if }\tau(K)>0.\end{cases}$

Also,

(5.3) $\epsilon(Q_{m,n}(K))=\begin{cases}0&\text{if }\tau(K)=\epsilon(K)=0,\\\
1&\text{otherwise}\end{cases}$

Figure 27. A lift of $\beta$ curve to the universal cover Figure 28. The
Alexander grading at various points of $\tilde{\beta}$ curve in the row
containing $c$

We follow the strategy outlined in Section 3, recovering the $\tau$ and
$\epsilon$ invariant of $Q_{m,n}(K)$ from its pairing diagram. Using the
notation from Section 3.2, we denote the lifts of $\alpha_{K}$ and
$\beta(Q_{m,n})$ in the universal cover of the doubly marked torus as
$\tilde{\alpha}$ and $\tilde{\beta}$, respectively.

###### Proof of Theorem 5.1.

When $\epsilon(K)$ = 0, $K$ is $\epsilon$-equivalent to the unknot $U$
[Hom14b]. Thus, $\tau(Q_{m,n}(K))=\tau(Q_{m,n}(U))$ and
$\epsilon(Q_{m,n}(K))=\epsilon(Q_{m,n}(U))$, so we have $\tau(Q_{m,n}(K))=0$
and $\epsilon(Q_{m,n}(K))=0$, as desired. For the rest of the proof, we assume
that $\epsilon(K)\neq 0$ and $m\geq n$.

We begin by a discussion of how to construct the pairing diagram of the
satellite $Q_{m,n}(K)$, which consists the two lifts $\tilde{\beta}$ and
$\tilde{\alpha}$, where the latter is pulled tight in the pegboard diagram.
Recall that Lemma 4.1 gives us a way to construct the Heegaard diagram
$\mathcal{H}$ of $Q_{m,n}$. We obtain $\tilde{\beta}$ by lifting the
$\beta$-curve in $\mathcal{H}$ into the universal cover, as shown in Figure
27. The lift spans $m+n$ columns in the universal cover, which we label by $1$
to $m+n$ from left to right. For the lift $\tilde{\alpha}$ of $\alpha_{K}$,
Lemma 3.1 describes the unstable segment of $\tilde{\alpha}$ and Figure 13
shows its relevant neighborhood. Note that each column contains a translated
copy of $\tilde{\alpha}$, but we only draw the copy contained in column $n$ to
column $n+1$, because this is the only relevant one containing the
distinguished generator in $\widehat{HF}(S^{3})$. For example, the pairing
diagram of $Q_{m,n}(K)$ in the case when $\tau(K)>0$ and $\epsilon(K)=1$ is
shown in Figure 29.

From the pairing diagram, we may determine the absolute Alexander grading of
the intersection points by Lemma 3.2 and the discussion thereafter. In
particular, note that if $x^{\prime}$ can be obtained from $x$ by shifting the
picture in the universal cover down a row, then
$A(x)-A(x^{\prime})=w(Q_{m,n})=-(m-n)$. Let $c$ be the fixed point under
hyperelliptic involution, with $A(c)=0$. For ease of computation, we determine
the grading of $x$ by first shifting the picture vertically so that the image
$x^{\prime}$ lies in the row containing $c$, and then determining the grading
of $x^{\prime}$ using Lemma 3.2. In Figure 28, we display the grading of
different points along $\tilde{\beta}$ in the row containing $c$.

We are now ready to recover $\tau(Q_{m,n}(K))$ and $\epsilon(Q_{m,n}(K))$. The
lift $\tilde{\alpha}$ has the form described in Lemma 3.1 and Figure 13, which
depends on $\tau(K)$ and $\epsilon(K)$, so it remains to inspect the following
four cases.

Figure 29. The pairing diagram for $Q_{m,n}(K)$ when $\tau(K)>0,\epsilon(K)=1$
${{y_{2n+1}}}$${{y_{2n-1}}}$${{x_{2n+1}}}$${{x_{2n-1}}}$${{x_{2n}}}$${{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bullet}}$${{x_{2n-2}}}$${{y_{3}}}$${\dots}$${{y_{1}}}$${{x_{3}}}$${{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bullet}}$${{x_{1}}}$${{x_{2}}}$$\scriptstyle{U^{m}}$$\scriptstyle{V}$$\scriptstyle{U^{n}}$$\scriptstyle{V}$$\scriptstyle{U^{m}}$$\scriptstyle{U^{m}}$$\scriptstyle{U^{n}}$$\scriptstyle{V}$$\scriptstyle{U^{m}}$$\scriptstyle{U^{n}}$$\scriptstyle{V}$
Figure 30. The relevant differentials in the $CFK_{\mathcal{R}}$ complex of
$Q_{m,n}(K)$ when $\tau(K)>0$, $\epsilon(K)=1$. Figure 31. The disks
highlighted in pink represent all the Whitney disks of filtration difference
$1$. Eliminating them by an isotopy, we get Figure 32.

Case 1: $\tau(K)>0$ and $\epsilon(K)=1$. The pairing diagram of $Q_{m,n}(K)$
is shown in Figure 29, where the fixed point $c$ is marked in red. Following
the strategy in Section 3.3, we first eliminate all the intersection points of
filtration difference 1. The Whitney disks connecting those points are
highlighted in pink in Figure 31, and the diagram after they are eliminated is
shown in Figure 32. If $m>n$, we eliminate the blue disks (with filtration
difference $n-1$), then the brown disks (with difference $n$), and finally the
yellow disks (with difference $m-1$). If $m=n$, we eliminate the blue and
yellow disks first, then the brown disks. Either way, $x_{1}$ is the remaining
intersection point after the isotopies.

Figure 32. The result of eliminating Whitney disks in Figure 31. If $m>n$, we
first eliminate the blue disks, then the brown disks, and finally the yellow
disks. If $m=n$, we first eliminate the blue and yellow disks, then the brown
disks. In both cases, we arrive at a complex with $x_{1}$ as the only
remaining intersection point. The small arrows with label $n$ or $n-1$
represent respectively $n$ or $n-1$ small arrows placed in parallel; these are
the A-buoys keeping track of the filtration difference

To recover the Alexander grading of $x_{1}$, we shift the diagram down
$\tau(K)-1$ rows to get $x_{1}^{\prime}$, and find that the algebraic
intersection $l_{c,x_{1}^{\prime}}\cdot\delta_{w,z}=-m$. It follows that

$\tau(Q_{m,n}(K))=-A(x_{1})=-(-(m-n)(\tau(K)-1)+l_{c,x_{1}^{\prime}}\cdot\delta_{w,z})=(m-n)\tau(K)+n.$

To compute $\epsilon(Q_{m,n}(K)$, we need to construct the subcomplex of the
$CFK_{\mathcal{R}}$ complex containing $x_{1}$. The relevant differentials of
this subcomplex are drawn in Figure 30. Note that the cycle
$\sum^{2n+1}_{i=1}x_{2i+1}$ survives in the homology $\widehat{HF}(S^{3})$, so
this cycle must be the distinguished element of some horizontally simplified
basis. It remains to look at its position in the vertical complex to recover
$\epsilon(Q_{m,n}(K)$. There is a vertical arrow from $y_{i}$ to $x_{i}$ for
each odd $i$, and together they give us a vertical arrow from
$\sum^{n}_{i=0}y_{2i+1}$ to $\sum^{n}_{i=0}x_{2i+1}$. Thus, the cycle
$\sum^{2n+1}_{i=0}x_{2i+1}$ is a boundary with respect to the vertical
differential, which means that $\epsilon(Q_{m,n}(K))=1$.

Case 2: $\tau(K)\geq 0$ and $\epsilon(K)=-1$. The pairing diagram is shown in
Figure 33. As before, we perform isotopies that eliminate intersection points
of the same filtration difference. If $\tau(K)=0$ and $m=n$, then the
intersection point that survives the spectral sequence is $x_{3}$; otherwise,
it is $x_{1}$. To compute the Alexander grading of $x_{3}$, we shift the
diagram down by $\tau(K)-1$ rows, and find that the the image $x_{3}^{\prime}$
has grading $-m$. Thus, when $\tau(K)=0$ and $m=n$, we have

$\tau(Q_{m,n}(K))=-A(x_{3})=-(-(m-n)(\tau(K)-1)-m)=m.$

Otherwise, we calculate the grading of $x_{1}$ using the same method, and
obtain that

$\tau(Q_{m,n}(K))=-A(x_{1})=-(-(m-n)\tau(K)-(m-1))=(m-n)\tau(K)+(m-1).$

We construct the subcomplex of $CFK_{\mathcal{R}}$ containing $x_{1}$ and
$x_{3}$, and the relevant part is shown in Figure 34. The analysis for
$\epsilon(Q_{m,n}(K))$ the same as the case when $\tau(K)>0$ and
$\epsilon(K)=1$, displayed in Figure 30. Hence, we have
$\epsilon(Q_{m,n}(K))=1$.

Figure 33. The pairing diagram for $Q_{m,n}(K)$ when $\tau(K)\geq
0,\epsilon(K)=-1$. Some bigons contributing to the differentials in the
subcomplex are shaded. From top to bottom, they correspond to the
differentials $x_{1}\xrightarrow{U^{m}}x_{2}$, $y_{3}\xrightarrow{V}x_{3}$,
and $x_{2n+1}\xrightarrow{U^{n}}x_{2n}$, respectively
${{y_{2n+1}}}$${{y_{2n-1}}}$${{x_{2n+1}}}$${{x_{2n-1}}}$${{x_{2n}}}$${{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bullet}}$${{x_{2n-2}}}$${{y_{3}}}$${\dots}$${{y_{1}}}$${{x_{3}}}$${{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bullet}}$${{x_{1}}}$${{x_{2}}}$$\scriptstyle{U^{m-1}}$$\scriptstyle{V}$$\scriptstyle{U^{n}}$$\scriptstyle{V}$$\scriptstyle{U^{m}}$$\scriptstyle{U^{m}}$$\scriptstyle{U^{n}}$$\scriptstyle{V}$$\scriptstyle{U^{m}}$$\scriptstyle{U^{n}}$$\scriptstyle{V}$
Figure 34. The relevant differentials in the $CFK_{\mathcal{R}}$ complex of
$Q_{m,n}(K)$ when $\tau(K)\geq 0$, $\epsilon(K)=-1$

Case 3: $\tau(K)\leq 0$ and $\epsilon(K)=1$. The pairing diagram is shown in
Figure 35. After eliminating all pairs of intersection points with the same
filtration difference, we are left with $x_{2n+1}$, which has Alexander
grading $-(m-n)\tau(K)$. Therefore, we have

$\tau(Q_{m,n}(K))=-A(x_{2n+1})=(m-n)\tau(K).$

Figure 36 shows the relevant differentials in the $CFK_{\mathcal{R}}$ complex
containing $x_{2n+1}$, and one observes that $x_{2n+1}$ is the distinguished
element in some horizontal simplified basis. Since there’s a vertical
differential pointing from $y_{1}$ to $U^{n}x_{2n+1}$, we have
$\epsilon(Q_{m,n}(K))=1$.

Figure 35. The pairing diagram for $Q_{m,n}(K)$ when $\tau(K)\leq
0,\epsilon(K)=1$
${{x_{2}}}$${{x_{1}}}$${{x_{4}}}$${{x_{3}}}$${{y_{2k+1}}}$${{y_{2k-1}}}$${\cdots}$${{y_{1}}}$${\vdots}$${\vdots}$${{y_{2k}}}$${\cdots}$${{y_{2}}}$${{x_{2n+1}}}$${{x_{2n}}}$${{x_{2n-1}}}$$\scriptstyle{U^{n}}$$\scriptstyle{U^{m}}$$\scriptstyle{U^{n}}$$\scriptstyle{U^{m}}$$\scriptstyle{U^{m}}$$\scriptstyle{U^{n}}$$\scriptstyle{U^{m}}$$\scriptstyle{V^{n}}$$\scriptstyle{U^{m-n}V^{n-1}}$$\scriptstyle{V^{n}}$$\scriptstyle{U^{m-n}V^{n-1}}$$\scriptstyle{V^{n}}$$\scriptstyle{U^{m-n}V^{n-1}}$
Figure 36. The relevant differentials in the $CFK_{\mathcal{R}}$ complex of
$Q_{m,n}(K)$ when $\tau(K)\leq 0$, $\epsilon(K)=1$

Case 4: $\tau(K)<0$ and $\epsilon(K)=-1$. The pairing diagram is displayed in
Figure 37, where the remaining intersection point is $x_{2n-1}$, which has
Alexander grading $-(m-n)(\tau(K)-1)$. This gives

$\tau(Q_{m,n}(K))=-A(x_{2n-1})=(m-n)(\tau(K))+(m-1))=(m-n)\tau(K)+(m-1).$

In Figure 38, we show the relevant differentials in the subcomplex containing
$x_{2n+1}$, which is a distinguished element in some horizontally simplified
basis. Since a vertical differential sends $y_{1}$ to $U^{n}x_{2n-1}$, we
conclude that $\epsilon(Q_{m,n}(K))=1$.

Figure 37. The pairing diagram for $Q_{m,n}(K)$ when
$\tau(K)<0,\epsilon(K)=-1$
${{x_{2}}}$${{x_{1}}}$${{x_{4}}}$${{x_{3}}}$${\vdots}$${\vdots}$${{y_{1}}}$${{x_{2n+1}}}$${{x_{2n}}}$${{x_{2n-1}}}$${{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bullet}}$$\scriptstyle{U^{n}}$$\scriptstyle{U^{m}}$$\scriptstyle{U^{n}}$$\scriptstyle{U^{m}}$$\scriptstyle{U^{m}}$$\scriptstyle{U^{n}}$$\scriptstyle{U^{m}}$$\scriptstyle{V}$
Figure 38. The relevant differentials in the $CFK_{\mathcal{R}}$ complex of
$Q_{m,n}(K)$ when $\tau(K)<0$, $\epsilon(K)=-1$

∎

The proof of Theorem 1.5 allows us to find the three-genus of the patterns
$Q_{m,n}$. Recall that by [OS04b], the genus of the knot is the largest
Alexander grading supporting non-zero Floer homology. Moreover, for a
satellite knot $P(K)$, we have a genus formula from [Sch53]

(5.4) $g(P(K))=|w(P)|g(K)+g(P),$

where $|w(P)|$ is the absolute value of the winding number.

######  Proposition 5.2.

For $m,n\geq 1$, the generalized Mazur patterns have genus
$g(Q_{m,n})=\min(m,n).$

###### Proof.

We take $P$ to be $Q_{m,n}$ and reverse the orientation (where without loss of
generality we assume that $m\geq n$), which has winding number $m-n$, and $K$
to be the right-handed trefoil $T_{2,3}$, which has
$\tau(T_{2,3})=\epsilon(T_{2,3})=g(T_{2,3})=1$. By Equation (5.4), we have

(5.5) $g(Q_{m,n}(T_{2,3}))=(m-n)+g(Q_{m,n}).$

We want to compute $g(Q_{m,n}(T_{2,3}))$, which is equal to the largest
Alexander grading for which $\widehat{HFK}(Q_{m,n}(K))$ is non-zero. Consider
the pairing diagram in Figure 29 from the proof of Theorem 5.1 and note that
$x_{1}$ has the lowest Alexander grading, but after taking the reverse this
generator has the largest Alexander grading. Since $x_{1}$ is also the
distinguished generator,

$g(Q_{m,n}(T_{2,3}))=A(x_{1})=\tau(Q_{m,n}(T_{2,3}))=|m-n|\tau(T_{2,3})+n=m.$

Plugging into Equation (5.5), we get $g(Q_{m,n})=n$. For any $m,n\geq 1$, we
have $g(Q_{m,n})=\min(m,n)$. ∎

##  Appendix A Computation of $\tau(Q_{m,n}(K))$ using bordered Heegaard Floer
homology

We give an alternative way of computing the $\tau$-invariant for satellites
along generalized Mazur patterns, using the ordinary bordered Heegaard Floer
homology. This is a generalization of the method used in [Lev16]. As expected,
we obtain the same value of $\tau(Q_{m,n}(K))$ as in Theorem 1.5.

The organization of the Appendix is as follows: in Section A.1, we describe
our strategy for computing $\tau(P(K))$; in Section A.2, we give a few
examples for computing $\tau(Q_{m,n}(K))$ for small values of $m$ and $n$; and
in Section A.3, we carry out the computation for general $m$ and $n$.

###  A.1. Bordered strategy to determine $\tau(P(K))$

In this section, we follow closely the notation in [Lev16]. Recall that an
alternative definition of the invariant $\tau(K)$ is

$\tau(K)=-\max\\{s\mid U^{n}\cdot HFK^{-}(K,s)\neq 0\textrm{ for all }n\geq
0\\}.$

In other words, $\tau(K)$ is minus the Alexander grading of the non-vanishing
generator for $\mathbb{F}[U]$ in $HFK^{-}(S^{3},K)$. Recall also the pairing
theorem [LOT18, Theorem 1.3]: for a satellite knot with pattern $P\subset V$
and companion $K$, we have

(A.1) $gCFK^{-}(P(K))\simeq CFA^{-}(V,P)\boxtimes\widehat{CFD}(X_{K}).$

Here, we may again assume that the pattern knot $P$ is a $(1,1)$-unknot
pattern.

Our goal is to directly compute the associated graded on the left hand side.
We discussed how to compute $CFA^{-}$ in Section 2.3, and
$\widehat{CFD}(X_{K})$ in Section 2.4. The box tensor product
$CFA^{-}(V,P)\boxtimes\widehat{CFD}(X_{K})$ is defined to be the
$\mathbb{F}$-vector space

$CFA^{-}(V,P)\boxtimes_{\mathcal{I}}\widehat{CFD}(X_{K})$

with the differential now given by the combinatorial formula

$\partial^{\boxtimes}(x\otimes
y)=\sum_{k+1}(x,\rho_{i_{1}},\cdots,\rho_{i_{k}})D_{i_{k}}\circ\cdots\circ
D_{i_{1}}(y)$

where the sum is taken over all $k$-element sequences $i_{1},\cdots,i_{k}$
(including the empty sequence when $k=0$) of elements in
$\\{\emptyset,1,2,3,12,23,123\\}$.

The box tensor product in Equation (A.1) consists of one direct summand whose
homology contains a $\mathbb{F}[U]$ part, obtained by box tensoring with a
unstable neighborhood in $\widehat{CFD}(X_{K})$, and other summands whose
homology is $U$-torsion. We are interested only in the summand with non-
vanishing homology, since the Alexander grading of its generator gives us
$-\tau(P(K))$.

It remains determine the absolute Alexander grading in $gCFK^{-}(S^{3},P(K))$,
which we denote by $A_{Q_{m,n}(K)}$. This can be achieved by the following
proposition in [Lev16].

######  Proposition A.1.

[Lev16, Proposition 2.2] Let $P\subset V$ be a based knot with winding number
$m$. For each element $a\in\mathrm{CFA}^{-}(V,P)\cdot\iota_{0}$, there exists
a constant $C_{a}$ with the following property: For any knot $K\subset S^{3}$,
and any homogeneous element
$x\in\iota_{0}\widehat{\mathrm{CFD}}\left(X_{K}\right)$, we have

(A.2) $A_{P(K)}(a\otimes x)=mA_{K}(x)+C_{a}.$

In what follows, we will set $P$ to be the generalized Mazur pattern
$Q_{m,n}$. Note that the constant $C_{a}$ in equation (A.2) is independent of
$K$. This allows us to determine $C_{a}$ by setting $K$ to be the unknot $O$.
Since $Q_{m,n}(O)$ is also the unknot, the tensor product complex
$CFA^{-}(V,Q_{m,n})\boxtimes\widehat{CFD(X_{O})}$ is generated over
$\mathbb{F}[U]$ by a single element with Alexander grading 0. Moreover, we
have $A_{O}(x)=0$ for any
$x\in\iota_{0}\widehat{\mathrm{CFD}}\left(X_{O}\right)$. Equation (A.2) then
allows us to determine the constant $C_{a}$ for certain generators
$a\in\mathrm{CFA}^{-}(V,Q_{m,n})$.

For any knot $K$, we may apply Proposition A.1 again to compute the absolute
Alexander gradings of the relevant generators of $CFK^{-}(S^{3},Q_{m,n}(K))$.
This computation is sufficient to determine $\tau(Q_{m,n}(K))$; we will carry
out this computation in more detail in Section A.3. Before that, we give some
examples of how this strategy may be applied for small values of $m$ and $n$.

###  A.2. Examples

######  Example A.2 (The pattern $Q_{3,1}$).

First, we obtain the following Heegaard diagram for the pattern $Q_{3,1}$ by
applying the inductive procedure in lemma 4.1. The resulting diagram is shown
in Figure 39. The indexing of the vertical column is $x_{1}$ to $x_{7}$ from
top to bottom, and the indexing of the horizontal column is $y_{1}$ to
$y_{12}$ from left to right.

Figure 39. The Heegaard diagram for $Q_{3,1}$

Next, we find all the pseudoholomorphic disks in the universal cover, shown in
Figure 40.

Figure 40. The universal cover of the bordered diagram for $Q_{3,1}$

The corresponding $A_{\infty}$ multiplications are:

  * •

$m_{1}(x_{1})=Ux_{6}$

  * •

$m_{2}(x_{1},\rho_{1})=y_{12}$

  * •

$m_{2}(x_{1},\rho_{12})=x_{7}$

  * •

$m_{2}(x_{1},\rho_{123})=Uy_{7}$

  * •

$m_{4}(x_{1},\rho_{3},\rho_{2},\rho_{1})=Uy_{1}$

  * •

$m_{1}(x_{2})=Ux_{5}$

  * •

$m_{2}(x_{2},\rho_{1})=y_{11}$

  * •

$m_{2}(x_{2},\rho_{12})=x_{6}$

  * •

$m_{4}(x_{2},\rho_{123},\rho_{2},\rho_{1})=Uy_{6}$

  * •

$m_{4}(x_{2},\rho_{3},\rho_{2},\rho_{1})=Uy_{2}$

  * •

$m_{1}(x_{3})=Ux_{4}$

  * •

$m_{2}(x_{3},\rho_{1})=y_{10}$

  * •

$m_{2}(x_{3},\rho_{12})=x_{5}$

  * •

$m_{4}(x_{3},\rho_{123},\rho_{2},\rho_{1})=Uy_{5}$

  * •

$m_{4}(x_{3},\rho_{3},\rho_{2},\rho_{1})=Uy_{3}$

  * •

$m_{2}(x_{4},\rho_{1})=y_{9}$

  * •

$m_{3}(x_{4},\rho_{12},\rho_{1})=y_{8}$

  * •

$m_{4}(x_{4},\rho_{12},\rho_{12},\rho_{1})=y_{7}$

  * •

$m_{4}(x_{4},\rho_{3},\rho_{2},\rho_{1})=Uy_{4}$

  * •

$m_{2}(x_{5},\rho_{1})=Uy_{8}$

  * •

$m_{3}(x_{5},\rho_{12},\rho_{1})=Uy_{7}$

  * •

$m_{4}(x_{5},\rho_{3},\rho_{2},\rho_{1})=Uy_{5}$

  * •

$m_{2}(x_{6},\rho_{1})=U^{2}y_{7}$

  * •

$m_{4}(x_{6},\rho_{3},\rho_{2},\rho_{1})=Uy_{6}$

  * •

$m_{2}(x_{7},\rho_{3})=Uy_{7}$

  * •

$m_{1}(y_{1})=Uy_{6}$

  * •

$m_{1}(y_{2})=Uy_{5}$

  * •

$m_{1}(y_{3})=Uy_{4}$

  * •

$m_{3}(y_{8},\rho_{2},\rho_{1})=y_{7}$

  * •

$m_{3}(y_{9},\rho_{2},\rho_{1})=y_{8}$

  * •

$m_{4}(y_{9},\rho_{2},\rho_{12},\rho_{1})=y_{7}$

  * •

$m_{2}(y_{10},\rho_{2})=x_{5}$

  * •

$m_{4}(y_{10},\rho_{23},\rho_{2},\rho_{1})=Uy_{5}$

  * •

$m_{1}(y_{10})=Uy_{9}$

  * •

$m_{2}(y_{11},\rho_{2})=x_{6}$

  * •

$m_{4}(y_{11},\rho_{23},\rho_{2},\rho_{1})=Uy_{6}$

  * •

$m_{1}(y_{11})=U^{2}y_{8}$

  * •

$m_{2}(y_{12},\rho_{2})=x_{7}$

  * •

$m_{2}(y_{12},\rho_{23})=Uy_{7}$

  * •

$m_{1}(y_{12})=U^{3}y_{7}$

This can be represented by the diagram below in Figure 41:

${x_{7}}$${y_{12}}$${x_{1}}$${y_{11}}$${x_{2}}$${y_{10}}$${x_{3}}$${y_{1}}$${y_{2}}$${y_{3}}$${y_{7}}$${x_{6}}$${y_{8}}$${x_{5}}$${y_{9}}$${x_{4}}$${y_{6}}$${y_{5}}$${y_{4}}$$\scriptstyle{U\rho_{3}}$$\scriptstyle{\rho_{2}}$$\scriptstyle{\begin{subarray}{c}U^{3}+\\\
U\rho_{23}\end{subarray}}$$\scriptstyle{\rho_{1}}$$\scriptstyle{\rho_{12}}$$\scriptstyle{U\rho_{123}}$$\scriptstyle{U}$$\scriptstyle{\rho_{2}}$$\scriptstyle{U^{2}}$$\scriptstyle{\rho_{1}}$$\scriptstyle{U}$$\scriptstyle{\rho_{12}}$$\scriptstyle{\rho_{2}}$$\scriptstyle{U}$$\scriptstyle{\rho_{1}}$$\scriptstyle{U}$$\scriptstyle{\rho_{12}}$$\scriptstyle{U}$$\scriptstyle{U}$$\scriptstyle{U}$$\scriptstyle{U^{2}\rho_{1}}$$\scriptstyle{\rho_{2}\rho_{1}}$$\scriptstyle{U\rho_{1}}$$\scriptstyle{U\rho_{12}\rho_{1}}$$\scriptstyle{\rho_{2}\rho_{1}}$$\scriptstyle{\rho_{2}\rho_{12}\rho_{1}}$$\scriptstyle{\rho_{1}}$$\scriptstyle{U\rho_{12}\rho_{1}}$$\scriptstyle{\rho_{12}\rho_{12}\rho_{1}}$
Figure 41. The $A_{\infty}$-module $CFA^{-}(Q_{3,1}(K))$

One may observe that a few differentials are not represented in the diagram
above, namely

$\begin{aligned} x_{1}&\stackrel{{\scriptstyle
U\rho_{3}\rho_{2}\rho_{1}}}{{\longrightarrow}}y_{1}\\\
y_{11}&\stackrel{{\scriptstyle
U\rho_{23}\rho_{2}\rho_{1}}}{{\longrightarrow}}y_{6}\\\
x_{2}&\stackrel{{\scriptstyle
U\rho_{123}\rho_{2}\rho_{1}}}{{\longrightarrow}}y_{6}\\\
x_{6}&\stackrel{{\scriptstyle
U\rho_{3}\rho_{2}\rho_{1}}}{{\longrightarrow}}y_{6}\\\
x_{2}&\stackrel{{\scriptstyle
U\rho_{3}\rho_{2}\rho_{1}}}{{\longrightarrow}}y_{2}\\\
\end{aligned}\qquad\qquad\qquad\begin{aligned} y_{10}&\stackrel{{\scriptstyle
U\rho_{23}\rho_{2}\rho_{1}}}{{\longrightarrow}}y_{5}\\\
x_{3}&\stackrel{{\scriptstyle
U\rho_{123}\rho_{2}\rho_{1}}}{{\longrightarrow}}y_{5}\\\
x_{5}&\stackrel{{\scriptstyle
U\rho_{3}\rho_{2}\rho_{1}}}{{\longrightarrow}}y_{5}\\\
x_{3}&\stackrel{{\scriptstyle
U\rho_{3}\rho_{2}\rho_{1}}}{{\longrightarrow}}y_{3}\\\
x_{4}&\stackrel{{\scriptstyle
U\rho_{3}\rho_{2}\rho_{1}}}{{\longrightarrow}}y_{4}\end{aligned}$

However, we may apply the following change of basis to eliminate these arrows:
let $x_{6}^{\prime}=x_{6}+U\rho_{3}\rho_{2}\rho_{1}y_{1}$,
$x_{5}^{\prime}=x_{5}+U\rho_{3}\rho_{2}\rho_{1}y_{2}$, and
$x_{4}^{\prime}=x_{4}+U\rho_{3}\rho_{2}\rho_{1}y_{3}$.

######  Example A.3 (The pattern $Q_{1,2}$).

The Heegaard diagram for $Q_{1,2}$ is shown in Figure 42.

Figure 42. The Heegaard diagram for $Q_{1,2}$

After counting pseudoholomorphic disks in the universal cover, we obtain the
following $A_{\infty}$ multiplications:

  * •

$m_{1}(x_{1})=Ux_{2}$

  * •

$m_{2}(x_{1},\rho_{1})=y_{8}$

  * •

$m_{2}(x_{2},\rho_{1})=y_{7}$

  * •

$m_{2}(x_{1},\rho_{12})=x_{3}$

  * •

$m_{2}(x_{1},\rho_{123})=U^{2}y_{7}$

  * •

$m_{2}(x_{3},\rho_{3})=U^{2}y_{7}$

  * •

$m_{2}(y_{8},\rho_{2})=x_{3}$

  * •

$m_{2}(y_{8},\rho_{23})=U^{2}y_{7}$

  * •

$m_{1}(y_{8})=U^{2}y_{7}$

  * •

$m_{4}(x_{1},\rho_{123},\rho_{2},\rho_{1})=Uy_{3}$

  * •

$m_{4}(x_{1},\rho_{3},\rho_{2},\rho_{1})=Uy_{1}$

  * •

$m_{5}(x_{1},\rho_{3},\rho_{2},\rho_{12},\rho_{1})=U^{2}y_{4}$

  * •

$m_{3}(x_{2},\rho_{12},\rho_{1})=y_{6}$

  * •

$m_{4}(x_{2},\rho_{3},\rho_{2},\rho_{1})=Uy_{2}$

  * •

$m_{5}(x_{2},\rho_{3},\rho_{2},\rho_{12},\rho_{1})=U^{2}y_{5}$

  * •

$m_{2}(x_{3},\rho_{1})=Uy_{6}$

  * •

$m_{4}(x_{3},\rho_{3},\rho_{2},\rho_{1})=Uy_{3}$

  * •

$m_{1}(y_{1})=Uy_{2}$

  * •

$m_{1}(y_{3})=Uy_{6}$

  * •

$m_{1}(y_{4})=Uy_{5}$

  * •

$m_{3}(y_{7},\rho_{2},\rho_{1})=y_{6}$

  * •

$m_{3}(y_{8},\rho_{23},\rho_{2},\rho_{1})=Uy_{3}$

  * •

$m_{3}(y_{1},\rho_{2}\rho_{1})=Uy_{4}$

  * •

$m_{3}(y_{2},\rho_{2}\rho_{1})=Uy_{5}$

which are represented by the $CFA^{-}$ complex below in Figure 43 with the
change of basis in Remark A.7 applied:

${y_{3}}$${x_{3}}$${y_{8}}$${x_{1}}$${y_{1}}$${y_{4}}$${y_{6}}$${y_{7}}$${x_{2}}$${y_{2}}$${y_{5}}$$\scriptstyle{U}$$\scriptstyle{U\rho_{3}\rho_{2}\rho_{1}}$$\scriptstyle{U\rho_{1}}$$\scriptstyle{U^{2}\rho_{3}}$$\scriptstyle{\rho_{2}}$$\scriptstyle{\begin{subarray}{c}U+\\\
U^{2}\rho_{23}\end{subarray}}$$\scriptstyle{\mathclap{U^{2}\rho_{123}}}$$\scriptstyle{\rho_{1}}$$\scriptstyle{\rho_{12}}$$\scriptstyle{U}$$\scriptstyle{U\rho_{123}\rho_{2}\rho_{1}}$$\scriptstyle{U}$$\scriptstyle{U}$$\scriptstyle{\rho_{2}\rho_{1}}$$\scriptstyle{\rho_{1}}$$\scriptstyle{\rho_{12}\rho_{1}}$
Figure 43. The $A_{\infty}$-module $CFA^{-}(Q_{1,2}(K))$

As before, one may check that the summands in the box tensor product with
$\widehat{CFD}(X_{K})$ look like the figures given for each value of $\tau(K)$
after plugging in $m=1$ and $n=2$. The only difference is that there are no
dashed arrows, since there are no $\rho_{2}$ arrows pointing to $x_{2}$ in
$CFA^{-}(V,Q_{1,2})$ that will pair nontrivially with the $D_{2}$ differential
fron $\lambda$ to $\xi_{0}$.

######  Example A.4 (The pattern $Q_{2,3}$).

We describe the $A_{\infty}$-module $CFA^{-}(Q_{2,3}(K)).$ Using the inductive
procedure outlined in Lemma 4.1, we obtain the Heegaard diagram in Figure 44.

Figure 44. The Heegaard diagram for $Q_{2,3}$

We provide a less exhaustive $CFA^{-}$ structure below in Figure 45, not
including the isolated components. There are also arrows from $y_{18}$ and
$x_{3}$ to $y_{16}$ featuring compositions of the arrows going through
$y_{17}$, which have also been omitted for clarity.

${y_{10}}$${y_{5}}$${x_{5}}$${y_{20}}$${x_{1}}$${y_{19}}$${x_{2}}$${y_{15}}$${y_{16}}$${y_{17}}$${x_{4}}$${y_{18}}$${x_{3}}$$\scriptstyle{U}$$\scriptstyle{U\rho_{2}\rho_{1}}$$\scriptstyle{U^{2}}$$\scriptstyle{U\rho_{3}\rho_{2}\rho_{1}}$$\scriptstyle{U^{2}\rho_{1}}$$\scriptstyle{U^{3}\rho_{3}}$$\scriptstyle{\rho_{2}}$$\scriptstyle{\begin{subarray}{c}U^{2}+\\\
U^{3}\rho_{23}\end{subarray}}$$\scriptstyle{\mathclap{U^{3}\rho_{123}}}$$\scriptstyle{\rho_{1}}$$\scriptstyle{\rho_{12}}$$\scriptstyle{U}$$\scriptstyle{U\rho_{123}\rho_{2}\rho_{1}}$$\scriptstyle{\rho_{2}}$$\scriptstyle{U}$$\scriptstyle{\rho_{1}}$$\scriptstyle{U}$$\scriptstyle{\rho_{12}}$$\scriptstyle{\rho_{2}\rho_{1}}$$\scriptstyle{\rho_{2}\rho_{12}\rho_{1}}$$\scriptstyle{\rho_{2}\rho_{1}}$$\scriptstyle{U\rho_{1}}$$\scriptstyle{\rho_{12}\rho_{1}}$$\scriptstyle{\rho_{2}\rho_{1}}$$\scriptstyle{\rho_{1}}$$\scriptstyle{\rho_{12}\rho_{1}}$
Figure 45. Part of the $A_{\infty}$-module $CFA^{-}(Q_{2,3}(K))$

###  A.3. Computation of $\tau(Q_{m,n}(K))$

In the case of general $m,n$, the invariant $\tau(Q_{m,n}(K))$ is given by the
following theorem.

######  Theorem A.5.

Let $Q_{m,n}$ be the generalized Mazur pattern embedded in the solid torus
$V$. If $m\neq n$, then for any knot $K\subset S^{3}$, we have

(A.3) $\tau(Q_{m,n}(K))=\begin{cases}|m-n|\tau(K)&\text{if }\tau(K)\leq
0\text{ and }\epsilon(K)\in\\{0,1\\},\\\ |m-n|\tau(K)+|m-n|&\text{if
}\tau(K)<0\text{ and }\epsilon(K)=-1,\\\ |m-n|\tau(K)+\min(m,n)&\text{if
}\tau(K)>0\text{ and }\epsilon(K)=1,\\\ |m-n|\tau(K)+\max(m,n)-1&\text{if
}\tau(K)\geq 0\text{ and }\epsilon(K)=-1.\end{cases}$

In the case where $m=n$, we have

(A.4) $\tau(Q_{m,m}(K))=\begin{cases}0&\text{if }\tau(K)<0,\\\ m-1&\text{if
}\tau(K)=0,\\\ m&\text{if }\tau(K)>0.\end{cases}$

The bordered Heegaard diagrams corresponding to the the generalized Mazur
patterns $Q_{m,n}$ is given in Lemma 4.1. We label the intersection points of
the $\alpha$-arcs and $\beta$-arcs as follows: let the intersection points
down the vertical arc be $x_{1}$ to $x_{1+2m}$, and the points along the
horizontal arc be $y_{1}$ to $y_{2m+2n+2nm-2}$. Reading off these diagrams,
the $A_{\infty}$ multiplications have $m$ squares which connect to each other,
and $mn+n-1$ isolated components. The first square contains a head, so one may
consider the structure to be as a train with $m$ carriages, with $\rho_{2}$
arrows connecting carriages. We may see this representation in Figure 46. We
index the squares by $r={1,...,m}$ unless otherwise specified.

######  Proposition A.6.

The generators $x_{1}$ to $x_{1+2m}$ lie in $\iota_{0}$, and the remaining
generators lie in $\iota_{1}$. The multiplications for the isolated components
are as follows:

  1. (1)

$y_{(2m+1)l+j}\xrightarrow[]{U}y_{(2m+1)(l+1)-j}$ for $j=1,...,m$, $l=0,1$

  2. (2)

$y_{(2m+1)l+(j-1)}\xrightarrow[]{U}y_{(2m+1)(l+1)-(j-1)}$ for $j=1,...,m+1$,
$2\leq l\leq n-1$.

When $n=1$, we only use Equation $1$, and $l=0$. When $n=2$, we use Equations
$1$ with both $l=0,1$. When $n\geq 3$, we use both equations.

The multiplications which make up the squares are as follows

  * •

$x_{r}\xrightarrow[]{\rho_{1}}y_{2m+2n+2mn-2-r+1}$ for $r=1,...,m$

  * •

$x_{r}\xrightarrow[]{U}x_{2m-r+1}$ for $r=1,...,m$

  * •

$y_{2m+2n+2mn-2-r+1}\xrightarrow[]{U^{m-r+1}}y_{2n+2mn-2+r}$ for $r=1,...,m$

  * •

$x_{1}\xrightarrow[]{U^{n}\rho_{123}}y_{2n+2mn-1}$

  * •

$y_{2m+2n+2mn-2}\xrightarrow[]{U^{n}\rho_{23}}y_{2n+2mn-1}$

  * •

$x_{1+2m}\xrightarrow[]{U^{n}\rho_{3}}y_{2n+2mn-1}$

  * •

$y_{2m+2n+2mn-2-r+1}\xrightarrow[]{\rho_{2}}x_{2m-r+2}$ for $r=1,...,m$

  * •

$x_{r+1}\xrightarrow[]{\rho_{12}}x_{2m-r+1}$ for $r=0,...,m-1$

  * •

$y_{2n+2mn-2+r}\xrightarrow[]{\rho_{2}\widehat{\rho_{12}}^{l-1}\rho_{1}}y_{2n+2mn-2+r-l}$
for $r=2,...,m$ and $l=1,...,r-1$

  * •

$x_{2m-r+1}\xrightarrow[]{U^{m-r}\widehat{\rho_{12}}^{l}\rho_{1}}y_{2n+2mn-2+r-l}$
for $r=1,...,m$ and $l=0,...,r-1$

  * •

$x_{2m+1}\xrightarrow[]{U\rho_{3}\rho_{2}\rho_{1}}y_{2m+1}$, when $n\geq 2$

  * •

$y_{2m+1}\xrightarrow[]{U^{n-1}}y_{2n+2mn-2}$, when $n\geq 2$

  * •

$x_{2m+1}\xrightarrow[]{U^{m}\rho_{1}}y_{2n+2mn-2}$, when $n\geq 2$

There are also the following multiplications which appear in both the isolated
components and the squares when $n\geq 2$:

  * •

$y_{j}\xrightarrow[]{U\rho_{2}\rho_{1}}y_{2m+1+j}$ for $j=1,...,2mn+n-2m-2$

Combinations of these elements also appear in the $A_{\infty}$ structure, as
long as the index of any of the terms do not exceed $2mn+n-1$.

One notices that by this, the isolated components are connected to each other,
but are isolated from the component containing the main squares.

Let $a=y_{2mn+n-2m-1}$, and $b=y_{2mn+2n-3}$. The $A_{\infty}$ module can be
represented by the following figure:

${a}$${y_{2m+1}}$${x_{2m+1}}$${y_{2mn+2m+2n-2}}$${x_{1}}$${\cdots}$${y_{2mn+m+2n-1}}$${x_{m}}$${b}$${y_{2n+2mn-2}}$${y_{2mn+2n-1}}$${x_{2m}}$${\cdots}$${y_{2mn+m+2n-2}}$${x_{m+1}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}U}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}U^{n-2}\rho_{2}\rho_{12}^{n-3}\rho_{1}}$$\scriptstyle{U^{n-1}}$$\scriptstyle{U\rho_{3}\rho_{2}\rho_{1}}$$\scriptstyle{U^{m}\rho_{1}}$$\scriptstyle{U^{n}\rho_{3}}$$\scriptstyle{\rho_{2}}$$\scriptstyle{\begin{subarray}{c}U^{m}+\\\
U^{n}\rho_{23}\end{subarray}}$$\scriptstyle{\mathclap{U^{n}\rho_{123}}}$$\scriptstyle{\rho_{1}}$$\scriptstyle{\rho_{12}}$$\scriptstyle{U}$$\scriptstyle{\rho_{2}}$$\scriptstyle{\rho_{2}}$$\scriptstyle{U}$$\scriptstyle{\rho_{1}}$$\scriptstyle{U}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rho_{2}\rho_{1}}$$\scriptstyle{\rho_{2}\rho_{1}}$$\scriptstyle{U^{m-1}\rho_{1}}$$\scriptstyle{\rho_{1}}$
Figure 46. The $CFA^{-}$ structure for generalized Mazur patterns, with some
of the differentials connecting the second row not included for clarity. The
dotted lines are included when $n\geq 2$, and the red dotted lines are
included when $n\geq 3$. The isolated components are not included here either,
although they may be easily determined by Proposition A.6. One in fact notices
that when $n\geq 3$, the isolated component $a\xrightarrow{U}b$ is no longer
isolated. See Section A.2 for examples of these relations

######  Remark A.7.

There are also the following multiplications:
$x_{j}\xrightarrow[]{\rho_{3}\rho_{2}\rho_{1}}y_{j}$ for $j=1,...,2m$. These
may be removed by the following change of basis:
$x_{2m-j+1}^{\prime}=x_{2m-j+1}+\rho_{3}\rho_{2}\rho_{1}y_{j}$. This isolates
$y_{j}\xrightarrow[]{U}y_{2n-j+1}$ for $j=1,...,n$. Furthermore, there are
also $U\rho_{2}\rho_{1}$ relations between the isolated components, outlined
in A.6. We may remove many of these by a similar change of basis as the one
above, with the appropriate substitutions.

From the orientation choice letting $w(Q_{m,n})=-(m-n)$ and Proposition A.1,
we have

$A_{Q_{m,n}(K)}(a\otimes x)=-(m-n)A_{k}(x)+C_{a}.$

The following lemma allows us to determine the coefficient $C_{a}$ for certain
$a$.

######  Lemma A.8.

The constants associated to the generators of
$CFA^{-}(V,Q_{m,n})\cdot\iota_{0}$ via Propsition A.1 are $C_{x_{m+1}}=0$,
$C_{x_{m}}=C_{x_{m+2}}=-1$,…, and $C_{x_{m+1-r}}=C_{x_{m+1+r}}=-r$ for
$r=1,...,m$.

###### Proof.

Let O $\subset S^{3}$ be the unknot, and $X_{O}$ its complement equipped with
the 0-framing. Note that $Q_{m,n}(O)$ is also the unknot. By the pairing
theorem, we have

$gCFK^{\infty}(Q_{m,n}(O))=CFA^{-}(V,Q_{m,n})\boxtimes\widehat{CFD}(X_{O}).$

The associated graded on the left hand side has a single generator in its
homology, with 0 Alexander grading since $\tau(Q(O))=0$. On the right hand
side, the type D structure $\widehat{CFD}(X_{O})$ has a single generator
$\xi_{0}$ in 0 grading. The tensor complex
$CFA^{-}(V,Q_{m,n})\boxtimes\widehat{CFD}(X_{O})$ has a summand:

${x_{1}\otimes\xi_{0}}$${\cdots}$${x_{m}\otimes\xi_{0}}$${x_{2m+1}\otimes\xi_{0}}$${x_{2m}\otimes\xi_{0}}$${\cdots}$${x_{m+1}\otimes\xi_{0}}$$\scriptstyle{U}$$\scriptstyle{U}$$\scriptstyle{U}$

It is easy to verify that this is indeed a summand. The only relations in
$CFA^{-}(V,Q_{m,n})$ that pair nontrivially with the type D structure are
arrows connecting $x_{1}$, $x_{2m}$ or $x_{2m+1}$ with labels $U^{p}$ or
$U^{p}\rho_{12}$ (for some integer $p\geq 0$), and all of them are captured
within the fundamental complex in Proposition A.6. One may observe further
that the homology of this summand is generated by $x_{m+1}\otimes\xi_{0}$,
which implies that $A(x_{m+1})=0$. The lemma follows. ∎

###### Proof of Theorem A.5.

Consider the box tensor product of $CFA^{-}(V,Q)$ with $\widehat{CFD}(X_{K})$,
where K is a knot in $S^{3}$, and $X_{K}$ is its exterior equipped with the
0-framing. Note that the generators of the “isolated” components do not affect
$\tau(Q_{m,n}(K))$ since their tensor products with $\widehat{CFD}(X_{K})$
produce summands of $CFK^{-}(Q_{m,n}(K))$ which are U-torsion. In the case
where $\epsilon(K)=0$, $\widehat{CFD}(X_{K})$ has a summand isomorphic to
$\widehat{CFD}(X_{O})$, so the tensor complex
$CFA^{-}(V,Q_{m,n})\boxtimes\widehat{CFD}(X_{K})$ has a summand isomorphic to
(diagram). It follows immediately that $\tau(Q_{m,n}(K))=\tau(Q_{m,n}(O))=0$.
Thus, we restrict our discussion to the cases where $\epsilon(K)=\pm 1$. Let
$s=2|\tau(K)|$, and apply the following change of variables for simplicity:

$w=y_{2n+2mn-2},\ \ y=y_{2m+2n+2mn-2},\ \ y^{\prime}=y_{2n+2mn-1},\ \
z=y_{2m+2n+2mn-3},\ \ z^{\prime}=y_{2n+2mn}.$

Moreover, we denote $\lambda_{l_{1}}^{1}$ as $\lambda$ and
$\kappa_{k_{1}}^{1}$ as $\kappa$ in the unstable neighborhood of $CFD(X_{K})$.

Recall that the homology of the tensor complex contains a $\mathbb{F}[U]$-free
part and a $U$-torsion part. Our goal is to determine the Alexander grading
for generator of the free part of the homology. Towards this end, we consider
three cases cases according to $\tau(K)$.

Case 1: When $\tau>0$, the unstable chain in $\widehat{CFD}(X_{K})$ (along
with the possible $D_{1}$ differential from $\kappa$ to $\eta_{0}$ if
$\epsilon(K)=-1$, and the $D_{2}$ differential from $\lambda$ to $\xi_{0}$ if
$\epsilon(K)=1$) is as follows

${\kappa}$${\eta_{0}}$${\mu_{1}}$${\cdots}$${\mu_{s}}$${\xi_{0}}$${\lambda.}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}D_{1}}$$\scriptstyle{D_{3}}$$\scriptstyle{D_{23}}$$\scriptstyle{D_{23}}$$\scriptstyle{D_{1}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D_{2}}$

The red dashed arrow is included if $\epsilon(K)=-1$ and $n>1$, while the blue
dashed arrow is included if $\epsilon(K)=1$ and $m>1$. The tensor complex has
the following direct summand:

${y_{2m+1}\otimes\kappa}$${x_{1+2m}\otimes\eta_{0}}$${y\otimes\mu_{1}}$${\cdots}$${y\otimes\mu_{s-1}}$${y\otimes\mu_{s}}$${x_{1}\otimes\xi_{0}}$${w\otimes\kappa}$${y^{\prime}\otimes\mu_{1}}$${\cdots}$${y^{\prime}\otimes\mu_{s-1}}$${y^{\prime}\otimes\mu_{s}}$${x_{2m}\otimes\xi_{0}}$${z^{\prime}\otimes\lambda}$${z\otimes\lambda}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}U^{n-1}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}U^{m}}$$\scriptstyle{U^{n}}$$\scriptstyle{U^{m}}$$\scriptstyle{U^{n}}$$\scriptstyle{U^{n}}$$\scriptstyle{U^{m}}$$\scriptstyle{U^{n}}$$\scriptstyle{U^{m}}$$\scriptstyle{U}$$\scriptstyle{U^{m-1}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}U^{m-1}}$

By Proposition A.1 and Lemma A.8, we have

$\displaystyle A(x_{1+2m}\otimes\eta_{0})$
$\displaystyle=-(m-n)A(\eta_{0})-m=(m-n)\tau(K)-m,$ $\displaystyle
A(x_{1}\otimes\xi_{0})$ $\displaystyle=-(m-n)A(\xi_{0})-m=-(m-n)\tau(K)-m,$
$\displaystyle A(x_{2m}\otimes\xi_{0})$
$\displaystyle=-(m-n)A(\xi_{0})-(m-1)=-(m-n)\tau(K)-(m-1),$

which allows us to compute that

$\displaystyle A(y^{\prime}\otimes\mu_{j})$
$\displaystyle=(m-n)\tau(K)+(n-m)j,$ $\displaystyle A(y\otimes\mu_{j})$
$\displaystyle=(m-n)\tau(K)+(n-m)j-m,$ $\displaystyle A(w\otimes\kappa)$
$\displaystyle=(m-n)\tau(K),$ $\displaystyle A(y_{2m+1}\otimes\kappa)$
$\displaystyle=(m-n)\tau(K)-(n-1),$ $\displaystyle
A(z^{\prime}\otimes\lambda)$ $\displaystyle=-(m-n)\tau(K),$ $\displaystyle
A(z\otimes\lambda)$ $\displaystyle=-(m-n)\tau(K)-(m-1).$

We then split our calculation into the following five subcases:

  * •

Case 1.1: If $\epsilon(K)=1$ and $m\geq n$, the free part of the homology is
generated by

$U^{n}z^{\prime}\otimes\lambda+\bigg{(}\sum_{r=1}^{s-1}U^{(m-n)(s-1-r)}y\otimes\mu_{r}\bigg{)}+U^{(m-n)(s-1)}x_{1+2m}\otimes\eta_{0},$

which has Alexander grading $-(m-n)\tau(K)-n$. This gives us

$\tau(Q_{m,n}(K))=(m-n)\tau(K)+n=|m-n|\tau(K)+n.$

  * •

Case 1.2: If $\epsilon(K)=1$ and $m<n$, the free part is generated by the
element

$x_{1+2m}\otimes\eta_{0}+\bigg{(}\sum_{r=1}^{s-1}U^{(n-m)r}y\otimes\mu_{r}\bigg{)}+U^{(n-m)s+1}x_{2m}\otimes\xi_{0},$

with Alexander grading $(m-n)\tau(K)-m$. In this case, we have

$\tau(Q_{m,n}(K))=-(m-n)\tau(K)+m=|m-n|\tau(K)+m.$

  * •

Case 1.3: If $\epsilon(K)=-1$ and $m>n$, the free part is either generated by

$x_{2m}\otimes\xi_{0}+U^{m-n-1}\bigg{(}\sum_{r=1}^{s-1}U^{(m-n)(s-1-r)}y\otimes\mu_{r}\bigg{)}+U^{(m-n)s-1}x_{1+2m}\otimes\eta_{0}$

if $n=1$, or by the above sum plus an additional term
$U^{(m-n)(s+1)}y_{1+2m}\otimes\kappa$ if $n>1$. In both cases, the generator
has Alexander grading $-(m-n)\tau(K)-(m-1)$, and

$\tau(Q_{m,n}(K))=(m-n)\tau(K)+(m-1)=|m-n|\tau(K)+(m-1).$

  * •

Case 1.4: If $\epsilon(K)=-1$ and $m<n$, the generator is

$y_{2m+1}\otimes\kappa+U^{n-m-1}x_{1+2m}\otimes\eta_{0}+U^{n-1-m}\bigg{(}\sum_{r=1}^{s-1}U^{(n-m)r}y\otimes\mu_{r}\bigg{)}+U^{(n-m)(s+1)}x_{2m}\otimes\xi_{0},$

which has Alexander grading $(m-n)\tau(K)-(n-1)$, giving us

$\tau(Q_{m,n}(K))=-(m-n)\tau(K)+(n-1)=|m-n|\tau(K)+(n-1).$

  * •

Case 1.5: If $\epsilon(K)=-1$ and $m=n$, the free part of the homology is
generated by

$Uy_{2m+1}\otimes\kappa+x_{1+2m}\otimes\eta_{0}+\bigg{(}\sum_{r=1}^{s-1}y\otimes\mu_{r}\bigg{)}+Ux_{2m}\otimes\xi_{0},$

with Alexander grading $(m-n)\tau(K)-m=-m$, and thus

$\tau(Q_{m,n}(K))=m.$

Taken together, when $\tau(K)>0$ and $m\neq n$, we have

$\tau(Q_{m,n}(K))=\begin{cases}|m-n|\tau(K)+\min(m,n)&\text{if
}\epsilon(K)=1,\\\ |m-n|\tau(K)+\max(m,n)-1&\text{if
}\epsilon(K)=-1.\end{cases}.$

Moreover, when $\tau(K)>0$ and $m=n$, we have

$\tau(Q_{m,n}(K))=m,\ \ \ \text{for all }\epsilon(K).$

Case 2: When $\tau<0$, the unstable chain in $\widehat{CFD}(X_{K})$ is the
following:

${\lambda}$${\xi_{0}}$${\mu_{1}}$${\cdots}$${\mu_{s}}$${\eta_{0}}$${\kappa}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D_{2}}$$\scriptstyle{D_{123}}$$\scriptstyle{D_{23}}$$\scriptstyle{D_{23}}$$\scriptstyle{D_{2}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}D_{1}}$

where again the red dashed arrow included if $\epsilon(K)=-1$ and $n>1$, and
the blue dashed arrow is included if $\epsilon(K)=1$ and $m>1$. The summand
with nonvanishing homology in the tensor complex is given by:

${z\otimes\lambda}$${x_{1}\otimes\xi_{0}}$${y\otimes\mu_{1}}$${\cdots}$${y\otimes\mu_{s-1}}$${y\otimes\mu_{s}}$${x_{1+2m}\otimes\eta_{0}}$${y_{1+2m}\otimes\kappa}$${z^{\prime}\otimes\lambda}$${x_{2m}\otimes\xi_{0}}$${y^{\prime}\otimes\mu_{1}}$${\cdots}$${y^{\prime}\otimes\mu_{s-1}}$${y^{\prime}\otimes\mu_{s}}$${w\otimes\kappa}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}U^{m-1}}$$\scriptstyle{U}$$\scriptstyle{U^{n}}$$\scriptstyle{U^{m}}$$\scriptstyle{U^{n}}$$\scriptstyle{U^{n}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}U}$$\scriptstyle{U^{m}}$$\scriptstyle{U^{n}}$$\scriptstyle{U^{m}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}U^{m}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}U^{n-1}}$

We calculate inductively that

$\displaystyle A(y^{\prime}\otimes\mu_{j})$
$\displaystyle=-(m-n)\tau(K)+j(n-m),$ $\displaystyle A(y\otimes\mu_{j})$
$\displaystyle=-(m-n)\tau(K)+j(n-m)-m.$

In the case when $\epsilon(K)=-1$ and $n>1$, we have

$\displaystyle A(w\otimes\kappa)$ $\displaystyle=(m-n)\tau(K),$ $\displaystyle
A(y_{1+2m}\otimes\kappa)$ $\displaystyle=(m-n)\tau(K)-(n-1),$

and in the case when $\epsilon(K)=1$ and $m>1$, we have

$\displaystyle A(z\otimes\lambda)$ $\displaystyle=-(m-n)\tau(K)-(m-1),$
$\displaystyle A(z^{\prime}\otimes\lambda)$ $\displaystyle=-(m-n)\tau(K).$

Next, we split our calculation into four cases, depending on the value of
$\epsilon(K)$ and whether $m\geq n$

  * •

Case 2.1: If $m\geq n$ and $\epsilon(K)=1$, the free part of the homology is
generated by $z^{\prime}\otimes\lambda$, which has Alexander grading

$A(z^{\prime}\otimes\lambda)=-(m-n)\tau(K),$

giving us

$\tau(Q_{m,n}(K))=(m-n)\tau(K)=|m-n|\tau(K).$

  * •

Case 2.2: If $m<n$ and $\epsilon(K)=1$, the free part is generated by
$y^{\prime}\otimes\mu_{s}$, with Alexander grading
$A(y^{\prime}\otimes\mu_{s})=(m-n)\tau(K)$. As such, we have

$\tau(Q_{m,n}(K))=-(m-n)\tau(K)=|m-n|\tau(K).$

  * •

Case 2.3: If $m\geq n$ and $\epsilon(K)=-1$, the free part is generated by
$x\otimes\xi_{0}$, with Alexander grading
$A(x\otimes\xi_{0})=-(m-n)\tau(K)-(m-1)$. This gives us

$\tau(Q_{m,n}(K))=(m-n)\tau(K)+(m-1)=|m-n|\tau(K)+(m-1).$

  * •

Case 2.4: If $m<n$ and $\epsilon(K)=-1$, the free part is generated by
$y_{1+2m}\otimes\kappa+U^{n-m-1}x_{1+2m}\otimes\eta_{0}$, with Alexander
grading $A(y_{1+2m}\otimes\kappa)=(m-n)\tau(K)-(n-1)$. Therefore, we have

$\tau(Q_{m,n}(K))=-(m-n)\tau(K)+(n-1)=|m-n|\tau(K)+(n-1).$

In summary, when $\tau(K)<0$, we have

$\tau(Q_{m,n}(K))=\begin{cases}|m-n|\tau(K)&\text{if }\epsilon(K)=1,\\\
|m-n|\tau(K)+\max(m,n)-1&\text{if }\epsilon(K)=-1.\end{cases}.$

Case 3: When $\tau=0$, the unstable neighborhood in $\widehat{CFD}(X_{K})$ is

${\lambda}$${\xi_{0}}$${\eta_{0}}$${\kappa,}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D_{2}}$$\scriptstyle{D_{12}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}D_{1}}$

and the differential in the tensor complex is

${z\otimes\lambda}$${x_{1}\otimes\xi_{0}}$${x_{1+2m}\otimes\eta_{0}}$${y_{2m+1}\otimes\kappa}$${z^{\prime}\otimes\lambda}$${x_{2m}\otimes\xi_{0}}$${w\otimes\kappa}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}U^{m-1}}$$\scriptstyle{U}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}U^{m}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}U^{n-1}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}U^{m-1}}$

where we include red dashed arrow if $\epsilon(K)=-1$ and $n>1$, and the blue
dashed arrow if $\epsilon(K)=1$ and $m>1$. One may check that this is indeed a
summand. We divide our computation into the following three cases

  * •

Case 3.1: If $\epsilon(K)=1$, either $z^{\prime}\otimes\lambda$ or
$x_{2m}\otimes\xi_{0}$ generates the free part of the homology, depending on
whether $m>1$ or $m=1$, respectively. In both cases, the Alexander grading of
the generator is 0, and therefore

$\tau(Q_{m,n}(K))=0=|m-n|\tau(K).$

  * •

Case 3.2: If $\epsilon(K)=-1$ and $m\geq n$, the free part of the homology is
generated by either $x_{2m}\otimes\xi_{0}+U^{m-n}y_{2m+1}\otimes\kappa$ if
$n>1$, or by $x_{2m}\otimes\xi_{0}$ if $n=1$. In both cases, the Alexander
grading of the generator is $-m+1$. Hence, we get

$\tau(Q_{m,n}(K))=m-1=|m-n|\tau(K)+\max(m,n)-1.$

  * •

Case 3.3: If $\epsilon(K)=-1$ and $m<n$, the free part is generated by
$U^{n-m}x_{2m}\otimes\xi_{0}+y_{2m+1}\otimes\kappa$, which has Alexander
grading $-n+1$, giving us

$\tau(Q_{m,n}(K))=n-1=|m-n|\tau(K)+\max(m,n)-1.$

All in all, when $\tau(K)=0$ we get

$\tau(Q_{m,n}(K))=\begin{cases}|m-n|\tau(K)&\text{if }\epsilon(K)=1,\\\
|m-n|\tau(K)+\max(m,n)-1&\text{if }\epsilon(K)=-1.\end{cases}$

∎

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