stmry"71 stmry"79 MnLargeSymbols’164 MnLargeSymbols’171

#  The Complexity of Shake Slice Knots

Charles Ransome Stine Brandeis University<EMAIL_ADDRESS>
###### Abstract.

We define a notion of complexity for shake-slice knots which is analogous to
the definition of complexity for h-cobordisms studied by Morgan-Szabó. We
prove that for each framing $n\neq 0$ and complexity $c\geq 1$, there is an
$n$-shake-slice knot with complexity at least $c$. Our construction makes use
of dualizable patterns, and we include a crash course in their constructions
and properties. We bound complexity by studying the behavior of the classical
knot signature and the Levine-Tristram signature of a knot under the operation
of twisting algebraically-one strands.

###### Key words and phrases:

shake slice, complexity, knot trace

###### 1991 Mathematics Subject Classification:

57K40 (Primary), 57K10 (Secondary); Date:

##  1\. Introduction

Perhaps the most well known difficulty in the study of smooth 4-manifolds is
the failure of the smooth h-cobordism theorem of Smale in 4-dimensions. The
consequences of this fact include the plethora of exotic smooth structures
which have been discovered on 4-manifolds as well as exotic diffeomorphisms
and exotically knotted surfaces in such 4-manifolds. The proof breaks down in
dimension four because one cannot gain sufficient control over the geometric
intersection data of certain embedded 2-spheres from the algebraic
intersection data assumed by the theorem. Essentially this boils down to the
question: given two embedded spheres in a smooth simply connected 4-manifold
with algebraic intersection number $\pm 1$, when can we find embedded spheres
in the same homology classes with geometric intersection number exactly $1$?

We study this question in what is perhaps the simplest possible context: the
existence of shake-slice knots. These are knots with the property that for
some $n\in\mathbb{Z}$ the associated knot trace $X_{n}(K)$ contains a smoothly
embedded sphere which generates $H_{2}$. If the knot $K$ is smoothly slice
then this condition is obviously satisfied by the sphere consisting of a slice
disk for $K$ in the 4-ball union the core of the attached 2-handle. Moreover
if we consider the intersection data between this sphere and the cocore of the
2-handle, we see the algebraic and geometric intersection numbers between them
are both one. If the answer to the previous question were yes then we would
expect this type of sphere to be the only way this phenomenon could occur; in
short that all shake-slice knots would be slice. Initial work of Akbulut
[Akb77] showed that there were shake slice knots for framings $n=1,2$ which
are not slice, and later work [Akb93], [CR16] showed that they exist for all
non-zero $n$. It has remained an interesting question to try to quantify how
far these knots are from being slice, and then to give examples of $n$-shake-
slice knots which are arbitrarily far from being slice.

Some work on this question was completed in [FMN+21]. The authors show that in
the case $n=\pm 1$, a knot is topologically shake-slice iff
$\operatorname{Arf}(K)=0$. They define a notion of complexity, the _shaking-
number_ , which is a re-normalization of the notion we will introduce here,
and show that it is bounded below by the signature. Since there are many
examples of knots with $\operatorname{Arf}(K)=0$ and large signature, this
provides a positive answer to the question. On the other hand, the surgery
theoretic methods employed in [FMN+21] are not well suited to constructing or
preserving smoothly embedded spheres, so it remains to be seen if there are
similar smoothly $(\pm 1)$-shake-slice examples. We give such examples for all
non-zero $n$ in this paper.

A generic transversality argument tells us that any smoothly embedded sphere
generating the homology of the trace of a shake-slice knot must intersect the
2-handle of the knot trace in some number of parallel copies of the core of
the 2-handle. If the sphere can be isotoped into a position where it intersect
the 2-handle in exactly one copy of the core then the lower part of the sphere
slices the knot and we are in the situation described above. It is natural to
consider a shake-slice knot to be far from being slice if it is such that any
smoothly embedded sphere which generates the second homology of its trace must
intersect the 2-handle in a large number of copies of the core. This is made
precise as follows: any generating sphere must intersect the 2-handle in
$1+2c$ parallel copies of the core for some
$c\in\mathbb{N}\cup\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}$.
Each copy of the core must intersect the cocore at exactly one point and so we
see one positive or negative intersection between the sphere and the cocore
and then $c$ algebraically cancelling pairs of intersections. We define the
complexity of an $n$-shake-slice knot to be the minimum number of
algebraically cancelling pairs of transverse intersection points between the
cocore of the 2-handle of its $n$-trace and a smoothly embedded sphere
generating its homology. We prove the following theorem:

######  Theorem 1.1.

Let
$n\in\mathbb{Z}\setminus\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}$
and $c\in\mathbb{N}$ be given. There exists a knot $K_{n,c}\subset{S}^{3}$
such that $K_{n,c}$ is smoothly $n$-shake-slice, yet the complexity of
$K_{n,c}$ is at least $c$. Moreover, the complexity bound applies to
topologically-locally-flatly embedded spheres generating the homology of the
$n$-trace as well as smoothly embedded ones.

We construct $K_{n,c}$ as a satellite knot using two dualizable-patterns and
we exploit the nice properties of dualizable patterns both in our proof that
our knots are smoothly $n$-shake-slice and in our computation of a lower bound
on complexity. Although dualizable patterns have been in the literature for
awhile already, there is no systematic review of their construction and
properties. We give such a review here, and we have ironed out a few kinks in
the notation, so as to produce a symbolic calculus which one can use to
manipulate them while preserving or changing certain of their properties in a
controlled way. In particular, we append a one page cheat sheet (cheatsheet)
which contains a concise summary of the calculus, and may be used as a
reference independently from the rest of this paper.

We give a topological argument following [FMN+21][Prop. 8.8] which shows that
the complexity of an $n$-shake-slice knot is bounded below by its 4-ball genus
(smoothly or topologically respectively) and thus any of the well known
4-genus bounds can be used to bound the complexity of an $n$-shake-slice knot.
We give a compatibility criterion for any such invariant, which, if satisfied,
allows us to compute the invariant for large families of knots constructed
using dualizable patterns. We observe that the Levine-Tristram signature of a
knot satisfies our compatibility criterion, and we give a general computation
of the LT signature of our knots $K_{n,c}$. The main technical point ends up
being an analysis of how the Levine-Tristram signature changes under the
operation of adding $n$ twists to $1+2c$ parallel strands of a knot, assuming
that all but one of the strands come with an oppositely oriented partner.

As a slight aside to the main result of this paper, we give a reformulation of
the Goeritz-Trotter method for computing the classical signature of a knot,
which, if not algorithmically quicker than procedure as described in [Lit79],
is we feel much easier to perform as a human looking at a diagram of a knot.
Using this reformulation, we are able to give a different, somewhat simpler
proof of the following theorem due to Tristram:

######  Theorem 1.2 ([Tri69], Cor. to Thm 3.2).

Let $K$ be a knot in ${S}^{3}$, and let $U$ be an unknot embedded in the
complement of $K$ such that the oriented linking number $lk(K,U)=1$. Let
$K_{n}$ be the result of applying $n$ full twists to all the strands of $K$
which pass through a spanning disk for $U$. Alternatively, $K_{n}$ can be
viewed as the knot in ${S}^{3}$ induced by $K$ after doing
$-\frac{1}{n}$-surgery on $U$. For all $n\in\mathbb{Z}$,
$\sigma(K_{n})=\sigma(K_{n+2})$, so the set
$\mathchoice{\left\\{\sigma(K_{n})\mid
n\in\mathbb{Z}\right\\}}{\\{\sigma(K_{n})\mid
n\in\mathbb{Z}\\}}{\\{\sigma(K_{n})\mid
n\in\mathbb{Z}\\}}{\\{\sigma(K_{n})\mid n\in\mathbb{Z}\\}}$ contains at most
two values.

This is in stark contrast to the behavior we see when we change the linking
condition to be $lk(K,U)=0$. Under this condition the signature of $K_{n}$
decreases steadily until it hits a fixed, stable value and the number of times
the signature decreases is roughly proportional to the number of algebraically
cancelling pairs of strands which pass through $U$. Instead, we see a totally
determined oscillation, which is independent of the number of algebraically
cancelling pairs of strands passing we are twisting. Tristram’s original proof
proceeds by constructing a family of very large Seifert forms for the knots
$K_{n}$ and then analyzing the behavior of the associated Hermitian forms. Our
proof uses a much simpler family of non-orientable spanning surfaces for the
knots $K_{n}$ which leads to significantly simpler algebra; however our method
cannot be generalized to the full signature function.

###  1.1. Organization

We start by reviewing the definition and properties of dualizable patterns. We
give two different diagrammatic methods for computing the dual of a pattern.
We give a universal, combinatorial method for constructing dualizable
patterns, and we give a systematic notation for building and manipulating
satellite knots with dualizable patterns. In section 3, we review the relevant
aspects of shake-slice knots. We prove that 4-genus bounds complexity, and we
give sufficient conditions on a dualizable pattern and a 4-genus invariant for
the construction of knots $K_{n,c}$. In section 4, we digress to give our
reformulation of Goeritz and Trotter’s method for computing the signature. We
use it to prove Theorem 4.1 and then we use that to deduce a proof of Theorem
1.1 for $n$ odd, using the classical signature. Finally, in section 5, we
verify the general sufficient conditions given in section 3 for a specific
dualizable pattern and the Levine-Tristram signature function. The proof of
Theorem 1.1 then follows immediately from Theorem 3.4 for both parities of
$n$, independently from the work in section 4.

###  1.2. Acknowledgements

This project is deeply indebted to Danny Ruberman, who suggested the question
and provided a great deal of support along the way, and Lisa Piccirillo for
many helpful conversations. The author also thanks Joshua Wang for helping to
fix an issue with the proof of Theorem 4.1, and Allison Miller for her
comments on an early draft.

##  2\. The Calculus of Dualizable Patterns

###  2.1. Definitions and Examples

####  2.1.1. The Definition of a Dualizable Pattern

Let $V:={D}^{2}\times{S}^{1}$ with preferred, oriented longitude
$\lambda_{V}:=\mathchoice{\left\\{pt\right\\}}{\\{pt\\}}{\\{pt\\}}{\\{pt\\}}\times{S}^{1}$
and oriented meridian
$\mu_{V}:=\partial{D}^{2}\times\mathchoice{\left\\{pt\right\\}}{\\{pt\\}}{\\{pt\\}}{\\{pt\\}}$
marked in $\partial V$. A pattern is a smooth embedding
$P:{S}^{1}\hookrightarrow\operatorname{Int}V$ of an oriented circle into the
interior of $V$. We define the winding number, $w(P):=lk(\mu_{V},P)$, where we
identify $P$ with its oriented image. We further define the wrapping number
$wr(P)=|P\cap({D}^{2}\times\mathchoice{\left\\{pt\right\\}}{\\{pt\\}}{\\{pt\\}}{\\{pt\\}})|$,
minimized over the oriented isotopy class of $P$. It is immediately clear that
that for any pattern $P$ we have that for some $c\in\mathbb{N}$:

$wr(P)=|w(P)|+2c$

where we can interpret $c$ as the number of pairs of algebraically cancelling
intersection points between $P$ and
$\mathchoice{\left\\{pt\right\\}}{\\{pt\\}}{\\{pt\\}}{\\{pt\\}}\times{D}^{2}$.
We define the complexity of the pattern, $c(P)$, to be the minimum such $c$
over all patterns isotopic to $P$ in $V$.

Given a pattern $P$ we denote $V\setminus\nu(P)$ by $V_{P}$ which comes with
two boundary components: $\partial V$ and $\partial\nu(P)$ both of which are
tori. We call $\partial V$ the ‘outer torus’ and $\partial\nu(P)$ the ‘inner
torus’ associated to $V_{P}$. The outer torus comes with a preferred
identification to ${S}^{1}\times{S}^{1}$ given by the pair
$(\mu_{V},\lambda_{V})$. We can also construct a similar identification for
the inner torus as follows: we glue $V$ into the complement of the unknot in
${S}^{3}$ so that $\mu_{V}$ is glued to the meridian of the unknot, and
$\lambda_{V}$ is glued to the unique longitude which bounds a disk in the
complement. This converts $P$ into an embedding
$P^{\prime}:{S}^{1}\hookrightarrow{S}^{3}$, thereby defining an oriented knot.
We get a triple embedding,

$\nu(P)\hookrightarrow V\hookrightarrow{S}^{3}$

and there is a unique null-homologous longitude on $\partial\nu(P)$ which
bounds an oriented surface in ${S}^{3}\setminus\nu(P)$. We call this curve
$\lambda_{P}\subset\partial\nu(P)$ and we orient it coherently with $P$ so
that $lk(\mu_{V},\lambda_{P})=lk(\mu_{V},P)$. After choosing any
identification of ${D}^{2}\times{S}^{1}$ with $\nu(P)$, we define $\mu_{P}$ to
be the image of
$\partial{D}^{2}\times\mathchoice{\left\\{pt\right\\}}{\\{pt\\}}{\\{pt\\}}{\\{pt\\}}$
under this identification. It is a standard exercise in 3-manifold topology
that this is well defined. We orient $\mu_{P}$ so that $lk(\mu_{p},P)=+1$.
Thus the pair $(\mu_{P},\lambda_{P})$ define an identification of the inner
torus of $V_{P}$ with the product of oriented circles, $\SS 11$.

######  Definition 2.1.

[MP18, Def. 3.1] A pattern $P$ is _dualizable_ with dual pattern $P^{*}\subset
V^{*}$ iff there exists an orientation preserving homeomorphism $*:V_{P}\to
V^{*}_{P^{*}}$ such that,

  1. (i)

$*$ maps the inner torus of $V_{P}$ to the outer torus of $V^{*}_{P^{*}}$, and
the outer torus to the inner torus.

  2. (ii)

$(*(\mu_{V}),*(\lambda_{V}))=(-\mu_{P^{*}},\lambda_{P^{*}})$

  3. (iii)

$(*(\mu_{P}),*(\lambda_{P}))=(-\mu_{V^{*}},\lambda_{V^{*}})$

Given a dualizable pattern $P$, we can obtain the dual pattern by Dehn filling
with slope $\infty$ the outer boundary (only) of $V_{P}$. The existence of $*$
implies that this manifold is homeomorphic to ${D}^{2}\times{S}^{1}$ so we
call it $V^{*}$ and we obtain $P^{*}$ as the core of the new Dehn filling,
oriented coherently with $\lambda_{V}$.

Whenever we draw explicit examples of patterns, their duals, and satellite
constructions, we will adopt the orientation convention that knot complements
in ${S}^{3}$ are oriented with the inward pointing normal first at the
boundary, while solid tori are oriented with the outward pointing normal first
at the boundary. Thus all our gluing maps are orientation preserving at the
boundary. This fixes an orientation convention on the exterior of a pattern
$V_{P}$. If we pick a random companion knot $K$ and consider the decomposition
of ${S}^{3}$ naturally associated to the satellite knot $P(K)$ we see
${S}^{3}$ split into three oriented, codimension-0 parts:

${S}^{3}=\nu(P)\underset{\partial\nu(P)}{\cup}V_{P}\underset{\partial
V}{\cup}E_{K}$

Since orientations must agree at both boundary tori across all three pieces
and we know that $\nu(P)$ is oriented outward-normal-first, while $E_{K}$ is
oriented inward-normal-first, we can deduce that $V_{P}$ must be oriented
inward-normal-first at $\partial\nu(P)$ and outward-normal-first at $\partial
V$. This leads to the rather pleasant sounding convention that $V_{P}$ is
oriented outward-first at its outer boundary and inward-first at its inner
boundary. In case this is confusing, we include a totally explicit picture of
$Wh^{-}(3_{1})\subset{S}^{3}$ with all longitudes, meridians, and orientation
arrows drawn:

2pt $\mu_{P}$ [l] at 122 207 ${S}^{3}\setminus\nu(3_{1})$ [ ] at 21 85
$\lambda_{V}$ [bl] at 68 238 $\lambda_{P}$ [tl] at 145 264 $\mu_{V}$ [l] at
506 301 $\nu(P)$ [br] at 52 382

Figure 1. Orientations and Surfaces in $Wh^{-}(3_{1})$

We now summarize some useful facts about dualizable patterns, all of which are
stated and proven in [MP18]. Given a knot $K\subset{D}^{2}\times{S}^{1}$, we
can construct an associated knot, $\widehat{K}\subset\SS 21$, by doubling
${D}^{2}\times{S}^{1}$ and letting $\widehat{K}$ be the composition of the
inclusions:

$\widehat{K}:K\hookrightarrow{D}^{2}\times{S}^{1}\hookrightarrow\SS 21$

Often $K$ will be either a pattern, $P$, or one of the framing curves from
$\partial V_{P}$, so that $V$ is the copy of ${D}^{2}\times{S}^{1}$ we are
doubling to get $\SS 21$.

######  Proposition 2.2.

[MP18, Prop 3.5] A pattern $P$ is dualizable iff $\widehat{P}$ is oriented-
isotopic to
$\widehat{\lambda_{V}}={S}^{1}\times\mathchoice{\left\\{pt\right\\}}{\\{pt\\}}{\\{pt\\}}{\\{pt\\}}$
in $\SS 21$.

######  Corollary 2.3.

[MP18, Cor. 3.6] A pattern $P$ with $w(P)=1$ is dualizable iff
$\mu_{P}\in\llangle\mu_{V}\rrangle$, the subgroup of $\pi_{1}(V_{P})$ normally
generated by $\mu_{V}$.

These statements are important because their proofs indicate both how to
construct dualizable patterns and, given a dualizable pattern, how to compute
its dual pattern. We will give a brief verbal description of these processes
here and include an example. Figure 2, Figure 3, and Figure 4 give the example
in two different notations simultaneously. We highly encourage the reader to
separate out the pages containing these figures so as to examine them side-by-
side with the verbal description below.

The key to understanding both points is to study the copies of $\SS 21$
obtained by doubling the solid tori $V$ and $V^{*}$ containing dualizable
patterns $P$ and $P^{*}$. They decompose naturally into three pieces:

(2.1) $\displaystyle\operatorname{double}(V)=\nu(P)\cup
V_{P}\cup{D}^{2}\times{D}^{1}$ (2.2)
$\displaystyle\operatorname{double}(V^{*})=\nu(P^{*})\cup
V^{*}_{P^{*}}\cup{D}^{2}\times{D}^{1}$

where we recall that the pair of curves $(\mu_{P},\lambda_{P})$ gives a
natural identification $\nu(P)={D}^{2}\times{S}^{1}$, and similarly for
$\nu(P^{*})$. We can equivalently describe this as:

(2.3)
$\displaystyle\operatorname{double}(V)=({D}^{2}\times{D}^{1})_{Inner}\cup
V_{P}\cup({D}^{2}\times{D}^{1})_{Outer}$ (2.4)
$\displaystyle\operatorname{double}(V^{*})=({D}^{2}\times{D}^{1})_{Inner}\cup
V^{*}_{P^{*}}\cup({D}^{2}\times{D}^{1})_{Outer}$

The homeomorphism $*:V_{P}\to V^{*}_{P^{*}}$, which swaps the inner and outer
boundaries, carries meridians to meridians and longitudes to longitudes. We
can extend it to a map,
$\widehat{*}:\operatorname{double}(V)\to\operatorname{double}(V^{*})$,
swapping the two solid tori filling in the inner and outer boundaries.
$\widehat{*}$ is clearly an orientation preserving self-homeomorphism of $\SS
21$, since both $V$ and $V^{*}$ come with preferred identifications to
${D}^{2}\times{D}^{1}$, and thus their doubles have such identifications to
$\SS 21$.

We are going to keep track of what happens to $\widehat{\lambda_{V}}$ and
$\widehat{\lambda_{P}}$, considered as oriented curves in $\SS 21$, under the
homeomorphism $\widehat{*}$. When we build $\SS 21$ by doubling $V$, we get
one of two canonical handle-body diagrams for $\SS 21$. The first one, which
we call the ‘2-ball diagram’, comes from viewing $\SS 21$ as surgery on a pair
of disjointly embedded 3-balls. We draw the complement of the two ${B}^{3}$’s
and we identify the two boundary ${S}^{2}$’s of the complement by the map
which reflects the complement across the sphere bisecting the line connecting
the two boundary spheres perpendicularly. The ${S}^{1}$ factor in this picture
is given by the line connecting the two balls in the diagram, and the
${S}^{2}$ factor is given by perpendicular bisector. For a more detailed
description of this diagram consult [GS99, page 114-115]. The second diagram,
which we will call the ‘dotted-circle diagram’, comes from viewing $\SS 21$ as
the boundary of zero-surgery on the unknot. We draw a round, circular unknot
in ${S}^{3}$ which we label with a $\langle 0\rangle$ to indicate performing
0-surgery on it. Here, the ${S}^{1}$ factor is given by a small meridian to
the surgered circle, while the ${S}^{2}$ factor is given by the disk which the
surgered circle bounds. Consult [GS99, page 167-168] for more details.

It follows from work of Gluck [Glu62] that there are only two orientation
preserving self-homeomorphisms of $\SS 21$ up to isotopy which are the
identity and the Gluck twist. Since the Gluck twist fixes the isotopy class of
any fixed copy of the ${S}^{1}$-factor, it follows from the construction of
$\widehat{*}$ that $\widehat{P}$ must be oriented isotopic to
$\mathchoice{\left\\{pt\right\\}}{\\{pt\\}}{\\{pt\\}}{\\{pt\\}}\times{S}^{1}$
in $\SS 21$ by an ambient isotopy. However this isotopy may change the framing
of $\widehat{P}$ since it may involve sliding $\widehat{P}$ over the ${S}^{2}$
factor. The algebraic intersection number of $\widehat{P}$ with the ${S}^{2}$
factor is $+1$ so each such slide will change the framing by $\pm 2$. After
the ambient isotopy, apply Gluck twisting around $\widehat{P}$ to correct the
framing, and let $\widehat{P^{*}}$ be the image of $\widehat{\lambda_{V}}$
under this isotopy and twisting. Lastly we recover $V^{*}$ as $\SS
21\setminus\widehat{\mu_{V}}$. This tells us how to dualize a pattern in
practice. Given a dualizable pattern $P$, we can obtain $P^{*}$ as follows:

  1. (1)

Draw $\widehat{P},\widehat{\mu_{V}},\widehat{\lambda_{V}}$ in either of the
standard diagrams of $\SS 21$ (2-ball or dotted-circle) and keep track of
their framings.

  2. (2)

Observe that removing a neighborhood of $\widehat{\mu_{V}}$ recovers $P\subset
V$.

  3. (3)

Draw an explicit, ambient isotopy of $\SS 21$ which moves $\widehat{P}$ to an
${S}^{1}$ factor, fixes $\widehat{\mu_{V}}$, and keeps $\widehat{\lambda_{V}}$
disjoint from the final position of $\widehat{P}$ in the diagram (what this
means will become clear in the examples).

  4. (4)

The ambient isotopy may change the framings of $\widehat{\lambda_{V}}$ and
$\widehat{P}$, so apply the Gluck twist map to the diagram to correct them
(again, what this means will become clear in the examples).

  5. (5)

Let $P^{*}$ be the image of $\widehat{\lambda_{V}}$ after the ambient isotopy
and Gluck twisting, considered in the complement of $\mu_{V}$.

The correctness of this process follows from the rough discussion proceeding
it, with a little care. A completely detailed account is given in [MP18]. We
will now give two examples of this computation which we will carry out in each
of the two diagrams for $\SS 21$ respectively.

####  2.1.2. An Example of Dualizing a Pattern

The following sequence of diagrams describe how to construct the dual of a
pattern using the method described in the previous subsection. The pattern in
question, $P$, is the main ingredient we use, along with its dual, $P^{*}$, to
construct the knots $K_{n,c}$ in Theorem 1.1. We do not actually include a
diagram of any of these knots in this paper but we give complete instructions
for how to draw them using the patterns $P$ and $P^{*}$.

This sequence of diagrams shows the isotopy which takes $\widehat{P}$ to
$\widehat{\lambda_{V}}$ and then the Gluck twisting which we apply to correct
the framings. We show the exact same isotopy and twisting in both ‘dotted-
circle’ and ‘2-ball’ notation. The dotted-circle computation is given in the
left-hand column and the 2-ball computation in the right hand column. We give
both because different stages of the process are relatively easier in each of
the two notations, and so it of great practical value to be familiar with
both. Moreover there is currently no fully worked out example in dotted-circle
notation in the literature, although there is a 2-ball example in [MP18].

Any isotopy of a curve in $\SS 21$ can be drawn in either of these diagrams as
a sequence of planar isotopies, Reidemeister moves, and slides of the curve
over the surgered regions of the diagram. We will not indicate the paths
traced out by a curve during planar isotopies or Reidemeister moves. We give
the start and end diagrams instead. For slides, we use a purple arrow to
indicate the band, which we assume to lie in the plane except where it crosses
over/under other strands in the diagram. After every handle slide or
diffeomorphism we will indicate the new framings of the slid curves by
integers in parentheses. We use also use a purple arrow to indicate the
rotation of an ${S}^{2}$ factor at the end when we are applying our Gluck
twist diffeomorphism to correct the framings. The components corresponding to
the surgery diagram of $\SS 21$ are drawn in red, $\widehat{P}$ is drawn in
blue, and $\widehat{\lambda_{V}}$ is drawn in green. Thus $P^{*}$ may be
obtained from the final dotted-circle diagram Figure 4 (left) by erasing the
blue curve and considering the green curve in the complement of the red curve.

2pt $\mathchoice{\left\langle 0\right\rangle}{\langle 0\rangle}{\langle
0\rangle}{\langle 0\rangle}$ [l] at 165 667 $\widehat{\lambda_{V}}\ (0)$ [r]
at 116 635 $\widehat{P}\ (0)$ [b] at 510 592 $\widehat{\lambda_{V}}\ (0)$ [t]
at 436 641 $\widehat{P}\ (0)$ [b] at 202 598 $(2)$ [br] at 27 550 $(2)$ [br]
at 329 555

Figure 2. Computing the dual of $P$, part I

2pt $(-2)$ [tl] at 165 584 $(-2)$ [tl] at 533 590 $(-2)$ [l] at 234 222 $(-2)$
[l] at 533 232 $\mathchoice{\left\langle 0\right\rangle}{\langle
0\rangle}{\langle 0\rangle}{\langle 0\rangle}$ [r] at 132 76 $\leftarrow$
Gluck Twisting $\to$ [] at 259 28

Figure 3. Computing the dual of $P$, part II

2pt $\widehat{P^{*}}$ [r] at 232 131 $(0)$ [l] at 174 30 $(0)$ [r] at 99 97
$\mathchoice{\left\langle 0\right\rangle}{\langle 0\rangle}{\langle
0\rangle}{\langle 0\rangle}$ [tr] at 124 47 $(0)$ [b] at 428 81
$\widehat{P^{*}}$ [r] at 524 122 $(0)$ [b] at 428 17 $+2$ [] at 173 63 $+2$ []
at 481 58

Figure 4. Computing the dual of $P$, part III

####  2.1.3. A Universal Construction

######  Proposition 2.4.

Any pattern, $P$, obtained from the following process is dualizable:

  1. (1)

Pick an oriented knot $K_{0}$ in ${S}^{3}$ and draw a positively oriented
meridian, $m_{0}$, around $K_{0}$.

  2. (2)

Draw unoriented parallel copies of $m_{0}$, labelled $m_{1},\ldots,m_{n}$, so
that all $m_{0\leq i\leq n}$ bound disjoint spanning disks.

  3. (3)

Draw a band $b_{i}$ connecting $K_{0}$ to $m_{i}$ for each $1\leq i\leq n$.
The bands are allowed to link in any way with $K_{0}$ and $m_{0}$, but they
are not allowed to intersect each other.

  4. (4)

Orient the new knot $K^{\prime}$ using the induced orientation from $K_{0}$.

  5. (5)

Let $P\subset V$ be the knot $K^{\prime}\subset{S}^{3}\setminus\nu(K_{0})$.

Moreover, given any dualizable pattern $Q$ we can take $K_{0}$ to be the
unknot and then choose bands so that the pattern, $P$, produced by this
process is isotopic to $Q$. Although the unknot is sufficient to produce any
dualizable pattern, it can be of great practical use to choose a non-trivial
base knot $K_{0}$.

###### Proof.

Let $P$ be obtained from the process of Proposition 2.4. We start by showing
that $P$ is dualizable. It suffices to give an explicit isotopy from
$\widehat{P}$ to ${S}^{1}$ in $\SS 21$ by Proposition 2.2. We consider this
problem in ‘dotted-circle’ diagrams. We obtain a picture of
$\widehat{P}\subset\SS 21$ by drawing $K^{\prime}\cup m_{0}$ and doing
0-surgery on $m_{0}$. Since the ends of the bands $b_{*}$ are the parallel
copies of $m_{0}$, we can slide each end once over the 0-surgered component
$m_{0}$. Each band then retracts back onto the original knot $\widehat{K_{0}}$
so we see that $\widehat{K^{\prime}}$ is isotopic to $\widehat{K_{0}}$ in $\SS
21$. It follows from the classical light bulb theorem in $\SS 21$ that
${K^{\prime}}$ is isotopic to ${S}^{1}$ since
$|\widehat{K_{0}}\cap{S}^{2}|=1$.

Let $Q$ be any fixed dualizable pattern. We must find a collection of bands to
attach to the unknot as in Proposition 2.4 which recover $Q$. Consider
$\widehat{Q}$ in the ‘dotted circle’ diagram of $\SS 21$. We know there is an
ambient isotopy $\mathcal{I}:\SS 21\times[0,1]\to\SS 21$ such that
$\mathcal{I}_{0}=\operatorname{Id}_{\SS 21}$ and
$\mathcal{I}_{1}(\widehat{Q})={S}^{1}$ by the characterization of dualizable
patterns given in Proposition 2.2. Using standard techniques from Kirby
calculus applied in dimension 3 to $\mathcal{I}$, we see it can be represented
in the ‘dotted circle’ diagram as an alternating sequence of slides over the
0-surgered region and isotopies in the complement of the 0-surgered region.
The key observation is that we can modify the isotopy in $\SS 21$ so that all
the handle-slides occur in sequence at the beginning, followed by an isotopy
in the complement of the 0-surgered region.

We can assume the first move in the sequence associated to $\mathcal{I}$ is a
handle-slide without loss of generality. We proceed until we come to the first
isotopy in the sequence, which we call $I_{1}$. At the end of $I_{1}$, there
is a small arc on $\widehat{P}$ where the next band for the next handle-slide,
$S_{1}$, is attached. We can pull this arc back along $I_{1}$ and keep track
of the path it traces out in $\SS 21$. It may be helpful to visualize this
path as a ribbon. Since this path itself is contained in a neighborhood of an
arc, we can modify $I_{1}$ so that the path intersects neither itself nor the
knot $\widehat{P}$. We create a new sequence, defining a new isotopy, by
removing $I_{1}$ and proceeding directly to $S_{1}$ except we append to the
band the path traced out by its attaching arc under $I_{1}$ thereby defining a
different slide $S^{\prime}_{1}$. The result of $S^{\prime}_{1}$ will be
isotopic to the result of $S_{1}\circ I_{1}$ by an isotopy called
$I^{\prime}_{1}$. We now construct a new isotopy $\mathcal{I}^{\prime}$ by
removing the subsequence $I_{1}$, $S_{1}$ from $\mathcal{I}$ and replacing it
with $S^{\prime}_{1}$, $I^{\prime}_{1}$. By proceeding inductively in this
manner, we can create a new isotopy which consists of a sequence of slides of
$\widehat{P}$ over the 0-surgered region, followed by a single isotopy in the
complement.

There is one more thing to check. We now have $\mathcal{I}^{\prime}$, which
consists of a sequence of handle-slides followed by an isotopy in the
complement of the 0-surgered region. The trick is that, because the slides are
done one after another and not simultaneously, it is possible that the band
from the $nth$ slide might intersect the bands from the slides $1,\ldots,n-1$.
We can fix this by applying the band-finger-move described in Figure 5 to
resolve the intersection of each band with all those of lower index. This
makes the bands significantly more complicated but it ensures that they are
disjoint. We can now play the isotopy backwards to see the ${S}^{1}$ factor of
$\SS 21$ being simultaneously slid some number of times over the 0-surgered
region to produce the pattern $Q$. This shows the existence of a sequence of
bands ${b_{1},\ldots,b_{n}}$ such that if we take $K_{0}$ to be the unknot
(really we should think of it as the ${S}^{1}$ factor) and attach the bands as
in Proposition 2.4 then we obtain $Q$. ∎

2pt $b_{i}$ [br] at 63 195 $b_{i+1}$ [bl] at 155 225 $b_{i}\cap b_{i+1}$ [l]
at 199 199 $b_{i+1}^{\prime}$ [bl] at 423 252 $b_{i}$ [br] at 301 204 $K_{0}$
[l] at 95 65 $K_{0}$ [l] at 345 19

Figure 5. Resolving intersections between bands with a finger move.

Since this construction happens in $\SS 21$, it is naturally compatible with
the method we give for computing the dual pattern. Given $K_{0}$ and
$b_{1},\ldots,b_{n}$, lift them to $\SS 21$, viewed as zero surgery on $m_{0}$
as before. Notice that $\lambda_{V}$ appears as a small meridian of $m_{0}$,
but it does not link with the parallel copies $m_{1},\ldots,m_{n}$ which make
up the ends of the bands $b_{1},\ldots,b_{n}$. We slide each band over $m_{0}$
which causes the end of each band to link once, geometrically, with
$\lambda_{V}$. We retract the bands back onto $K_{0}$, which effectively
attaches the same set of bands to $\lambda_{V}$. We slide $K_{0}$ over $m_{0}$
until it has become an unknot (possibly further tangling the bands now
attached to $\lambda_{V}$). We slide every strand of $\lambda_{V}$ which
passes through $K_{0}$ over $m_{0}$ then apply the Gluck twist (spinning the
sphere composed of the obvious spanning disk for $m_{0}$ with the core of the
0-surgery) to correct the framings. A very simple example of this is given in
Figure 6.

2pt $\widehat{\lambda_{V}}$ [br] at 25 284 $(0)$ [tr] at 27 252 $b_{1}$ [bl]
at 172 332 $(0)$ [b] at 125 235 $K_{0}$ [br] at 159 245 $m_{0}$ [b] at 70 285
$\mathchoice{\left\langle 0\right\rangle}{\langle 0\rangle}{\langle
0\rangle}{\langle 0\rangle}$ [tr] at 70 252 $m_{1}$ [bl] at 121 280
$\widehat{P}$ [ ] at 89 330 $(+2)$ [r] at 289 314 $(-2)$ [b] at 33 134 $(0)$
[b] at 240 293 $(0)$ [bl] at 434 292 $(+2)$ [b] at 523 308 $(+2)$ [b] at 134
151 $(-2)$ [br] at 225 158 $(+2)$ [r] at 346 94 $(0)$ [br] at 443 151 $P^{*}$
[bl] at 566 166 $(0)$ [br] at 543 70 $+2$ at 461 112 Gluck Twisting [t] at 398
23 $\mu_{V^{*}}$ [t] at 485 83 $\lambda_{V^{*}}$ [tr] at 555 130

Figure 6. Working out $P^{*}$ from a base-knot-with-bands description of $P$.
Going from bottom-middle to bottom-right is a simple isotopy of the three
component link in the diagram. The framings of the blue and green components
change because we are also applying the Gluck twisting map here, however,
because each of the two components only intersects a spanning disk for the red
circle at one point, we can spin blue clockwise and green counter-clockwise
without causing them to twist together.

###  2.2. Properties of Dualizable Patterns

We will define all of the notation we need to manipulate dualizable patterns
and knot invariants which are compatible with them in this section. These
objects and their properties form a pleasing calculus which we wish to express
in its entirety. We include a singe-page cheat sheet at the end of the paper
which can be printed out or clipped separately from the rest. We hope it may
serve as a useful reference.

####  2.2.1. Definitions and Notation

######  Definition 2.5.

Let $K\subset{S}^{3}$ be a knot with exterior $E_{K}$ and let $P\subset V$ be
a dualizable pattern. Let $E_{K}$ come with oriented meridian $\mu_{K}$ and
oriented, null-homologous longitude $\lambda_{K}$ on its boundary. Observe
that $E_{K}\cup V$ is diffeomorphic to ${S}^{3}$ if we glue the pair
$(\mu_{K},\lambda_{K})$ to $(\mu_{V},\lambda_{V})$. Let the knot $P(K)$ be the
image of $P$ under this diffeomorphism. We call the knot $P(K)$ the _satellite
knot_ with _pattern_ $P$ and _companion_ $K$.

If we have a sequence of patterns $P_{1},\ldots,P_{k}$, we can apply them
iteratively to a companion knot $K$ to obtain the knot, $P_{1}\circ\ldots\circ
P_{k}(K)$. If we draw this knot in ${S}^{3}$ in the conventional way, we see
that the leftmost pattern in the composition notation, $P_{1}$, appears to be
the innermost ‘shape’ in the diagram of the satellite knot. Conversely, the
companion, $K$, appears to be the outermost ‘shape’.

######  Definition 2.6.

Given a pattern $P$, let $P_{n}$ denote the pattern obtained by taking the
image of $P$ under the self-diffeomorphism of $V$ which twists a single
${D}^{2}$ factor $n$ times in the direction specified by the orientation of
$\mu_{V}$.

######  Definition 2.7.

Given a pattern $P$, let $\overline{P}$ denote the pattern obtained by
reversing the orientation of $V$, $P$, $\lambda_{V}$, and $\mu_{V}$. We
formally define the new orientation to be outward-normal-first just as $P$
was. This is equivalent to reversing all the crossings and orientation arrows
in a diagram for $P$.

######  Definition 2.8.

Given a pattern $P$, let the same symbol, $P$, also denote the knot $P(U)$ in
appropriate contexts. For example, if $Q$ is another pattern then $Q(P)$
denotes the knot $Q\circ P(U)$.

######  Definition 2.9.

Given a knot $K\subset{S}^{3}$, let $K_{\\#}$, called ‘K pound’, denote the
unique wrapping number one pattern such that $K(U)=K$.

We often combine this definition with the previous one, so that for a general
pattern, $P$, $P_{\\#}$ is the unique wrapping number one pattern such that
$P_{\\#}(U)=P=P(U)$. The action of a pound pattern is to take the connected
sum of the companion knot with $P(U)$, which explains the notation.

######  Definition 2.10.

Given a dualizable pattern $P$, let $P^{*}$ denote the dual pattern with
outward normal first orientation (as usual).

######  Definition 2.11.

Given a pattern $P$, let $P^{n}$ denotes the pattern $P$ composed with itself
$n$ times.

######  Definition 2.12.

Given a dualizable pattern $P$, let $P^{-1}$ denote the pattern
$\overline{P^{*}}$. The name comes from the fact that $P^{-1}(P)$ and
$P(P^{-1})$ are always concordant to the unknot.

Since we often wish to combine one or more of these operations, we adopt the
following conventions concerning their order in the absence of parentheses:

  1. (1)

$P^{*}_{n}=(P^{*})_{n}$ not $(P_{n})^{*}$ (star goes inside twisting)

  2. (2)

$\overline{P}_{n}=(\overline{P})_{n}$ not $\overline{(P_{n})}$ (bar goes
inside twisting)

  3. (3)

$P_{n}^{m}=(P^{m})_{n}$ not $(P_{n})^{m}$ (composition goes inside twisting)

  4. (4)

$P^{*}_{\\#}=(P^{*})_{\\#}$ not $(P_{\\#})^{*}$ (star goes inside pound)

  5. (5)

$\overline{P}_{\\#}=(\overline{P})_{\\#}$ not $\overline{(P_{\\#})}$ (bar goes
inside pound)

  6. (6)

$P_{\\#}^{m}=(P^{m})_{\\#}$ not $(P_{\\#})^{m}$ (composition goes inside
pound)

  7. (7)

These conventions are chosen so that if a pattern is written with a sequence
of superscripts and a sequence of subscripts, then the operations should be
applied in the order: superscripts from left to right followed by subscripts
from left to right (bar counts as the leftmost superscript!).

Now that we have clearly established the notation we give several relations,
all of which are proven, if not in this notation, in [MP18].

######  Proposition 2.13.

The following properties hold for all dualizable patterns $P,Q$, knots $K,J$,
and $n,m\in\mathbb{Z}$:

  1. (i)

$K_{\\#}(J)=K\mathbin{\\#}J=J\mathbin{\\#}K=J_{\\#}(K)$,
$P(K)=P(K_{\\#}(U))=(P\circ K_{\\#})(U)=P\circ K_{\\#}$

  2. (ii)

$(P^{*})^{*}=P$, $(P_{n})_{m}=P_{(n+m)}$, $P=P_{0}$,
$\overline{(\overline{P})}=P$

  3. (iii)

$P^{*}_{n}=(P^{*})_{n}=(P_{-n})^{*}$, $\overline{(P_{n})}=(\overline{P})_{-n}$

  4. (iv)

$P^{-1}=\overline{(P^{*})}=\overline{P}^{*}$

  5. (v)

$(P\circ P^{-1})(K)\sim K\sim(P^{-1}\circ P)(K)$

  6. (vi)

$(P\circ Q)^{*}=Q^{*}\circ P^{*}$

  7. (vii)

$(P\circ Q)_{n}=(P_{n}\circ Q_{n})$

  8. (viii)

If the wrapping number of $P$ is one, then:

$P^{*}=P,\hskip 14.22636ptP^{-1}=\overline{P},\hskip 14.22636ptP_{n}=P,\hskip
14.22636ptP=P(U)_{\\#},\hskip 14.22636ptP(K)=P(U)\mathbin{\\#}K$

This proposition shows that conventions (3) and (5) are actually unnecessary
since the operations they govern commute. Nevertheless, we include these
conventions for completeness.

####  2.2.2. Traces and Surgeries

######  Definition 2.14.

Let $K\subset{S}^{3}$ be an oriented knot, then let $X_{n}(K):=h^{0}_{4}\cup
h^{2}_{4}(K,n)$, the union of a 4-dimensional 0-handle and a 4-dimensional
2-handle attached along the knot $K$ with $n$-framing. We denote the boundary
of $X_{n}(K)$ in the usual way by ${S}^{3}_{n}(K)$.

######  Theorem 2.15.

$X_{0}(P)\cong X_{0}(P^{*})$ for any dualizable pattern $P$. It follows that
${S}^{3}_{0}(P)\cong{S}^{3}_{0}(P^{*})$.

The proof consists of checking that both three manifolds can be obtained as
fillings of $V_{P}$ and $V^{*}_{P^{*}}$ respectively, and that $*$ extends to
these fillings. In fact this diffeomorphism extends to a diffeomorphism of the
4-dimensional traces. Understanding this is a good exercise for the reader,
and the details are written in [MP18]. Once the reader understands the
argument for $n=0$, it should be reasonably clear how to adjust the result to
arbitrary $n$. The stronger theorem is:

######  Theorem 2.16.

$X_{n}(P)\cong X_{n}(P^{*}_{n})$ for any dualizable pattern $P$ and
$n\in\mathbb{Z}$. Thus ${S}^{3}_{n}(P)\cong{S}^{3}_{n}(P^{*}_{n})$.

The the reader may be surprised that the left hand side is not
$X_{n}(P^{*}_{-n})$. The trick is that the sign of $n$ changes twice during
the natural process by which we construct the diffeomorphism. The first change
comes from the fact that $*:V_{P}\to V^{*}_{P^{*}}$ reverses the orientations
of the meridians but not the longitudes, which sends the framing $n\mapsto-n$.
This shows $X_{n}(P)\cong X_{n}((P_{-n})^{*})$. The second sign change comes
from reversing the order of the twisting and the dualization.

Already, it should be clear that combining Proposition 2.4, Proposition 2.13,
and Theorem 2.16 gives a formidable repertoire of tools with which we can
construct pairs of knots $(K,J)$ such that $X_{n}(K)\cong X_{n}(J)$ for some
fixed $n\in\mathbb{Z}$. When $(K,J)$ share this property, we will call $J$ an
_n-retrace_ of $K$, and the act of constructing $J$ from $K$ will be called
_n-retracing_ $K$.

####  2.2.3. Compatible Invariants

######  Definition 2.17.

Let $\mathscr{I}$ be an knot invariant taking values in an Abelian group
$\mathcal{A}$ (usually $\mathbb{Z}$ in practice). We say $\mathscr{I}$ is
_compatible_ with dualizable patterns iff

  1. (1)

$\mathscr{I}:\mathcal{C}\to\mathcal{A}$ is a well-defined homomorphism from
the smooth (or TOP) concordance group.

  2. (2)

If $X_{0}(K)\cong X_{0}(J)$ then $\mathscr{I}(K)=\mathscr{I}(J)$.

for any two knots $K,J\subset{S}^{3}$.

The point is that these two properties combine to give a satellite formula for
$\mathscr{I}$ whenever the pattern used in the satellite construction is
dualizable.

######  Theorem 2.18.

Let $\mathscr{I}$ be a knot invariant which is compatible with dualizable
patterns then for any dualizable pattern $P$ and companion knot $K$ we have,

$\mathscr{I}(P_{n}(K))=\mathscr{I}(K)+\mathscr{I}(P_{n})$

in particular for $n=0$ we have,

$\mathscr{I}(P(K))=\mathscr{I}(K)+\mathscr{I}(P)$

where the pattern with no input should be interpreted as the pattern applied
to the unknot.

###### Proof.

Assume the hypotheses of the theorem. We begin by calculating:

$\displaystyle X_{0}(P_{n}(K))$ $\displaystyle\cong X_{0}(P_{n}\circ K_{\\#})$
$\displaystyle\cong X_{0}((P_{n}\circ K_{\\#})^{*})$ $\displaystyle\cong
X_{0}(K_{\\#}^{*}\circ(P_{n})^{*})$ $\displaystyle\cong
X_{0}(K_{\\#}\circ(P_{n})^{*})$ $\displaystyle\cong
X_{0}(K\mathbin{\\#}(P_{n})^{*})$

therefore $K\mathbin{\\#}(P_{n})^{*}$ is a 0-retrace of $P_{n}(K)$, and it
follows from the two properties included in the definition of compatibility
that,

$\mathscr{I}(P_{n}(K))=\mathscr{I}(K\mathbin{\\#}(P_{n})^{*})=\mathscr{I}(K)+\mathscr{I}((P_{n})^{*})={\mathscr{I}}(K)+{\mathscr{I}}(P_{n})$

The last equality comes from the fact that $X_{0}((P_{n})^{*})\cong
X_{0}(P_{n})$ so ${\mathscr{I}}((P_{n})^{*})={\mathscr{I}}(P_{n})$. Taking
$n=0$ recovers the second statement in the theorem. ∎

So when is an invariant compatible with dualizable patterns? The best known
example of such an invariant is the classical signature and, more generally,
the Levine-Tristram signature at a root of unity, $\sigma_{K}(\omega)$. Both
are compatible with dualizable patterns, which can be easily deduced from two
of their many, equivalent definitions. That both are concordance homomorphisms
comes from defining them as the signatures of certain branched covers of the
knot $K$ with coefficients computed in $\mathbb{Q}[\omega]$. One checks that
connect-summing the knots corresponds to connect-summing the covers and then
quotes Novikov additivity for signatures. That both are invariant under
0-retracing follows from defining the signature in terms of the
$\omega$-twisted cohomology of ${S}^{3}_{0}(K)$. The details can be found in
the excellent survey article [Con21].

We end the section by encouraging the reader to make use of the one-page cheat
sheet (cheatsheet) which we have appended to this paper. It summarizes the
whole calculus of dualizable patterns including the construction and
dualization process. We recommend the reader to separate it from the rest of
the paper and use it as a side-by-side reference to follow the calculations in
the coming sections.

##  3\. Shake Genus and Complexity Bounds

###  3.1. Definitions of the Shake-Genus and Complexity

######  Definition 3.1.

We define the _$n$ -shake-genus_ of a knot $K$, denoted $g^{sh}_{n}(K)$, to be
the minimum genus over all orientable surfaces embedded in $X_{n}(K)$
representing the canonical generator of $H_{2}(K,\mathbb{Z})$ (we assume $K$
is oriented). A knot is _$n$ -shake-slice_ iff its $n$-shake-genus is zero.

This definition is well known already in the literature and leads to other
definitions in a similar spirit such as $n$-shake-concordance, see [CR16].

######  Definition 3.2.

Let $K$ be an $n$-shake-slice knot. We define the _$n$ -complexity_ of $K$,
denoted by $c(K,n)$ or just $c(K)$ when $n$ is understood from the context, to
be the minimum non-negative integer, $c$, satisfying,

$\mathchoice{\left\lvert
S\cap\operatorname{cocore}(h^{2}_{4}(K,n))\right\rvert}{\lvert
S\cap\operatorname{cocore}(h^{2}_{4}(K,n))\rvert}{\lvert
S\cap\operatorname{cocore}(h^{2}_{4}(K,n))\rvert}{\lvert
S\cap\operatorname{cocore}(h^{2}_{4}(K,n))\rvert}=1+2c$

occurring among all smoothly embedded spheres, $S$, generating the second
homology of $X_{n}(K)$. We can also distinguish between whether the sphere $S$
is smoothly embedded or topologically-locally-flatly embedded, which allows us
to define smooth and topological complexity respectively.

We note that this definition is a renormalization of the _n-shaking-number_
[FMN+21]. Since the oriented intersection number $S\cdot{D}^{2}=1$ for all $S$
we see that the extra intersections must occur in oppositely oriented pairs
and $c$ is normalized to count the number of pairs. This definition was
inspired by the similar notion of complexity for h-cobordisms defined in
[MS99].

###  3.2. 4-Genus Bounds Complexity

######  Proposition 3.3.

[FMN+21][Prop. 8.8] Let $K$ be an $n$-shake-slice knot then $g^{4}(K)\leq
c(K)$. This holds both in the smooth and topological setting.

###### Proof.

The proof is identical in both the smooth and topological setting so pick one
without loss of generality. Let $K$ be as in the proposition and let $S$ be an
embedded sphere realizing the minimal complexity. Assume to the contrary that
$c(K)<g(K)$. We will use $S$ to construct a surface $\Sigma\subset{B}^{4}$
with boundary $K$ and $g(\Sigma)=c(K)$ thereby creating a contradiction. By
working in a chart which contains the 2-handle of $X_{n}(K)$ in its interior
and applying general position to $S$ and the cocore of the 2-handle, we can
assume that after a small perturbation $S$ intersects the 2-handle in
$2c(K)+1$ parallel copies of the core. The homological condition on $S$
implies that exactly $c(K)+1$ of these parallel copies must be oriented
coherently with the core and the remaining $c(K)$ copies must be oriented
oppositely to the core so that the oriented intersection number
$S\cdot\operatorname{cocore}(h^{2}_{4}(K,n))=1$. If we consider the boundary
of the 4-ball to which the 2-handle is attached, we observe that the parallel
copies of the core intersect the boundary of the 4-ball in parallel copies of
the knot, twisted so that the linking number of each with the knot is $\pm n$
depending on the orientation of the copy (as unoriented parallels they all
twist with the same handedness). Let this link be called
$K^{c}_{n}\subset{S}^{3}$. Consider $S\cap{B}^{4}$. We know $S\cap
h^{2}_{4}(K,n)$ is a disjoint union of $2c(K)+1$ disks so it follows that
$S\cap{B}^{4}$ is a $(2c(K)+1)$-times punctured sphere whose boundary is
$K^{c}_{n}$. We can pair up all but one of the components of $K^{n}_{c}$ into
oppositely oriented pairs, with each pair bounding a disjoint annulus $A_{i}$
for $1\leq i\leq c(K)$ in the complement of the link. This is clear from
Figure 7. Let $\Sigma:=(S\cap{B}^{4})\cup(\bigcup_{i}A_{i})$. It is
immediately clear that $\partial\Sigma=K$, that $\Sigma\subset{B}^{4}$, and
that $g^{4}(\Sigma)=c(K)$.

2pt $h^{2}_{4}(K,n)$ [bl] at 123 154 ${S}^{2}$ [tl] at 133 69 $h^{0}_{4}\
(={B}^{4})$ [ ] at 28 34 $\leftarrow X_{n}(K)\to$ [ ] at 182 130 $n$ [tl] at
345 19 $A_{1}$ [br] at 227 152

Figure 7. Left: a schematic of the perturbed sphere, indicating its
intersection with the 2-handle. Right: the link $K^{0}_{1}$ for $K=4_{1}$.
Notice that the green components of $K^{0}_{1}$ are coherently oriented while
the orange are oppositely oriented. Clearly we could pair up the innermost
green component with the orange component and draw the obvious oriented band
which they cobound in the diagram. This is $A_{1}$ for this example.

∎

###  3.3. A Sufficient Condition for High Complexity

We now give a general process for constructing $n$-shake-slice knots of
arbitrary complexity using dualizable patterns and a compatible invariant.

######  Theorem 3.4.

Let ${\mathscr{I}}$ be an integer-valued compatible invariant whose magnitude
provides a lower bound on the 4-genus of a knot. Let $Q$ be a dualizable
pattern, let $n\in
Z\setminus\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}$, and let
$c\in\mathbb{N}$. Assume:

${\mathscr{I}}(Q_{n})-{\mathscr{I}}(Q)\neq 0$

It follows that the knot $K$ defined by,

$K:=(\overline{Q}^{*}_{n})^{c}\circ Q^{c}$

is smoothly $n$-shake-slice, and has smooth or topological complexity at least
$c$ depending on whether ${\mathscr{I}}$ bounds the smooth or topological
4-genus.

###### Proof.

Let $K$ be given satisfying the hypotheses of the theorem. We will establish
that $K$ is $n$-shake-slice by showing that it is an $n$-retracing of a slice
knot. We will bound the complexity of $K$ by showing $|{\mathscr{I}}(K)|\geq
c$ and then quoting Proposition 3.3. Clearly the knot
$\overline{(Q^{c})}\mathbin{\\#}Q^{c}$ is smoothly slice since it is the
connected sum of a knot with its mirror. Using the calculus of dualizable
patterns we show that $K$ is an $n$-retracing of it.

$\displaystyle X_{n}(\overline{(Q^{c})}\mathbin{\\#}Q^{c})$
$\displaystyle\cong X_{n}(Q^{c}_{\\#}\circ\overline{Q}^{c})$
$\displaystyle\cong X_{n}((Q^{c}_{\\#}\circ\overline{Q}^{c})^{*}_{n})$
$\displaystyle\cong X_{n}((\overline{Q}^{c\,*}\circ(Q^{c}_{\\#})^{*})_{n})$
$\displaystyle\cong X_{n}((\overline{Q}^{*\,c}\circ Q^{c}_{\\#})_{n})$
$\displaystyle\cong X_{n}(\overline{Q}^{*\,c}_{n}\circ(Q^{c}_{\\#})_{n})$
$\displaystyle\cong X_{n}((\overline{Q}^{*}_{n})^{c}\circ Q^{c})$
$\displaystyle\cong X_{n}(K)$

Clearly $X_{n}(\overline{(Q^{c})}\mathbin{\\#}Q^{c})$ has a sphere generating
its second homology group, which is given by capping off a slice disk for the
knot with the core of the 2-handle. Taking the image of this sphere under the
diffeomorphism to $X_{n}(K)$ shows that $K$ is $n$-shake-slice. Next we
compute:

$\displaystyle{\mathscr{I}}((\overline{Q}^{*}_{n})^{c}\circ Q^{c})$
$\displaystyle={\mathscr{I}}((\overline{Q}^{*}_{n})^{c})+{\mathscr{I}}(Q^{c})$
$\displaystyle=c{\mathscr{I}}(\overline{Q}^{*}_{n})+c{\mathscr{I}}(Q)$
$\displaystyle=c{\mathscr{I}}((\overline{Q}_{-n})^{*})+c{\mathscr{I}}(Q)$
$\displaystyle=c{\mathscr{I}}(\overline{Q}_{-n})+c{\mathscr{I}}(Q)$
$\displaystyle=c{\mathscr{I}}(\overline{(Q_{n})})+c{\mathscr{I}}(Q)$
$\displaystyle=-c{\mathscr{I}}(Q_{n})+c{\mathscr{I}}(Q)$
$\displaystyle=c({\mathscr{I}}(Q)-{\mathscr{I}}(Q_{n}))$
$\displaystyle\implies|{\mathscr{I}}(K)|$
$\displaystyle=c|{\mathscr{I}}(Q)-{\mathscr{I}}(Q_{n})|\geq c$

and it follows from Proposition 3.3 that $c\leq c(K)$. ∎

Our main result will follow from taking ${\mathscr{I}}$ to be the Levine-
Tristram signature and the pattern $Q$ to be $P_{1}$ where $P$ is the pattern
from the first example in section 2. The rest of the paper is concerned with
verifying the condition that ${\mathscr{I}}(Q_{n})-{\mathscr{I}}(Q)\neq 0$.

##  4\. A Theorem about Classical Signatures

We digress for a moment in this section to prove the following theorem about
the classical signature. We will apply it in the next section to verify the
conditions of Theorem 3.4 for our pattern $Q:=P_{1}$ and $n$ odd. The case of
$n$ even and non-zero will be handled by a different method.

######  Theorem 4.1.

Let $K$ be a knot, and let $U$ be an unknot embedded in the complement of $K$
such that the oriented linking number $lk(K,U)=1$. Let $K_{n}$ be the result
of applying $n$ full twists to all the strands of $K$ which pass through a
spanning disk for $U$. For all $n\in\mathbb{Z}$, $\sigma(K)=\sigma(K_{2})$, so
the set $\mathchoice{\left\\{\sigma(K_{n})\mid
n\in\mathbb{Z}\right\\}}{\\{\sigma(K_{n})\mid
n\in\mathbb{Z}\\}}{\\{\sigma(K_{n})\mid
n\in\mathbb{Z}\\}}{\\{\sigma(K_{n})\mid n\in\mathbb{Z}\\}}$ contains at most
two values.

The proof will proceed by using the method of Goeritz and Trotter, as
described in [GL78], to calculate the classical signature of $K$ from a non-
orientable spanning surface. We will show that there is a canonical way to
convert a Seifert surface for $K$ into a non-orientable spanning surface for
$K_{2}$. Moreover, the Goeritz form for $K_{2}$ can be computed from the
Seifert form for $K$ and then diagonalized in full generality. The correction
term can be easily computed and we will show that the correction term and
change of signature offset each other, which proves the result.

###  4.1. Goeritz Forms and the Correction Term

Let $K$ be a knot and let $F$ be a spanning surface, possibly non-orientable,
for $K$. Let $x_{1},\ldots,x_{n}$ be an oriented basis for
$H_{1}(F,\mathbb{Z})$ and let $x^{\tau}_{i}$ denote the double cover of
$x_{i}$ obtained by locally parametrizing the normal bundle as $F\times[-1,1]$
and lifting $x_{i}$ to
$x_{i}\times\mathchoice{\left\\{-1\right\\}}{\\{-1\\}}{\\{-1\\}}{\\{-1\\}}\cup
x_{i}\times\mathchoice{\left\\{+1\right\\}}{\\{+1\\}}{\\{+1\\}}{\\{+1\\}}$ in
each chart. If $x_{i}$ is an orientation preserving loop in $F$ then
$x_{i}^{\tau}$ is the disjoint union of two parallel copies of $x_{i}$, one on
either side of $F$, and both oriented coherently with $x_{i}$. If $x_{i}$ is
an orientation reversing loop then $x_{i}^{\tau}$ is a single circle which
wraps twice around $x_{i}$ just as the boundary of a Mobius band wraps around
its central circle, and which is oriented coherently with $x_{i}$.

We define the Goeritz matrix associated to the pair $(K,F)$ to be the form
$\mathscr{G}_{ij}:=lk(x_{i},x_{j}^{\tau})$. We observe that this form recovers
the symmetrized Seifert form if $F$ is orientable since,

$lk(x_{i},x_{j}^{\tau})=lk(x_{i},x_{j}^{+})+lk(x_{i},x_{j}^{-})=lk(x_{i},x_{j}^{+})+lk(x_{j},x_{i}^{+})$

For a proof that $\mathscr{G}$ is symmetric when $F$ is non-orientable see
[GL78].

We define the correction term associated to $(K,F)$ as follows. Let $K_{F}$
denote a coherently oriented parallel copy of $K$ which is chosen to be
disjoint from $F$, and observe that there is a unique such choice up to
isotopies supported in a neighborhood of $K$. The correction term associated
to $(K,F)$ is defined by $\eta:=\frac{1}{2}lk(K,K^{\prime})$ which is always
an integer. The central theorem of [GL78] is:

######  Theorem 4.2.

Let $K$ be a knot in ${S}^{3}$ and let $F$ be a possibly non-orientable
spanning surface for $K$ with Goeritz form $\mathscr{G}$ and correction term
$\eta$. It follows that the classical knot signature of $K$ can be computed
from $F$ by the formula:

$\sigma(K)=\operatorname{sign}(\mathscr{G})-\eta$

We give the following example to demonstrate the practical utility of this
theorem. The $(2,2n+1)$-torus knot has an obvious spanning surface
homeomorphic to the Mobius band. The Goeritz form associated to this surface
is the one-by-one form $[2n+1]$ and the correction term is also $2n+1$,
therefore,

$\sigma(T_{2,2n+1})=\operatorname{sign}([2n+1])-(2n+1)=1-2n-1=\boxed{-2n}$

The amazing brevity of this calculation will not be lost on those who have
tried to compute the signature of torus knots using orientable spanning
surfaces.

The most practical way to approach this calculation for a specific knot $K$ is
to perform an isotopy of the pair $(K,F)$ in ${S}^{3}$ to make $F$ look like a
small disk in the plane with several bands attached. Of course the bands will
twist and link with each other for a generic knot, but we can arrange them so
that they lie in the same plane as the disk, except when one of them passes
over another, or when one twists around its core arc. We label the bands
$B_{1},\ldots,B_{2n},N_{1},\ldots,N_{m}$ so that the $B_{*}$ are the
orientable bands and the $N_{*}$ are the non-orientable bands. Each band
yields a generator of $H_{1}(F,\mathbb{Z})$, obtained by orienting the core
arc of the band and connecting the ends of the arc with a small, boundary-
parallel arc in the disk. The orientation of the core arc can be chosen
arbitrarily for the orientable bands, but for the non-orientable bands it is
convenient to choose the orientation to be coherent with the orientation of
the edges of the band (which both point in the same direction because the band
is non-orientable!). These generators are called
$b_{1},\ldots,b_{2n},n_{1},\ldots,n_{m}$ respectively and form a basis. We can
compute the Goeritz form in this basis easily by defining the writhe, linking
number, and twisting number of the bands then expressing the entries of the
Goeritz form in terms of these numbers.

######  Definition 4.3.

Let $F\subset{S}^{3}$ be a connected, possibly non-orientable surface with a
single boundary component, $K$. Assume that $F$ is embedded as a flat disk
with bands, $B_{1},\ldots,B_{2n},N_{1},\ldots,N_{m}$, attached to it and
assume that all bands come with oriented core arcs as in the previous
paragraph. We give the following definitions for any bands $X_{i},X_{j}$ of
$F$:

  1. (1)

The _writhe_ of $X_{i}$, denoted $W(X_{i})$, is twice the signed count of
self-crossings of the band $X_{i}$ where the sign is determined by the sign of
the induced self-crossing of the core arc of $X_{i}$.

  2. (2)

The _twisting number_ of $X_{i}$, denoted $T(X_{i})$ is the signed count of
half twists in the band relative to the plane. The sign is determined by the
handedness of the twisting as one travels along the core arc (note: this is
independent of the orientation of the core arc!).

  3. (3)

The _linking number_ of $X_{i}$ with $X_{j}$, denoted $Lk(X_{i},X_{j})$, is
the signed count of crossings between the bands $X_{i}$ and $X_{j}$ with the
signs determined by the orientations of their cores.

The linking number of $X_{i}$ with itself is defined by
$Lk(X_{i},X_{i}):=W(X_{i})+T(X_{i})$. Note that this is twice what the reader
might expect if we compare it to the standard definition of the linking number
between knots.

These definitions yield the following:

######  Proposition 4.4.

In the setting of the previous definition, the Goeritz form and correction
term associated to the pair $(K,F)$ can be computed by the following formulas:

$\mathscr{G}=[Lk(X_{i},X_{j})]_{i,j=1}^{2n+m}\hskip
56.9055pt\eta=\sum_{i,j=1}^{n}Lk(N_{i},N_{j})$

In particular, one can compute $\eta$ directly from $\mathscr{G}$ by summing
all the entries in $\mathscr{G}$ which correspond to a pair of non-orientable
bands.

###### Proof.

2pt $K$ [b] at 222 146 $K$ [l] at 116 146 $x_{j}$ [bl] at 145 198 $x_{i}$ [br]
at 12 193 $x^{\tau}_{i}$ [l] at 121 59 $x^{\tau}_{i}$ [l] at 109 39
$x^{\tau}_{j}$ [t] at 28 39 $x^{\tau}_{j}$ [t] at 39 27 $x^{\tau}_{j}$ [b] at
266 162 $x^{\tau}_{j}$ [b] at 284 161 $x_{i}$ [t] at 277 71 $x_{j}$ [r] at 157
63 $K$ [] at 392 188 $x^{\tau}_{i}$ [b] at 422 206 $x^{\tau}_{i}$ [b] at 442
198 $x_{i}$ [t] at 336 39 (A) [ ] at 78 200 (B) [ ] at 229 200 (C) [ ] at 376
200

Figure 8. (A) bands crossing each other, (B) arcs and their push-offs meeting
in the disk, (C) a band with a half twist. The cores of the bands are orange,
the push-offs of the cores in green. In (A), (C) red dots mark green-orange
crossings, and purple dots mark orange-orange and blue-blue crossings
respectively.

We will rely on the combinatorial definition of linking number in ${S}^{3}$
which says that if $L,L^{\prime}$ are disjoint, oriented links then we can
compute the linking number $lk(L,L^{\prime})$ as half the sum of the signed
count of crossings between the components of $L$ and the components of
$L^{\prime}$. We use links here because, in the case of a generator $b_{i}$,
corresponding to an orientable band, we see $b_{i}^{\tau}$ is a disjoint union
of two coherently oriented, parallel circles. We begin by verifying the
formula for the Goeritz form in the basis given by the oriented cores of the
bands union small arcs in the disk. By definition,
$\mathscr{G}_{ij}=lk(x_{i},x_{j}^{\tau})$ so we must find a way to
systematically locate and count the crossings between these two links. There
are two different types of crossing. The first occurs when the band $X_{i}$
passes over/under the band $X_{j}$, Figure 8 (A), in which case both strands
of $x_{j}^{\tau}$ pass over/under $x_{i}$: this contributes $\pm 2$ to the
signed count of crossings between them and is exactly double the amount which
it contributes to the computation of $Lk(X_{i},X_{j})$. In places where the
band $X_{i}$ crosses itself this same analysis holds, and each full twist in
the band contributes $\pm 4$ to the count of crossings between $x_{i}$ and
$x_{i}^{\tau}$. It follows that the contributions to the computation of
$lk(x_{i},x_{j}^{\tau})$ given by the bands is exactly double the contribution
given to $Lk(X_{i},X_{j})$. What remains is to understand the contribution
given by the points in the central disk of $F$ where the curves $x_{i}$ and
$x_{j}$ intersect each other on the surface $F$. By examining the local
picture of $x_{i}$ and $x_{j}^{\tau}$ near such an intersection, Figure 8 (B),
we see that the two crossings always come with opposite signs and so the
contribution is zero. Thus we can always assume both links have no crossings
on the disk during the computation of $lk(x_{i},x_{i}^{\tau})$. It follows
that $lk(x_{i},x_{j}^{\tau})=Lk(X_{i},X_{j})=\mathscr{G}_{ij}$.

Next we analyze the correction term. Since $K^{\prime}$ is disjoint from $F$,
it follows that $K^{\prime}$ is isotopic to the Seifert framed parallel
$K^{\prime\prime}$ as $K^{\prime}$ runs around the base disk and the
orientable bands. This means the signed count of the crossings between $K$ and
$K^{\prime}$ will agree with the count between $K$ and $K^{\prime\prime}$ near
the disk and orientable bands. This can be easily seen to be zero since we can
clearly surger out the non-orientable bands from $(K,F)$ which makes $F$ into
an orientable spanning surface and $K^{\prime}$ into a disjoint parallel which
must therefore have linking number zero with the new knot. It follows that
$lk(K,K^{\prime})$ can be computed entirely from counting crossings between
$K^{\prime}$ and the boundaries of the non-orientable bands. Where two non-
orientable bands cross, the count changes by $\pm 4$ and therefore the linking
number changes by $\pm 2$ and the correction term changes by $\pm 1$. This
change is exactly the contribution the crossing makes to the linking number
between the two bands. Wherever the band has a half twist, each of the strands
of $K^{\prime}$ on either side of the band must cross over the band which adds
four crossings of the same sign, so that the count changes again by $\pm 4$
and thus the correction term changes by $\pm 1$, Figure 8 (C). Again, this is
exactly the contribution which the half twist makes to the writhe of the band
and therefore to the band’s linking number with itself. It follows that the
correction term is exactly the sum of the entries of $\mathscr{G}$ which
correspond to pairs of non-orientable bands.

∎

###  4.2. Adding Pairs of Twists

We will need the following lemma:

######  Lemma 4.5.

Let $K\subset Y^{3}$ be a null-homologous knot in an oriented 3-manifold. Let
$D\subset Y^{3}$ be an embedded disk with oriented boundary $U$ such that
$U\cdot K=+1$ and therefore,

$\mathchoice{\left\lvert K\cap D\right\rvert}{\lvert K\cap D\rvert}{\lvert
K\cap D\rvert}{\lvert K\cap D\rvert}=1+2c$

for some
$c\in\mathbb{N}\cup\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}$.
It follows that there exists an orientable spanning surface $F\subset Y^{3}$
for $K$ such that $F\cap D$ consists of $c$ arcs on $D$ connecting oppositely
oriented intersection points between $K$ and $D$ and one arc connecting the
remaining positive intersection point to a point on $U=\partial D$. Thus the
intersection of $F$ with a neighborhood of $D$ looks like the upper left of
Figure 10.

2pt $F_{1}$ [l] at 37 69 $F$ [l] at 252 70 $K$ [tr] at 166 10 $K$ [tr] at 387
20 $K\cap F_{1}$ [bl] at 104 86

Figure 9. Resolving a ribbon intersection of a Seifert Surface.

###### Proof.

We can pair up the strands of $K$ which intersect $D$ into oppositely oriented
pairs, with one strand left over. We connect each pair of strands by an arc in
$D$ which connects their intersection points, ensuring that all the arcs are
disjoint. We then connect the final intersection point to the boundary $U$ by
an arc disjoint from the rest. Each of the $c+1$ arcs we have just defined can
be projected slightly out of the disk, one copy on each side. We can then
surger the knot by resolving each pair of strands which pass through the disk
using the pair of arcs which we have constructed. This transforms the knot
into a $(c+1)$-component oriented link and leaves one pair of projected arcs
unused: the pair which come from the arc connected to the boundary $U$.
Consider the corresponding ends of this pair which came from the point on the
arc which touches $U$. We can connect these two ends with a small arc that
goes around half a meridian to $U$ and is therefore disjoint from $D$. We can
then replace the portion of $K$ which passes through $D$ with this new arc we
have defined. The reader may recognize this as the result of doing a finger
move. This gives us an oriented link $L\subset Y^{3}\setminus\nu(D)$. Clearly
$L$ is also null-homologous in $Y^{3}\setminus\nu(D)$ and it follows that $L$
has a connected, oriented spanning surface $F_{0}\subset
Y^{3}\setminus\nu(D)$. We can recover $K$ from $L$ and simultaneously convert
$F_{0}$ into an immersed spanning surface for $K$ by attaching to $F$ the
obvious ribbons which undo the resolutions and the finger move we used to
produce $L$. Let the new, immersed, spanning surface for $K$ which we obtain
by this process be called $F_{1}$. Since $F_{1}$ was obtained from an embedded
surface by attaching ribbons, it follows that the only self-intersections of
$F_{1}$ with itself are ribbon intersections. These can be resolved according
to the local model shown in Figure 9. Let $F$ be the result of resolving the
self-intersections of $F_{1}$ in this manner. It now follows that $F$
satisfies the conditions of the lemma ∎

2pt $U\ (=\partial D)$ [bl] at 154 370 $F$ [ ] at 57 251 $K$ [b] at 206 284
$B_{i}$ [l] at 232 309 $B_{j}$ [l] at 222 365 $2$ [ ] at 119 120 $2$ [ ] at
447 243 $K_{+2}$ [l] at 225 86 $K_{+2}$ [b] at 613 173 $F_{+2}$ [ ] at 325 102
$n_{1}$ [tl] at 563 125 $B_{i}$ [l] at 629 211 $B_{j}$ [l] at 630 299 $N_{1}$
[b] at 488 357

Figure 10. Adding two left-handed twists to $K$ and modifying its spanning
surface.

We can now give the proof of Theorem 4.1:

###### Proof of Theorem 4.1.

Consider a local neighborhood of the spanning disk for $U$ and its
intersection with $K$. We see $U$ as a standard unknot and $2n+1$ strands of
$K$ passing through it, $n+1$ passing positively and $n$ passing negatively.
We can pair up each positive strand with a negative strand and we will have
one positive left over. We can assume without loss of generality that there is
a Seifert surface $F$ for $K$ such that each oppositely oriented pair of
strands passing through $U$ is the boundary of an orientable band of $F$ by
the lemma. We can further assume that the intersection between the disk and
$F$ is an arc connecting the left-over strand to $U$ and several disjoint
intervals corresponding to the bands. Now consider $K_{2}$. We can see from
Figure 10 that $K_{2}$ is equal to $K$ outside a neighborhood of the spanning
disk for $U$, and moreover there is a natural way to modify $F$ to obtain a
non-orientable spanning surface for $K_{2}$ which we will call $F_{2}$. Notice
that the orientable bands of $F_{2}$ correspond to the bands of $F$ in an
obvious way, while exactly one non-orientable band has been added.

We will compare the Goeritz form ${\mathscr{G}}$ of $F$ to the form
${\mathscr{G}}_{2}$ of $F_{2}$. Let’s order the bands of $F$ by putting the
ones which don’t pass through the spanning disk for $U$ first then those that
do. If $B,B^{\prime}$ are a bands with self-linking numbers $b,b^{\prime}$
which pass through the disk $l,l^{\prime}$ times, counted with sign, then the
self-linking of the corresponding bands of $F_{2}$ will be
$b+4l^{2},b+{4l^{\prime}}^{2}$ respectively. If $Lk(B,B^{\prime})=p$ then the
linking number of the corresponding bands in $F_{2}$ will be $p+4ll^{\prime}$.
From the diagram, we can see that the self-linking of the new, non-orientable
band, $N$, is $+1$ and of course its linking numbers with $B,B^{\prime}$ are
$2l,2l^{\prime}$. Thus the intersection of the three rows and columns
corresponding to $B,B^{\prime},N$ form the submatrix given on the left below.
We can diagonalize this submatrix using the pairs of row and column operations
indicated by the arrows below.

$\begin{pmatrix}b+4l^{2}&p+4ll^{\prime}&2l\\\
p+4l^{\prime}l&b^{\prime}+{4l^{\prime}}^{2}&2l^{\prime}\\\
2l&2l^{\prime}&1\end{pmatrix}\buildrel((2)-2l^{\prime}\cdot(3))\mapsto(2)\over{\leadsto}\begin{pmatrix}b+4l^{2}&p&2l\\\
p&b^{\prime}&0\\\
2l&0&+1\end{pmatrix}\buildrel((1)-2l\cdot(3))\mapsto(1)\over{\leadsto}\begin{pmatrix}b&p&0\\\
p&b^{\prime}&0\\\ 0&0&1\end{pmatrix}$

Since $B,B^{\prime}$ could have been any two bands from $F$ we see that this
sequence of operations applied to each disjoint pair of corresponding bands in
$F_{2}$ gives a matrix congruence from ${\mathscr{G}}_{2}$ to
${\mathscr{G}}\oplus[+1]$. It follows that
$\operatorname{sign}({\mathscr{G}}_{2})=\operatorname{sign}({\mathscr{G}})+1$.
Since there is only one non-orientable band on $F_{2}$, we see that the
correction term is its self-intersection which gives $\eta=+1$. Thus,

$\sigma(K_{2})=\operatorname{sign}({\mathscr{G}}_{2})-\eta=\operatorname{sign}({\mathscr{G}})+1-1=\operatorname{sign}({\mathscr{G}})+0=\sigma(K)$

The result now follows from the obvious inductive argument on $n$.

∎

##  5\. The Examples; Proof of the Main Theorem

###  5.1. The Diagrams

Consider the pattern $P$ defined in Figure 11.

2pt $P$ [ ] at 219 167 $P^{*}$ [ ] at 472 141 $\mu_{V}$ [t] at 413 37 $+2$ [ ]
at 455 60 $\mu_{V}$ [t] at 140 61

Figure 11. The patterns $P$ and $P^{*}$.

We computed $P^{*}$ earlier. We will analyze the Levine-Tristram signatures of
$P_{n}$ for $n>1$, although $P$ will not be the base pattern for the
construction of the knots. Rather we will use $Q:=P_{1}$ as the base pattern,
so that:

$Q_{n}=P_{1+n},\qquad Q^{*}_{n}=(P_{1})^{*}_{n}=P^{*}_{-1+n}$

We will show that this pattern $Q$ combined with the classical knot signature
satisfies the hypotheses of Theorem 3.4 for $n>0$ and odd.

######  Proposition 5.1.

Let $P$ be the pattern described in Figure 11 and let $Q:=P_{1}$. Letting
$k\in\mathbb{Z}$,

$\sigma(Q_{2k})=0,\qquad\sigma(Q_{1+2k})=2$

It follows that for $n=2k+1$,

$\sigma(Q_{n})-\sigma(Q)\neq 0$

and so the pair $(Q,\sigma)$ satisfy the hypotheses of Theorem 3.4 for all odd
integers $n$.

###### Proof.

2pt $1$ [br] at 49 134 $2$ [br] at 22 166 $3$ [br] at 87 192 $4$ [b] at 131
118 $Q_{1}=P_{2}$ [ ] at 227 164 $1$ [br] at 332 132 $2$ [br] at 304 167 $3$
[br] at 372 190 $4$ [b] at 388 109 $5$ [b] at 419 109 $6$ [b] at 452 106
$Q_{2}=P_{3}$ [ ] at 510 162

Figure 12. The Seifert forms for $Q$ and $Q_{1}$. The numbers correspond to
the row/column of the Seifert form which corresponds to that basis element.

We apply Theorem 4.1 to obtain:

$\sigma(Q_{2k})=\sigma(Q)\hskip 28.45274pt\sigma(Q_{1+2k})=\sigma(Q_{1})$

and so it will suffice to compute $\sigma(Q)$ and $\sigma(Q_{1})$. We will do
this explicitly by drawing Seifert surfaces for $Q$ and $Q_{1}$ then writing
down a Seifert form and computing the signature. We see from Figure 12 that
$Q$ and $Q_{1}$ have Seifert forms $A$ and $A_{1}$ respectively given by,

$A=\begin{pmatrix}1&1&1&0\\\ 0&0&1&-1\\\ 1&2&0&0\\\
0&0&-1&0\end{pmatrix}\hskip 28.45274ptA_{1}=\begin{pmatrix}1&1&1&0&0&0\\\
0&0&1&0&0&-1\\\ 1&2&0&0&0&0\\\ 0&0&-1&0&0&0\\\ 0&0&0&1&0&0\\\
0&0&0&0&1&0\end{pmatrix}$

after symmetrizing and computing signatures we see,

$\displaystyle\sigma(Q)=\sigma(A+A^{T})$
$\displaystyle=\sigma\begin{pmatrix}2&1&2&0\\\ 1&0&3&-1\\\ 2&3&0&-1\\\
0&-1&-1&0\end{pmatrix}=0$ $\displaystyle\sigma(Q_{1})=\sigma(A_{1}+A^{T}_{1})$
$\displaystyle=\sigma\begin{pmatrix}2&1&2&0&0&0\\\ 1&0&3&0&0&-1\\\
2&3&0&-1&0&0\\\ 0&0&-1&0&1&0\\\ 0&0&0&1&0&1\\\ 0&-1&0&0&1&0\end{pmatrix}=2$

which concludes the proof. ∎

This gives smoothly $n$-shake-slice knots with arbitrarily high topological
complexity for any odd $n$. We need the greater generality of the Levine-
Tristram signatures to tackle even $n$. The main thrust of our analysis will
be the following. We will construct a family of Seifert surfaces $F_{n}$ for
$P_{n}$ which yield Seifert forms $A_{n}$. We will be able to give a general
formula for the matrix $A_{n}$ and deduce a general formula for the
symmetrized Alexander polynomial $\Delta_{n}(t)$ of the knot $P_{n}$. The
Levine-Tristram signature is known to change exactly at those complex numbers
on the unit circle where the symmetrized Alexander polynomial passes through
zero. We will show explicitly that $\Delta_{1}(t)>0$ for all
$t\in{S}^{1}\subset\mathbb{C}$ and therefore $\sigma(Q,\omega)\equiv 0$
(recall $Q:=P_{1}$). Since $\sigma(-,\omega)$ is a compatible invariant for
fixed root of unity $\omega$, it will suffice to show that there exists a root
of unity $\omega_{n}$ depending on $n$ such that $\sigma(Q_{n},\omega_{n})\neq
0$. We know that the symmetrized Alexander polynomial of any knot has
evaluates to $+1$ at $t=+1$, and we will show that for any $n>1$ there is a
value on the unit circle at which $\Delta_{n}(t)$ is negative. This shows that
$\Delta_{n+1}(t)$ must pass through zero somewhere on the unit circle and so
there is some $\omega_{n}$ such that $\sigma(Q_{n},\omega_{n})\neq 0$ for any
fixed $n>0$. This means that the pair $(Q,\sigma(-,\omega_{n}))$ satisfies the
hypotheses of Theorem 3.4.

###  5.2. Adding Twists around Algebraically One Strand

We will now explain how to create a family of Seifert surfaces for $Q_{n}$
starting from a given Seifert surface for $Q_{-1}=P$. We use the following
extension of Lemma 4.5.

######  Lemma 5.2.

Let $K\cup U\subset{S}^{3}$ be as in Lemma 4.5, and let $K_{n}$ be as in
Theorem 4.1. Let $F$ be the Seifert surface for $K$ obtained from Lemma 4.5.
We may convert the surface $F$ into a new Seifert surface $F_{n}$ for $K_{n}$
by making a finite number of local modifications to $F$ as indicated in Figure
13.

2pt $F$ [ ] at 59 362 $K$ [b] at 231 420 $B_{i}$ [l] at 248 444 $B_{j}$ [l] at
248 492 $F^{\prime}$ [ ] at 323 362 $(-)$ [ ] at 382 484 $(+)$ [ ] at 418 484
$B_{j}$ [l] at 510 490 $B_{i}$ [l] at 510 444 $D$ [ ] at 72 251 $U$ [bl] at 87
293 $B_{i}$ [l] at 523 271 $K$ [b] at 500 240 $F$ [ ] at 72 182 $n$ [ ] at 286
166 $D$ [ ] at 73 95 $U$ [br] at 63 139 $(+)$ [ ] at 138 130 $(-)$ [ ] at 198
130 $(+)$ [ ] at 372 130 $(-)$ [ ] at 437 130 $K$ [ ] at 506 94 $B_{i}$ [l] at
527 116 $F^{\prime}$ [ ] at 65 33 $K$ [ ] at 495 427

Figure 13. The top two figures show how to locally modify the Seifert surface
$F$ without changing the knot $K$ into a surface $F^{\prime}$ which is
compatible with adding $\pm 1$ twist. Switching all the band-over-band
crossings on the $(\pm)$ side of the red box adds $\pm 1$ full twist to $K$.
Concatenating $n$ such modifications creates a Seifert surface for $K_{\pm
n}$. This is demonstrated in the lower two figures for a knot with one band
passing through $D$.

The proof of the lemma is obvious from the figure up to one small point. Since
we are encoding the two sides of the Seifert surface by color (Indigo/Orange),
the modification indicated in Figure 13 only applies when all the bands
passing through $U$ have the same side facing out of the page as the portion
of the surface which intersects $U$. This can always be achieved by putting a
Reidemeister II move in each band which has the wrong side facing outwards
(one of the new crossings goes on each side of the disk $D$ then we shrink the
neighborhood of $D$ so that neither crossing appears). Applying the lemma to
an appropriate Seifert surface for $P$ yields surfaces for $P_{n}$ and $Q_{n}$
for all values of $n$. These surfaces are given in Figure 14 and we call them
$F_{n}$. We orient the cores of all the handles of $F_{n}$ as shown then
connect the ends of the cores by arcs in the central disk of $F_{n}$. We
obtain a basis for $H_{1}(F_{n},\mathbb{Z})$ which we label
$b_{1},\ldots,b_{2n+2}$.

2pt $1$ [br] at 75 193 $2$ [br] at 39 239 $3$ [br] at 121 270 $4$ [br] at 113
137 $5$ [r] at 155 129 $2n+1$ [b] at 214 151 $2n+2$ [b] at 244 152
$P_{n}=Q_{n-1}$ [ ] at 312 236 $n$ [b] at 194 175 $F_{n}$ [ ] at 68 53

Figure 14. A Seifert surface, $F_{n}$, for $P_{n}$ with a labeled basis,
$(b_{1},\ldots,b_{2n+2})$, for $H_{1}(F_{n},\mathbb{Z})$.

We see from the diagram that the genus of $F_{n}$ is $n+1$. The associated
Seifert form, $A_{n}:=[lk(b_{i},b_{j}^{+})]$, is thus a $(2n+2)\times(2n+2)$
matrix for $n\geq 1$. We see by inspecting the diagram that $A_{n}$ has the
following form:

$A_{n}=\left(\begin{array}[]{cccc|ccccccccc|c}1&1&1&0&0&0&0&0&\cdots&0&0&0&0&0\\\
0&0&1&0&0&0&0&0&\cdots&0&0&0&0&-1\\\ 1&2&0&0&0&0&0&0&\cdots&0&0&0&0&0\\\
0&0&-1&0&0&0&0&0&\cdots&0&0&0&0&0\\\ \hline\cr
0&0&0&1&0&0&0&0&\cdots&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0&\cdots&0&0&0&0&0\\\
0&0&0&0&0&1&0&0&\cdots&0&0&0&0&0\\\ 0&0&0&0&0&0&1&0&\cdots&0&0&0&0&0\\\
0&0&0&0&0&0&0&1&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&0&0&0&0&0\\\
0&0&0&0&0&0&0&\cdots&0&1&0&0&0&0\\\ 0&0&0&0&0&0&0&\cdots&0&0&1&0&0&0\\\
0&0&0&0&0&0&0&\cdots&0&0&0&1&0&0\\\ \hline\cr
0&0&0&0&0&0&0&\cdots&0&0&0&0&1&0\\\ \end{array}\right)$

which we describe in terms of the nine sub-matrices indicated in the figure:

  * —

The central block of $A_{n}$ is a $(2n-3)\times(2n-3)$ matrix consisting of
ones down the diagonal connecting the $(2,1)$-position to the
$(2n-3,2n-4)$-position and zeroes elsewhere (the positions are given relative
to the central block not $A_{n}$).

  * —

The top and right edge blocks consist of zeroes everywhere.

  * —

The bottom and left edge blocks have a $1$ in their upper-rightmost entry and
zeroes elsewhere.

  * —

The corner blocks are independent of $n$.

Note that the dimensions of the edge blocks are scaled appropriately to the
center. The correctness of this description of $A_{n}$ may take a moment to
check directly from the figure, but it is clear that $A_{n}$ will have this
form for any $n\geq 1$. We give the cases $n=1$ and $n=6$ here for additional
clarity:

$\displaystyle A_{1}$ $\displaystyle=\left(\begin{array}[]{cccc}1&1&1&0\\\
0&0&1&-1\\\ 1&2&0&0\\\ 0&0&-1&0\\\ \end{array}\right),\qquad A_{6}$
$\displaystyle=\left(\begin{array}[]{cccc|ccccccccc|c}1&1&1&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&1&0&0&0&0&0&0&0&0&0&0&-1\\\ 1&2&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&-1&0&0&0&0&0&0&0&0&0&0&0\\\ \hline\cr 0&0&0&1&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&1&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&1&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&1&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&1&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&1&0&0\\\ \hline\cr
0&0&0&0&0&0&0&0&0&0&0&0&1&0\\\ \end{array}\right)$

We will use $A_{n}$ to compute the symmetrized Alexander polynomial and
Levine-Tristram signature function associated to the knots $P_{n}$. We briefly
review the relationship between these two knot invariants before proceeding
with the calculation.

######  Proposition 5.3.

Let $K$ be a knot and $A$ be a Seifert form for $K$. The symmetrized Alexander
polynomial of $K$ can be computed by the formula:

$\Delta_{K}(t)=t^{-\dim(A)/2}\det(tA-A^{T})=\det(\sqrt{t}A-\frac{1}{\sqrt{t}}A^{T})$

and $\Delta_{K}(t)\in\mathbb{R}$ for any $t\in\mathbb{C}$ satisfying
$\mathchoice{\left\lvert t\right\rvert}{\lvert t\rvert}{\lvert t\rvert}{\lvert
t\rvert}=1$. The Levine-Tristram signature of $K$ at a unit-norm complex
number $t$ can be computed by the formula:

$\sigma(K,t)=\sigma((1-t)A+(1-\frac{1}{t})A^{T})$

Moreover, we have the following equality:

$(t^{-1/2}-t^{1/2})^{\dim(A)}\Delta_{K}(t)=\det((1-t)A+(1-\frac{1}{t})A^{T})$

from which it follows that the function
$\sigma(K,-):({S}^{1}\subset\mathbb{C})\to\mathbb{Z}$ is piecewise constant,
changes at least wherever
$\Delta_{K}(-):({S}^{1}\subset\mathbb{C})\to\mathbb{R}$ changes signs, and can
only change at values where $\Delta_{K}(t)=0$.

###### Proof.

The content of this proposition is all well known already, nevertheless we
will review the main ideas for completeness’ sake. We know $\det(tA-A^{T})$
computes $\Delta_{K}(t)$ up to normalization by some power of $t$. We show our
normalization factor is correct by showing the expression we give is invariant
under the replacement $t\mapsto t^{-1}$:

$\det(\sqrt{t}A-\frac{1}{\sqrt{t}}A^{T})=\det(-(\sqrt{t}A-\frac{1}{\sqrt{t}}A^{T})^{T})=\det(\frac{1}{\sqrt{t}}A-\sqrt{t}A^{T})$

since the dimension of $A$ is even and determinants are unaffected by
transposition. The relationship between $\Delta_{K}(t)$ and $\sigma(K,t)$ is
explained in [GL15] although it was known much earlier. The key observation is
that,

$(\frac{1}{t}-1)^{\dim(A)}\det(tA-A^{T})=\det((\frac{1}{t}-1)(tA-A^{T}))=\det((1-t)A+(1-\frac{1}{t})A^{T})$

from which we can easily confirm our constant is correct and non-zero for
$t\in\mathbb{C}\setminus\mathchoice{\left\\{1\right\\}}{\\{1\\}}{\\{1\\}}{\\{1\\}}$
and $\mathchoice{\left\lvert t\right\rvert}{\lvert t\rvert}{\lvert
t\rvert}{\lvert t\rvert}=1$. We can interpret the right-hand side of the
equation above as the product of the eigenvalues of the form
$(1-t)A+(1-t^{-1})A^{T}$. The signature of this form can only change at values
of $t$ for which this product is zero. Moreover the signature is forced to
change wherever an odd number of its eigenvalues change sign which must happen
wherever the product changes sign. We can also deduce that both the
normalization constant $(t^{-1/2}-t^{1/2})^{\dim(A)}$ and $\Delta_{K}(t)$ are
real-valued for $t\in{S}^{1}\subset\mathbb{C}$. These follow from the
observation that $1/t=\overline{t}$ for any such $t$ and that $\dim(A)$ is
even. We compute:

$\Delta_{K}(t)=a_{0}+\sum_{k=1}^{N}a_{k}(t^{k}+\frac{1}{t^{k}})=a_{0}+\sum_{k=1}^{N}a_{k}(t^{k}+\overline{t^{k}})=a_{0}+\sum_{k=1}^{N}a_{k}(2\Re(t^{k}))\in\mathbb{R}$

for some sequence
$\mathchoice{\left\\{a_{0},\ldots,a_{N}\right\\}}{\\{a_{0},\ldots,a_{N}\\}}{\\{a_{0},\ldots,a_{N}\\}}{\\{a_{0},\ldots,a_{N}\\}}\subset\mathbb{Z}$
and,

$(t^{-1/2}-t^{1/2})^{\dim(A)}=(\overline{t^{1/2}}-t^{1/2})^{\dim(A)}=(2\Re(t)-2)^{\dim(A)/2}\in\mathbb{R}$

∎

######  Proposition 5.4.

Let $n\geq 0$ and let $\Delta_{n}(t)$ denote the symmetrized Alexander
polynomial of $P_{n}$. It is given by the formula:

$\Delta_{n}(t)=\frac{1}{t^{n+1}}-\frac{2}{t^{n}}+\frac{1}{t^{n-1}}-\frac{1}{t}+3-t+t^{n-1}-2t^{n}+t^{n+1}$

Clearly $\Delta_{n}(1)=1$, but for $t^{n}=1$ and $2\Re(t)<1$ we obtain
$\Delta_{n}(t)<0$. It follows that for $n>0$ there exists some root of unity
$t\in{S}^{1}\subset\mathbb{C}$ which depends on $n$ such that
$\sigma(P_{n+1},t)\neq 0$.

###### Proof.

We will prove the formula for $\Delta_{n}(t)$ by giving a general process for
reducing $tA_{n}-A^{T}$ into the block sum of a 4-by-4 matrix depending on $n$
and the matrix $-\mathbb{1}_{2n-2}$. This process consists of a sequence of
elementary row and column operations followed by cyclically permuting an odd
number of rows and so the determinant remains unchanged. We will easily be
able to compute the determinant of the 4-by-4 for general $n$ and this will
give us the formula for $\Delta_{n}(K)$. We will verify the remaining claims
by direct computation.

Let $n\geq 1$ be fixed and let:

$\displaystyle B_{n}$
$\displaystyle:=tA_{n}-A_{n}^{T}=\left(\begin{array}[]{cccc|ccccccccc|c}t-1&t&t-1&0&0&0&0&0&0&\cdots&0&0&0&0\\\
-1&0&t-2&0&0&0&0&0&0&\cdots&0&0&0&-t\\\
t-1&2t-1&0&1&0&0&0&0&0&\cdots&0&0&0&0\\\ 0&0&-t&0&-1&0&0&0&0&\cdots&0&0&0&0\\\
\hline\cr 0&0&0&t&0&-1&0&0&0&\cdots&0&0&0&0\\\
0&0&0&0&t&0&-1&0&0&\cdots&0&0&0&0\\\ 0&0&0&0&0&t&0&-1&0&\cdots&0&0&0&0\\\
0&0&0&0&0&0&t&0&-1&\ddots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&0&0&0&0&t&0&\ddots&0&0&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&-1&0&0&0\\\
0&0&0&0&0&0&0&\cdots&0&t&0&-1&0&0\\\ 0&0&0&0&0&0&0&\cdots&0&0&t&0&-1&0\\\
0&0&0&0&0&0&0&\cdots&0&0&0&t&0&-1\\\ \hline\cr
0&1&0&0&0&0&0&\cdots&0&0&0&0&t&0\\\ \end{array}\right)$

We note that $B_{n}$ is a $(2n+2)\times(2n+2)$-matrix. Observe that the
highest non-zero entry in columns (5) to $(2n+1)$ is a $-1$. Starting with
column $(2n+1)$ and working backwards to column (5) one column at a time, we
can perform elementary row operations which use the $-1$ in each column to
clear all the other non-zero entries below it. Notice that doing this in
column $(i)$ will create more non-zero entries in the column $(i-2)$; this is
why we start with column $(2n+1)$ and work backwards. After completing this
process we get the matrix $B_{n}^{\prime}$ given by:

$B^{\prime}_{n}:=\left(\begin{array}[]{cccc|ccccccccc|c}t-1&t&t-1&0&0&0&0&0&0&0&\cdots&\cdots&0&0\\\
-1&0&t-2&0&0&0&0&0&0&0&\cdots&\cdots&0&-t\\\
t-1&2t-1&0&1&0&0&0&0&0&0&\cdots&\cdots&0&0\\\
0&0&-t&0&-1&0&0&0&0&0&\cdots&\cdots&0&0\\\ \hline\cr
0&0&0&t&0&-1&0&0&0&0&\cdots&\cdots&0&0\\\
0&0&-t^{2}&0&0&0&-1&0&0&0&\cdots&\cdots&0&0\\\
0&0&0&t^{2}&0&0&0&-1&0&0&\cdots&\cdots&0&0\\\
0&0&-t^{3}&0&0&0&0&0&-1&0&\cdots&\cdots&0&0\\\
0&0&0&t^{3}&0&0&0&0&0&-1&\ddots&&\vdots&\vdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\vdots&\vdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&&\ddots&\ddots&0&0\\\
0&0&-t^{n-1}&0&0&0&0&0&0&\cdots&\cdots&0&-1&0\\\
0&0&0&t^{n-1}&0&0&0&0&0&\cdots&\cdots&0&0&-1\\\ \hline\cr
0&1&-t^{n}&0&0&0&0&0&0&\cdots&\cdots&0&0&0\\\ \end{array}\right)$

Next, we consider the rows $(5)$ to $(2n+1)$. The rightmost non-zero entry in
each of these rows is a $-1$, so we can perform an elementary column operation
which uses the $-1$ to clear the power of $t$ appearing in either the third or
forth entry of the row. We perform these column operations in order starting
with the one for row $(2n+1)$ and ending with the one for row $(5)$. After
that, we perform a row operation which uses the $-1$ in the
$(2n+1,2n+2)$-position to clear the $-t$ in the $(2,2n+2)$-position. Thus we
obtain the matrix $B_{n}^{\prime\prime}$ given by:

$B^{\prime\prime}_{n}:=\left(\begin{array}[]{cccc|ccccccccc|c}t-1&t&t-1&0&0&0&0&0&0&0&\cdots&\cdots&0&0\\\
-1&0&t-2&-t^{n}&0&0&0&0&0&0&\cdots&\cdots&0&0\\\
t-1&2t-1&0&1&0&0&0&0&0&0&\cdots&\cdots&0&0\\\
0&0&0&0&-1&0&0&0&0&0&\cdots&\cdots&0&0\\\ \hline\cr
0&0&0&0&0&-1&0&0&0&0&\cdots&\cdots&0&0\\\
0&0&0&0&0&0&-1&0&0&0&\cdots&\cdots&0&0\\\
0&0&0&0&0&0&0&-1&0&0&\cdots&\cdots&0&0\\\
0&0&0&0&0&0&0&0&-1&0&\cdots&\cdots&0&0\\\
0&0&0&0&0&0&0&0&0&-1&\ddots&&\vdots&\vdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\vdots&\vdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&&\ddots&\ddots&0&0\\\
0&0&0&0&0&0&0&0&0&\cdots&\cdots&0&-1&0\\\
0&0&0&0&0&0&0&0&0&\cdots&\cdots&0&0&-1\\\ \hline\cr
0&1&-t^{n}&0&0&0&0&0&0&\cdots&\cdots&0&0&0\\\ \end{array}\right)$

We finish by cyclically permuting rows $(4)-(2n+2)$, which is achieved by an
even number of row transpositions, therefore leaving the determinant
unchanged. This gives us the matrix $B^{\prime\prime\prime}_{n}$ which clearly
decomposes as a block sum in the way described earlier:

$B^{\prime\prime\prime}_{n}:=\left(\begin{array}[]{cccc|ccccccccc|c}t-1&t&t-1&0&0&0&0&0&0&0&\cdots&\cdots&0&0\\\
-1&0&t-2&-t^{n}&0&0&0&0&0&0&\cdots&\cdots&0&0\\\
t-1&2t-1&0&1&0&0&0&0&0&0&\cdots&\cdots&0&0\\\
0&1&-t^{n}&0&0&0&0&0&0&0&\cdots&\cdots&0&0\\\ \hline\cr
0&0&0&0&-1&0&0&0&0&0&\cdots&\cdots&0&0\\\
0&0&0&0&0&-1&0&0&0&0&\cdots&\cdots&0&0\\\
0&0&0&0&0&0&-1&0&0&0&\cdots&\cdots&0&0\\\
0&0&0&0&0&0&0&-1&0&0&\cdots&\cdots&0&0\\\
0&0&0&0&0&0&0&0&-1&0&\cdots&\cdots&0&0\\\
0&0&0&0&0&0&0&0&0&-1&\ddots&&\vdots&\vdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\vdots&\vdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&&\ddots&\ddots&0&0\\\
0&0&0&0&0&0&0&0&0&\cdots&\cdots&0&-1&0\\\ \hline\cr
0&0&0&0&0&0&0&0&0&\cdots&\cdots&0&0&-1\\\ \end{array}\right)$

We are now able to verify the formula for $\Delta_{n}(t)$.

$\displaystyle\det(B_{n})=\det(B^{\prime\prime\prime}_{n})$
$\displaystyle=\det\left(\left(\begin{array}[]{cccc}t-1&t&t-1&0\\\
-1&0&t-2&-t^{n}\\\ t-1&2t-1&0&1\\\
0&1&-t^{n}&0\end{array}\right)\oplus-\mathbb{1}_{2n-2}\right)$
$\displaystyle=\det\left(\begin{array}[]{cccc}t-1&t&t-1&0\\\
-1&0&t-2&-t^{n}\\\ t-1&2t-1&0&1\\\ 0&1&-t^{n}&0\end{array}\right)$
$\displaystyle=t^{2n}+3t^{n+1}-t^{n+2}-2t^{2n+1}+t^{2n+2}-t^{n}+t^{2}-2t+1$
$\displaystyle\implies\Delta_{n}(t)=\frac{1}{t^{n+1}}\det(B_{n})$
$\displaystyle=\frac{1}{t^{n+1}}-\frac{2}{t^{n}}+\frac{1}{t^{n-1}}-\frac{1}{t}+3-t+t^{n-1}-2t^{n}+t^{n+1}$

which confirms the formula claimed in the proposition. We leave it to the
reader to check that $\Delta_{n}(1)=1$. It remains to show that for any
$t\in{S}^{1}\subset\mathbb{C}$ satisfying $t^{n}=1$ and $2\Re(t)<1$, we have
$\Delta_{n}(t)<0$. Assuming the hypotheses on $t$ we compute:

$\displaystyle\Delta_{n}(t)$
$\displaystyle=\frac{1}{t^{n+1}}-\frac{2}{t^{n}}+\frac{1}{t^{n-1}}-\frac{1}{t}+3-t+t^{n-1}-2t^{n}+t^{n+1}$
$\displaystyle=\frac{1}{t^{n}}\left(\frac{1}{t}-2+t\right)-\frac{1}{t}+3-t+t^{n}\left(\frac{1}{t}-2+t\right)$
$\displaystyle=(2\Re(t)-2)+(3-2\Re(t))+(2\Re(t)-2)$ $\displaystyle=2\Re(t)-1$
$\displaystyle 2\Re(t)<1$ $\displaystyle\implies 2\Re(t)-1<0$
$\displaystyle\implies\Delta_{n}(t)<0$

Lastly we must check that for each $n>0$ there is a root of unity
$t\in{S}^{1}\subset\mathbb{C}$ such that $\sigma(P_{n+1},t)\neq 0$. Pick any
$n>0$ it follows that there is an $(n+1)$-root of unity with real part less
than $1/2$ and thus $\Delta_{n+1}(t)$ attains both positive and negative
values on ${S}^{1}\subset\mathbb{C}$. The intermediate value theorem
guarantees there is some $t_{*}$ at which $\Delta_{n+1}$ changes sign.
Proposition 5.3 guarantees that $\sigma(P_{n+1},t)$ must have different values
on either side of $t_{*}$ and so $\sigma(P_{n+1},t)\neq 0$ on at least one
side of $t_{*}$. Since roots of unity are dense in ${S}^{1}$, we can pick one
arbitrarily close to $t_{*}$ on the side at which $\sigma(P_{n+1},t)$ is non-
zero. ∎

We are now in a position to complete the proof that for each $n\in\mathbb{N}$
there is a root of unity $\omega_{n}$ such that
$\sigma(Q_{n},\omega_{n})-\sigma(Q,\omega_{n})\neq 0$. Recall that $Q:=P_{1}$
so $Q_{n}=P_{1+n}$. We verify that $\sigma(Q,-)\equiv 0$. The proof of
Proposition 5.3 gives us:

$\displaystyle\Delta(Q,t)=\Delta_{1}(t)$
$\displaystyle=t^{2}+\frac{1}{t^{2}}-3t-\frac{3}{t}+5$
$\displaystyle=\left(t+\frac{1}{t}\right)^{2}-3\left(t+\frac{1}{t}\right)+3$
$\displaystyle=(2\Re(t))^{2}-3(2\Re(t))+3$
$\displaystyle=4\Re(t)^{2}-6\Re(t)+3$ $\displaystyle\implies\Delta(Q,t)$
$\displaystyle>0,\hskip 14.22636pt(\forall\,t\in{S}^{1})$
$\displaystyle\implies\sigma(Q,\omega)$ $\displaystyle=0,\hskip
14.22636pt(\forall\,\omega\in{S}^{1})$

where the final implication follows from Proposition 5.3. Finally, Proposition
5.4 gives a root of unity $\omega_{n}$ such that $\sigma(Q_{n},\omega_{n})\neq
0$. We have now verified the hypotheses of Theorem 3.4 for the pair
$(Q,\sigma)$ which proves that there exist smoothly $n$-shake-slice knots with
arbitrary topological complexity for all
$n\in\mathbb{Z}\setminus\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}$.

### The Calculus of Dualizable Patterns: a Cheat Sheet

[Charles Stine, 2021]

Let $K\subset{S}^{3}$. We adopt the convention that
$E_{K}:={S}^{3}\setminus\nu(K)$ is oriented INF at its boundary, while
${D}^{2}\times{S}^{1}(\cong\nu(K))$ is oriented ONF at its boundary.

Definitions of patterns and dual patterns:

$\displaystyle P\subset V\ (\cong{D}^{2}\times{S}^{1}),\hskip
14.22636ptV_{P}:=V\setminus\nu(P)$ $\displaystyle V_{P}\ \text{is ort'd INF at
$\partial\nu(P)$, ONF at $\partial V$}$ $\displaystyle P^{*}\subset V^{*}\
(\cong{D}^{2}\times{S}^{1}),\qquad V^{*}_{P^{*}}:=V^{*}\setminus\nu(P^{*})$
$\displaystyle*:V_{P}\to V^{*}_{P^{*}}\text{ (preserving orientation)}$
$\displaystyle(\mu_{P},\lambda_{P},\mu_{V},\lambda_{V})\buildrel*\over{\longmapsto}(-\mu_{V^{*}},\lambda_{V^{*}},-\mu_{P^{*}},\lambda_{P^{*}})$

Ways to modify patterns:

  * •

$P^{*}$ — the dual pattern

  * •

$P_{n}$ — add $n$ meridonal twists

  * •

$\overline{P}$ — reverse all crossings and the orientation of $P$

  * •

$P^{m}$ — self-compose $m$ times

  * •

$P^{-1}$ — $(:=\overline{P}^{*})$, the concordance inverse

  * •

$P_{\\#}$ — connect-sum pattern, $wr(P_{\\#})=1$

Composition conventions:

  * •

$P^{*}_{n}=(P^{*})_{n}$ not $(P_{n})^{*}$

  * •

$\overline{P}_{n}=(\overline{P})_{n}$ not $\overline{(P_{n})}$

  * •

$P_{n}^{m}=(P^{m})_{n}$ not $(P_{n})^{m}$

  * •

$P^{*}_{\\#}=(P^{*})_{\\#}$ not $(P_{\\#})^{*}$

  * •

$\overline{P}_{\\#}=(\overline{P})_{\\#}$ not $\overline{(P_{\\#})}$

  * •

$P_{\\#}^{m}=(P^{m})_{\\#}$ not $(P_{\\#})^{m}$

These conventions are chosen so that if a pattern is written with a sequence
of superscripts and a sequence of subscripts, then the operations should be
applied in the order: superscripts from left to right followed by subscripts
from left to right (bar counts as the leftmost superscript!).

Basic Identities ($P,Q$ dualizable patterns, $J,K$ knots in ${S}^{3}$, and
$m,n\in\mathbb{Z}$):

$\displaystyle(P^{*})^{*}=P,\quad\overline{(\overline{P})}=P,\quad P_{0}=P$
$\displaystyle(P_{n})^{*}=P^{*}_{-n},\quad\overline{(P_{n})}=\overline{P}_{-n},\quad\overline{(P^{*})}=\overline{P}^{*}$
$\displaystyle(P\circ P^{-1})(K)\sim K\sim(P^{-1}\circ P)(K)$
$\displaystyle(P\circ Q)^{*}=Q^{*}\circ P^{*}$ $\displaystyle(P\circ
Q)_{n}=(P_{n}\circ Q_{n})$ $\displaystyle(P_{n})_{m}=P_{(n+m)}$ $\displaystyle
K_{\\#}(J)=K\mathbin{\\#}J=J\mathbin{\\#}K=J_{\\#}(K)$ $\displaystyle
P(K)=P(K_{\\#}(U))=(P\circ K_{\\#})(U)=P\circ K_{\\#}$

For patterns satisfying $wr(P)=1$:

$\displaystyle P^{*}=P,\quad P^{-1}=\overline{P},\quad P_{n}=P$ $\displaystyle
P=P(U)_{\\#},\quad P(K)=P(U)\mathbin{\\#}K$

Retracing Theorems ($P$ dualizable):

$\displaystyle X_{0}(P)\cong X_{0}(P^{*}),\quad X_{n}(P)\cong
X_{n}(P^{*}_{n})$ $\displaystyle X_{n}(P\mathbin{\\#}\overline{P}^{*})\cong
X_{n}(\overline{P}_{n}\circ P)$ $\displaystyle\implies(\overline{P}_{n}\circ
P)\ \text{is $n$-shake-slice!}$

Let ${\mathscr{I}}:\mathcal{C}\to\mathcal{A}$ be a homomorphism from the
smooth or TOP concordance group to an Abelian group $\mathcal{A}$ (usually
$\mathbb{Z}$). If ${\mathscr{I}}$ is 0-trace invariant, meaning,

$X_{0}(K)\cong X_{0}(J)\implies{\mathscr{I}}(K)={\mathscr{I}}(J)$

then, for any dualizable $P\subset V$, $K\subset{S}^{3}$, and
$n\in\mathbb{Z}$,

${\mathscr{I}}(P_{n}(K))={\mathscr{I}}(K)+{\mathscr{I}}(P_{n})$

We call such invariants _compatible_ with dualizable patterns.

2pt $\widehat{\lambda_{V}}$ [br] at 25 284 $(0)$ [tr] at 27 252 $b_{1}$ [bl]
at 172 332 $(0)$ [b] at 125 235 $K_{0}$ [br] at 159 245 $m_{0}$ [b] at 70 285
$\mathchoice{\left\langle 0\right\rangle}{\langle 0\rangle}{\langle
0\rangle}{\langle 0\rangle}$ [tr] at 70 252 $m_{1}$ [bl] at 121 280
$\widehat{P}$ [ ] at 89 330 $(+2)$ [r] at 289 314 $(-2)$ [b] at 33 134 $(0)$
[b] at 240 293 $(0)$ [bl] at 434 292 $(+2)$ [b] at 523 308 $(+2)$ [b] at 134
151 $(-2)$ [br] at 225 158 $(+2)$ [r] at 346 94 $(0)$ [br] at 443 151 $P^{*}$
[bl] at 566 166 $(0)$ [br] at 543 70 $+2$ at 461 112 Gluck Twisting [t] at 398
23 $\mu_{V^{*}}$ [t] at 485 83 $\lambda_{V^{*}}$ [tr] at 555 130

Figure 15. Constructing and dualizing a pattern via an isotopy in $\SS 21$. A
number in parentheses indicates the new framing of a curve after a slide move
or diffeomorphism.

## References

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