MnLargeSymbols’164 MnLargeSymbols’171

# Torus knots, the A-polynomial, and $\SL(2,\C)$

John A. Baldwin  Department of Mathematics
Boston College<EMAIL_ADDRESS>and  Steven Sivek  Department of
Mathematics
Imperial College London<EMAIL_ADDRESS>
###### Abstract.

The A-polynomial of a knot is defined in terms of $\SL(2,\C)$ representations
of the knot group, and encodes information about essential surfaces in the
knot complement. In 2005, Dunfield–Garoufalidis and Boyer–Zhang proved that it
detects the unknot using Kronheimer–Mrowka’s work on the Property P
conjecture. Here we use more recent results from instanton Floer homology to
prove that a version of the A-polynomial distinguishes torus knots from all
other knots, and in particular detects the torus knot $T_{a,b}$ if and only if
one of $|a|$ or $|b|$ is $2$ or both are prime powers. These results enable
progress towards a folklore conjecture about boundary slopes of non-torus
knots. Finally, we use similar ideas to prove that a knot in $S^{3}$ admits
infinitely many $\SL(2,\C)$-abelian Dehn surgeries if and only if it is a
torus knot, affirming a variant of a conjecture due to Sivek–Zentner.

##  1\. Introduction

One of the most fruitful ways to probe the geometry and topology of a
3-manifold $Y$ is to study the space of homomorphisms from its fundamental
group to another group $G$,

$R_{G}(Y)=\mathrm{Hom}(\pi_{1}(Y),G).$

However $R_{G}(Y)$ sometimes fails to capture any more information about $Y$
than does its first homology. It is interesting to study this failure, which
prompts the definition: a 3-manifold $Y$ is called _$G$ -abelian_ if every
homomorphism in $R_{G}(Y)$ has abelian image. These can be viewed as the
simplest 3-manifolds from the perspective of the representation variety.

For example, a longstanding conjecture [Kir97, Problem 3.105] which has
received recent attention [LPCZ23, BS18] holds that $S^{3}$ is the only
$\mathrm{SU}(2)$-abelian homology 3-sphere. Zentner proved this with
$\SL(2,\C)$ in place of $\mathrm{SU}(2)$ in [Zen18]. There has also been
considerable interest in studying $\mathrm{SU}(2)$-abelian Dehn surgeries on
knots, as in Kronheimer and Mrowka’s work on the Property P conjecture and its
offshoots [KM04b, KM04a, BS23, BLSY21].

One of our two main results concerns a conjecture by Sivek and Zentner on this
theme [SZ22], which asserts that torus knots are the only knots in $S^{3}$
with infinitely many $\mathrm{SU}(2)$-abelian Dehn surgeries. We prove this
with $\SL(2,\C)$ in place of $\mathrm{SU}(2)$:

######  Theorem 1.1.

Let $K\subset S^{3}$ be a knot. Then $S^{3}_{r}(K)$ is $\SL(2,\C)$-abelian for
infinitely many $r\in\mathbb{Q}$ if and only if $K$ is a torus knot.

This is ultimately a theorem about $\SL(2,\C)$-representation varieties of
knot complements, as is our next result on the A-polynomial of Cooper, Culler,
Gillet, Long, and Shalen. Recall that the _A-polynomial_ of a knot $K\subset
S^{3}$ [CCG+94] is a polynomial $A_{K}(M,L)\in\mathbb{Z}[M,L]$ that describes
the pairs of values taken by representations

$\pi_{1}(S^{3}\setminus K)\to\SL(2,\C)$

on a meridian and longitude of $K$. The representation variety always contains
a component of abelian representations, contributing a factor of $L-1$;
discarding this component, we get the _enhanced A-polynomial_

$\tilde{A}_{K}(M,L)\in\mathbb{Z}[M,L],$

which is related to the original by

(1.1) $A_{K}(M,L)=\operatorname{lcm}(\tilde{A}_{K}(M,L),L-1).$

The Newton polygon of $\tilde{A}_{K}(M,L)$ is a convex polygon
$\mathcal{N}(K)\subset\mathbb{R}^{2}$, and we say that $K$ is _thin_ if this
polygon is contained in a line. Dunfield and Garoufalidis [DG04] and Boyer and
Zhang [BZ05] independently proved that $K$ is the unknot if and only if
$\mathcal{N}(K)$ is a point. Our other main result is a classification of all
remaining thin knots:

######  Theorem 1.2.

A knot $K\subset S^{3}$ is thin if and only if it is a torus knot.

That is, $\tilde{A}_{K}(M,L)$ distinguishes torus knots from all other knots.
In addition, Theorem 1.2 enables us to completely determine which torus knots
are detected by the enhanced A-polynomial (in particular, it detects all
$T_{2,2n+1}$ torus knots):

######  Corollary 1.3.

$\tilde{A}_{K}(M,L)$ detects the nontrivial torus knot $T_{a,b}$ if and only
if:

  * •

$|a|=2$ or $|b|=2$, or

  * •

$|a|$ and $|b|$ are powers of different primes.

This greatly improves upon our previous work in [BS23, Corollary 10.16], which
showed that the enhanced A-polynomial detects a much more limited set of torus
knots. Theorem 1.2 further implies that the combination of the enhanced
A-polynomial and the degree of the Alexander polynomial detects _every_ torus
knot:

######  Corollary 1.4.

The pair $\big{(}\tilde{A}_{K}(M,L),\deg\Delta_{K}(t)\big{)}$ detects every
torus knot.

This is a substantial strengthening of results by Ni and Zhang, who showed in
[NZ17] that the combination of the enhanced A-polynomial and _knot Floer
homology_ detects each torus knot. Among other things, they use the fact that
knot Floer homology detects fibered knots to argue that a knot with the same
combination of these invariants as a torus knot is fibered. By contrast, we
know directly that thin knots are fibered via our work on instanton L-space
surgeries [BS23]. We only use the degree of the Alexander polynomial (which is
encoded by but is a much weaker invariant than knot Floer homology) to
distinguish torus knots $T_{a,b}\neq T_{c,d}$ with $ab=cd$, which
$\tilde{A}_{K}(M,L)$ cannot generally do by itself, per Corollary 1.3.

While Theorem 1.2 shows that the _enhanced_ A-polynomial distinguishes torus
knots from all other knots, it does not quite prove that the A-polynomial
does. Indeed, if $A_{K}(M,L)=A_{T_{a,b}}(M,L)$ then (1.1) implies that either

$\tilde{A}_{K}(M,L)=\tilde{A}_{T_{a,b}}(M,L)\,\textrm{ or
}\,\tilde{A}_{K}(M,L)=(L-1)\tilde{A}_{T_{a,b}}(M,L).$

In the first case, Theorem 1.2 implies that $K$ is a torus knot. The problem
is that we cannot rule out the second case at present. We therefore pose the
following:

######  Question 1.5.

Does $A_{K}(M,L)$ distinguish torus knots from all other knots?

One of the most powerful features of the A-polynomial is its relationship with
boundary slopes of incompressible surfaces in the knot complement. We describe
below an application of Theorem 1.2 along these lines, after a bit of
background and motivation.

Recall that a _boundary slope_ for a knot is an isotopy class of curves on the
boundary of a tubular neighborhood of the knot arising as the intersection of
the boundary torus with an incompressible surface in the knot complement. It
was shown in [CCG+94] that the slope of each side of the Newton polygon of
$A_{K}$ is a boundary slope for $K$.

For instance, a nontrivial torus knot $T_{a,b}$ has two boundary slopes: the
cabling slope $ab$ and the Seifert slope 0. This is reflected in the fact that
$\mathcal{N}(T_{a,b})$ is a line segment of slope $ab$, which implies that the
Newton polygon of its A-polynomial is a parallelogram with sides of slopes
$ab$ and $0$, where the slope $0$ sides come from the extra $L-1$ factor. This
example motivates the following strengthening of Question 1.5:

######  Question 1.6.

Is it true that $K$ is a nontrivial torus knot if and only if the Newton
polygon of $A_{K}(M,L)$ is a parallelogram?

A longstanding folklore conjecture asserts that nontrivial torus knots are in
fact the _only_ knots in $S^{3}$ with two boundary slopes. This would follow
from an affirmative answer to Question 1.6. With Theorem 1.2, we are able to
make the following partial progress:

######  Theorem 1.7.

If $K\subset S^{3}$ is a nontrivial fibered knot but not a torus knot, then
either:

  * •

there is a connected, separating, incompressible surface in the complement of
$K$ which meets the boundary torus in curves of slope 0, or

  * •

$K$ has at least 3 boundary slopes.

Finally, we note that the A-polynomial is defined for knots in arbitrary
3-manifolds, and ask whether our main theorems have analogues in this more
general setting:

######  Question 1.8.

Do analogues of Theorems 1.1 and 1.2 hold for knots in other 3-manifolds?

Suppose, for instance, that $K\subset Y$ is a knot with irreducible
complement. Is it true that $K$ is thin, or admits infinitely many
$\SL(2,\C)$-abelian surgeries, if and only if its complement is Seifert
fibered over the disk with at most two singular fibers?

###  1.1. Proof sketch

The proofs of Theorems 1.1 and 1.2 are similar in spirit; we hint here at our
proof of the latter. Very briefly, suppose that $K\subset S^{3}$ is a
nontrivial thin knot. Then $K$ is not hyperbolic [CCG+94], and Sivek–Zentner
prove in [SZ22] that $K$ admits infinitely many $\mathrm{SU}(2)$-abelian
surgeries (Proposition 2.15). It then follows from a combination of
Sivek–Zentner’s work and our own in [BS18, BS23] that $K$ is an instanton
L-space knot and hence fibered (Theorems 2.1 and 2.7). Moreover, our work in
[BS22], combined with newer work of Li–Ye [LY21] on the surgery formula in
framed instanton homology, places strong restrictions on the coefficients of
the Alexander polynomial of $K$ (Corollary 2.3). We use the above results to
prove that $K$ is not a satellite knot: if it were, then we show that there
would be another thin satellite knot whose pattern is a nontrivial torus knot,
but we prove that any such knot would violate the Alexander polynomial
constraints (Proposition 3.1). Therefore, since $K$ is neither hyperbolic nor
a satellite, $K$ must be a torus knot.

###  1.2. Organization

In §2, we provide a review of instanton L-spaces, $\mathrm{SU}(2)$-averse
knots, and the A-polynomial, and prove some preliminary results. In §3, we
prove new results about instanton L-space knots which are satellites. In §4,
we apply these results to prove Theorem 1.2 and Corollaries 1.3, 1.4, and
Theorem 1.7. We also prove as a consequence of Theorem 1.2 that only torus
knots are $\mathrm{PSL}(2,\C)$-averse. In §5, we prove Theorem 1.1.

###  1.3. Acknowledgments

We thank Nathan Dunfield and Xingru Zhang for helpful correspondence. The
first author was supported by NSF CAREER Grant DMS-1454865.

##  2\. Preliminaries

In this section we provide background and establish some preliminary results
which will be important in our proofs of both Theorems 1.1 and 1.2. Here and
throughout, we adopt the standard convention that the unknot is a torus knot
but not a satellite knot. Moreover, when writing that $K=P(C)$ is a satellite
knot with pattern $P\subset S^{1}\times D^{2}$ and companion $C\subset S^{3}$,
we assume that $C$ is nontrivial and that $P$ is not a core of the solid
torus.

###  2.1. Instanton L-space knots

Recall that a 3-manifold $Y$ is called an _instanton L-space_ if it is a
rational homology 3-sphere and its _framed instanton homology_ $I^{\\#}(Y)$,
defined by Kronheimer and Mrowka in [KM11], satisfies

$\operatorname{rank}I^{\\#}(Y)=|H_{1}(Y)|.$

A knot $K\subset S^{3}$ is called an _instanton L-space knot_ if the Dehn
surgery $S^{3}_{r}(K)$ is an instanton L-space for some positive
$r\in\mathbb{Q}$. We proved the following in [BS23, Theorem 1.15]:

######  Theorem 2.1.

If $K\subset S^{3}$ be a nontrivial instanton L-space knot then:

  * •

$K$ is fibered and strongly quasipositive, and

  * •

$S^{3}_{r}(K)$ is an instanton L-space if and only if $r\geq 2g(K)-1$.

Further restrictions on instanton L-space knots come from _instanton knot
homology_

$\mathit{KHI}(K)=\bigoplus_{i=-g(K)}^{g(K)}\mathit{KHI}(K,i)$

defined in [KM10b], whose graded Euler characteristic agrees (up to sign) with
the Alexander polynomial,

$\sum_{i}\chi\big{(}\mathit{KHI}(K,i)\big{)}\cdot t^{i}=\pm\Delta_{K}(t),$

by [KM10a, Lim10]. In particular, Li and Ye proved the following in [LY21,
Theorem 1.9]:

######  Theorem 2.2.

If $K\subset S^{3}$ is an instanton L-space knot then

$\operatorname{rank}\mathit{KHI}(K,j)=0\text{ or }1$

for each $j$. The coefficients of the Alexander polynomial $\Delta_{K}(t)$
therefore all lie in $\\{-1,0,1\\}$. Moreover, the nonzero coefficients
alternate in sign.

Combined with our previous work in [BS22], this yields the corollary below:

######  Corollary 2.3.

If $K\subset S^{3}$ is an instanton L-space knot of genus $g\geq 1$, then the
Alexander polynomial of $K$ has leading terms

$\pm\Delta_{K}(t)=t^{g}-t^{g-1}+\ldots,$

where the omitted terms all have degree at most $g-2$.

###### Proof.

Theorem 2.1 says that $K$ is fibered, from which it follows that

$\operatorname{rank}\mathit{KHI}(K,g)=1$

by [KM10b, Corollary 7.19]. Since $K$ is a nontrivial fibered knot, we also
know from [BS22, Theorem 1.7] that

$\operatorname{rank}\mathit{KHI}(K,g-1)\geq 1,$

in which case Theorem 2.2 says that

$\operatorname{rank}\mathit{KHI}(K,g-1)=1,$

and so the $t^{g-1}$-coefficient of $\Delta_{K}(t)$ is $\pm 1$. Since the
nonzero coefficients of $\Delta_{K}(t)$ alternate in sign, the signs of the
$t^{g}$\- and $t^{g-1}$-coefficients must be $+1$ and $-1$ in some order. ∎

We will be particularly interested in instanton L-space knots which are
satellites, and we study these in more detail in §3. Here, we establish the
following preliminary lemma:

######  Lemma 2.4.

Let $K=P(C)$ be a satellite knot with pattern $P$ and companion $C$, and let
$w\geq 0$ denote the winding number of $P$. If $K$ is an instanton L-space
knot, then:

  * •

$w\geq 1$,

  * •

$P(U)$ is not the unknot,

  * •

$P(U)$ and $C$ are both fibered.

###### Proof.

First note that $K$ is fibered by Theorem 2.1. This implies that $w\neq 0$ and
that both $P(U)$ and $C$ are fibered by a combination of [BZ03, Corollary
4.15] and Stallings’ theorem that a knot is fibered if and only if the
commutator subgroup of its knot group is finitely generated [BZ03, Theorem
5.1]. It remains to prove that $P(U)$ is nontrivial.

Suppose for a contradiction that $P(U)$ is the unknot, and let $g=g(K)$. Then

$\Delta_{K}(t)=\Delta_{P(U)}(t)\cdot\Delta_{C}(t^{w})=\Delta_{C}(t^{w}).$

The left side has the form $\pm(t^{g}-t^{g-1}+\dots)$ by Corollary 2.3, while
the right side can only have that form if $w=1$. On the other hand, Hirasawa,
Murasugi, and Silver [HMS08, Corollary 1] proved that if $w=1$ and $P(U)=U$,
and if $K$ is fibered, then $P$ is isotopic to the core of the solid torus, in
which case $K$ is not a satellite knot, a contradiction. ∎

######  Remark 2.5.

It should be true that if $K=P(C)$ is an instanton L-space knot then $P(U)$
and $C$ are as well. The Heegaard Floer analogue of this claim is a theorem of
Hanselman, Rasmussen, and Watson [HRW24, Theorem 1.15], but the instanton
version remains open.

###  2.2. $\mathrm{SU}(2)$-averse knots

Given a group $G$, recall from the introduction that a 3-manifold $Y$ is said
to be _$G$ -abelian_ if every homomorphism

$\pi_{1}(Y)\to G$

has abelian image. We will be interested in knots with infinitely many
$G$-abelian surgeries, for $G=\mathrm{SU}(2)$ and $\SL(2,\C)$, leading to the
shorthand:

######  Definition 2.6.

A knot $K\subset S^{3}$ is _$G$ -averse_ if it is nontrivial and there are
infinitely many rational numbers $r$ for which $S^{3}_{r}(K)$ is $G$-abelian.

Sivek and Zentner proved the following in [SZ22, Theorem 1.1], tying together
the notions of $\mathrm{SU}(2)$-averse knots and instanton L-spaces:

######  Theorem 2.7.

If $K\subset S^{3}$ is $\mathrm{SU}(2)$-averse then the set of
$\mathrm{SU}(2)$-abelian slopes is bounded and has a single accumulation point
$r(K)$. Moreover:

  * •

$r(K)$ is rational and satisfies $|r(K)|>2$, and

  * •

if $r(K)>0$ then $S^{3}_{s}(K)$ is an instanton L-space for all $s\geq\lceil
r(K)\rceil-1$.

For example, every nontrivial torus knot $T_{a,b}$ is $\mathrm{SU}(2)$-averse
with $r(T_{a,b})=ab$, because $(nab+1)/n$-surgery on $T_{a,b}$ is a lens
space, and hence $\mathrm{SU}(2)$-abelian, for all $n\in\mathbb{Z}$. Sivek and
Zentner conjecture [SZ22] that these are in fact the _only_
$\mathrm{SU}(2)$-averse knots. Our Theorem 1.1 is the statement that their
conjecture holds with $\SL(2,\C)$ in place of $\mathrm{SU}(2)$.

Note that if $K$ is $\mathrm{SU}(2)$-averse then so is its mirror
$\overline{K}$, and $r(\overline{K})=-r(K)$. In particular, Theorem 2.7
implies the following, which leads to strong restrictions on
$\mathrm{SU}(2)$-averse knots by the results in the previous section:

######  Corollary 2.8.

If $K\subset S^{3}$ is $\mathrm{SU}(2)$-averse then $K$ or $\overline{K}$ is
an instanton L-space knot.

We will use this in combination with the results of the previous section to
study $\SL(2,\C)$-averse knots in §5, noting that a knot which is
$\SL(2,\C)$-averse is also $\mathrm{SU}(2)$-averse.

###  2.3. The A-polynomial

The A-polynomial $A_{K}(M,L)$ of a knot $K\subset S^{3}$, introduced by
Cooper, Culler, Gillet, Long, and Shalen [CCG+94], is defined in terms of
representations

$\rho:\pi_{1}(S^{3}\setminus\nu(K))\to\SL(2,\C)$

that restrict to diagonal matrices on the peripheral subgroup, meaning that

$\displaystyle\rho(\mu)$ $\displaystyle=\begin{pmatrix}M&0\\\
0&M^{-1}\end{pmatrix},$ $\displaystyle\rho(\lambda)$
$\displaystyle=\begin{pmatrix}L&0\\\ 0&L^{-1}\end{pmatrix},$

where $\mu$ and $\lambda$ are a meridian and longitude on the boundary of the
tubular neighborhood $\nu(K)$ of the knot. The tuples $(M,L)$ which arise in
this way define a subset of $\C^{\ast}\times\C^{\ast}$. The 1-dimensional
components of the Zariski closure of this subset form an affine plane curve
$V(K)$, and we take

$A_{K}(M,L)\in\mathbb{Z}[M,L]$

to be the defining polynomial of $V(K)$. It is normalized to have integer
coefficients whose greatest common divisor is 1, and no repeated factors, and
is thus well-defined up to sign.

One of the components $X_{\mathrm{red}}$ of $V(K)$ corresponds to the abelian
representations

$\rho_{M}:\pi_{1}(S^{3}\setminus\nu(K))\twoheadrightarrow
H_{1}(S^{3}\setminus\nu(K))\to\SL(2,\C),\qquad M\in\C^{\ast}$

that send a meridian $\mu$ to the matrix $\left(\begin{smallmatrix}M&0\\\
0&M^{-1}\end{smallmatrix}\right)$. Then $\rho_{M}(\lambda)$ is always the
identity matrix since $\lambda$ is nullhomologous, so the image of this
component in $\C^{\ast}\times\C^{\ast}$ is precisely the line $\\{L=1\\}$.
This shows that $L-1$ always divides $A_{K}(M,L)$.

In 2005, Dunfield and Garoufalidis [DG04] and Boyer and Zhang [BZ05] used
Kronheimer and Mrowka’s work in [KM04a] to prove that $A_{K}(M,L)$ detects the
unknot:

######  Theorem 2.9.

$A_{K}(M,L)=L-1$ if and only if $K$ is the unknot.

Our Theorem 1.2 vastly generalizes this result. Let $\tilde{V}(K)$ denote the
plane curve defined in the same way as $V(K)$, except that we only consider
pairs $(M,L)$ coming from _irreducible_ representations. Following Ni and
Zhang [NZ17], we define an enhancement

$\tilde{A}_{K}(M,L)\in\mathbb{Z}[M,L]$

to be the defining polynomial of $\tilde{V}(K)$. As mentioned in the
introduction, this _enhanced_ A-polynomial is related to the original by

$A_{K}(M,L)=\operatorname{lcm}(\tilde{A}_{K}(M,L),L-1).$

Indeed, $\tilde{A}_{K}(M,L)$ no longer _has_ to be a multiple of $L-1$, but
this can still happen if there are many irreducible representations which send
$\lambda$ to the identity matrix. An immediate consequence of Theorem 2.9 is
that $\tilde{A}_{K}(M,L)$ detects the unknot as follows:

######  Corollary 2.10.

$\tilde{A}_{K}(M,L)=1$ if and only if $K$ is the unknot.

Given a knot $K\subset S^{3}$, let

$\mathcal{N}(K)\subset\mathbb{R}^{2}$

denote the _Newton polygon_ of $\tilde{A}_{K}(M,L)$, which is by definition
the convex hull of all points $(a,b)\in\mathbb{Z}^{2}$ such that the
$M^{b}L^{a}$-coefficient of $\tilde{A}_{K}(M,L)$ is nonzero.

######  Remark 2.11.

The fact that $(a,b)$ corresponds to a monomial $M^{b}L^{a}$, rather than to
$M^{a}L^{b}$, is not a typo: this allows us to directly relate the edge slopes
of $\mathcal{N}(K)$ to slopes in the knot complement, rather than to their
reciprocals. We will return to this point in §4.

Corollary 2.10 is equivalent to the statement that $\mathcal{N}(K)$ is a point
if and only if $K$ is the unknot. The next simplest convex polygon after a
point is a line segment, which leads to:

######  Definition 2.12.

Let $r\in\mathbb{Q}$. We say that a knot $K\subset S^{3}$ is _$r$ -thin_ if
$\mathcal{N}(K)$ is contained in a line segment of slope $r$. We say that $K$
is _thin_ if it is $r$-thin for some $r\in\mathbb{Q}$.

The enhanced A-polynomial is very difficult to compute in general, but this
computation is fairly easy for torus knots. The results of these computations
are described in the example below, which shows in particular that a
nontrivial torus knot $T_{a,b}$ is $ab$-thin.

######  Example 2.13.

Let $|a|>b\geq 2$. Then the enhanced A-polynomial of the torus knot $T_{a,b}$
is given by

$\tilde{A}_{T_{a,b}}(M,L)=\begin{cases}1+M^{2a}L,&b=2,\,a>0\\\
M^{-2a}+L,&b=2,\,a<0\\\ -1+M^{2ab}L^{2},&b>2,\,a>0\\\
-M^{-2ab}+L^{2},&b>2,\,a<0\\\ \end{cases}$

as stated in [NZ17, Lemma 2.1], for instance.

Our Theorem 1.2 asserts that torus knots are the _only_ thin knots. Then

$\tilde{A}_{K}(M,L)=\tilde{A}_{T_{ab}}(M,L)$

implies that $K=T_{p,q}$ where $pq=ab$. Corollary 1.3 will follow from this
together with the corollary below, which is an immediate consequence of the
calculations in Example 2.13:

######  Corollary 2.14.

$\tilde{A}_{K}(M,L)$ detects $T_{a,b}$ among nontrivial torus knots if and
only if:

  * •

$|a|=2$ or $|b|=2$, or

  * •

$|a|$ and $|b|$ are powers of different primes.

The next result, which we proved in [BS23, §10], is important because (1) it
says that in thinking about thin knots we do not have to consider hyperbolic
knots, and (2) it connects thinness with the notions of
$\mathrm{SU}(2)$-averse knots and instanton L-spaces. This will allow us to
study thin knots using instanton Floer techniques and the results in previous
sections.

######  Proposition 2.15.

If $K\subset S^{3}$ is a nontrivial $r$-thin knot then:

  * •

$K$ is not hyperbolic,

  * •

$K$ is $\mathrm{SU}(2)$-averse with $r(K)=r$.

###### Proof.

The first claim is [BS23, Proposition 10.7], which is really just a
reinterpretation of [CCG+94, Proposition 2.6]. The second claim is [BS23,
Proposition 10.10].∎

Proposition 2.15 says in particular that any thin knot is either a torus knot
or a satellite knot. The torus knot case is completely understood, as
described above. Our focus for the remainder of this paper will therefore be
on thin satellite knots, and we record the following:

######  Proposition 2.16.

Suppose $K=P(C)$ is an $r$-thin satellite knot and let $w$ be the winding
number of the pattern $P$. Then:

  * •

$P(U)$ is nontrivial and $w\geq 2$,

  * •

$P(U)$ is $r$-thin,

  * •

the companion $C$ is $(r/w^{2})$-thin.

###### Proof.

Proposition 2.15 says that $K$ is $\mathrm{SU}(2)$-averse. Then Corollary 2.8
says that either $K$ or $\overline{K}$ is an instanton L-space knot, in which
case Lemma 2.4 implies that $P(U)$ is nontrivial and that $w\geq 1$. The
second and third claims are then part of [BS23, Lemma 10.12]. It remains to
prove that $w\geq 2$, for which it suffices to show that $w\neq 1$.

Since $P(U)$ and $C$ are $r$\- and $(r/w^{2})$-thin, they are
$\mathrm{SU}(2)$-averse and thus up to mirroring are nontrivial instanton
L-space knots, again by Proposition 2.15 and Corollary 2.8. Suppose their
genera are $g\geq 1$ and $h\geq 1$, respectively. Note that

$\Delta_{K}(t)=\Delta_{P(U)}(t)\cdot\Delta_{C}(t^{w}).$

If $w=1$, then Corollary 2.3 says that the right side of this equation has the
form

$\pm\big{(}t^{g}-t^{g-1}+O(t^{g-2})\big{)}\big{(}t^{h}-t^{h-1}+O(t^{h-2})\big{)}=\pm\big{(}t^{g+h}-2t^{g+h-1}+O(t^{g+h-2})\big{)}.$

But then the $t^{g+h-1}$-coefficient of $\Delta_{K}(t)$ would be $\pm 2$,
contradicting Theorem 2.2. Therefore $w\neq 1$, completing the proof. ∎

##  3\. L-space satellite knots with torus knot pattern

The results of the previous section show that understanding thin knots
requires us to understand instanton L-space knots that are satellite knots. In
this section, we prove new results about such knots; these will be key in our
proofs of both Theorems 1.1 and 1.2. The proposition below is the main
technical result of this section:

######  Proposition 3.1.

Let $K=P(C)$ be a satellite knot whose pattern $P$ has winding number $w\geq
1$. If $P(U)=T_{a,b}$ is a nontrivial torus knot such that $ab$ is a multiple
of $w^{2}$, and $C$ or $\overline{C}$ is an instanton L-space knot, then
neither $K$ nor $\overline{K}$ is an instanton L-space knot.

We will prove Proposition 3.1 by examining the Alexander polynomial of $K$,
and showing that it cannot have the form guaranteed by Theorem 2.2: either
some coefficient has absolute value at least 2, or two consecutive nonzero
coefficients have the same sign. We begin by studying the leading terms of the
Alexander polynomials of torus knots:

######  Lemma 3.2.

Let $g=g(T_{p,q})=\frac{1}{2}(p-1)(q-1)$, where $p>q\geq 2$ and $\gcd(p,q)=1$.
Then

$\Delta_{T_{p,q}}(t)=\sum_{i=0}^{\lfloor
p/q\rfloor}(t^{g-iq}-t^{g-iq-1})+O(t^{g-p}).$

###### Proof.

Noting that $2g-1=pq-p-q$, we write

$\displaystyle(t^{p}-1)t^{g}\Delta_{T_{p,q}}(t)$
$\displaystyle=\frac{(t^{pq}-1)(t-1)}{t^{q}-1}$
$\displaystyle=(t-1)\left(t^{(p-1)q}+t^{(p-2)q}+\dots+t^{q}+1\right)$
$\displaystyle=(t-1)\left(t^{2g+p-1}+t^{2g+p-1-q}+\dots+t^{2g+p-1-q(p-1)}\right)$

and so

(3.1)
$(t^{p}-1)\Delta_{T_{p,q}}(t)=(t-1)\left(t^{g+p-1}+t^{g+p-1-q}+\dots+t^{g+p-1-q(p-1)}\right).$

Similarly, if we let $r=\lfloor\frac{p}{q}\rfloor$, then we have $q(r+1)>p$
and $1\leq r\leq p-1$. (The bound $r\leq p-1$ is immediate from $q\geq 2$,
which implies that $r<p$.) We compute that

$\displaystyle(t^{p}-1)\sum_{i=0}^{r}(t^{g-iq}-t^{g-iq-1})$
$\displaystyle=(t^{p}-1)(t-1)\sum_{i=0}^{r}t^{g-1-iq}$
$\displaystyle=(t-1)\sum_{i=0}^{r}t^{g+p-1-iq}-\underbrace{(t-1)\sum_{i=0}^{r}t^{g-1-iq}}_{=\,O(t^{g})}.$

In the first term on the right, since $r\leq p-1$ we can extend the summation
from $i=r$ to $i=p-1$ by adding additional terms, all of which are no bigger
than

$(t-1)\cdot O(t^{g+p-1-q(r+1)})=O(t^{g}).$

Similarly the entire second term on the right is $O(t^{g})$, so overall we
have

(3.2)
$(t^{p}-1)\sum_{i=0}^{r}(t^{g-iq}-t^{g-iq-1})=\sum_{i=0}^{p-1}t^{q+p-1-iq}+O(t^{g}).$

Comparing (3.1) and (3.2) gives

$(t^{p}-1)\left(\Delta_{T_{p,q}}(t)-\sum_{i=0}^{r}(t^{g-iq}-t^{g-iq-1})\right)=O(t^{g}),$

and so by comparing degrees we conclude that

$\Delta_{T_{p,q}}(t)-\sum_{i=0}^{r}(t^{g-iq}-t^{g-iq-1})=O(t^{g-p}).\qed$

######  Lemma 3.3.

Suppose that $K=P(C)$ is a satellite knot, where

  * •

$P(U)=T_{a,b}$ for some relatively prime integers $a$ and $b$, with $a>b\geq
2$;

  * •

and either $C$ or $\overline{C}$ is an instanton L-space knot.

If either $K$ or $\overline{K}$ is an instanton L-space knot, and if $P$ has
winding number $w<a$, then $w$ is a positive multiple of $b$.

###### Proof.

We note that $w\geq 1$ by Lemma 2.4, so $1\leq w\leq a-1$. Now Lemma 3.2 says
that if $g=g(T_{a,b})$ then

$\Delta_{T_{a,b}}(t)=\sum_{i=0}^{\lfloor
a/b\rfloor}(t^{g-ib}-t^{g-ib-1})+O(t^{g-a}).$

Certainly $t^{g-a}=O(t^{g-w-1})$ since $a\geq w+1$, so we can read the
$t^{g-w}$-coefficient of $\Delta_{T_{a,b}}(t)$ directly off of this sum: it is

$c_{g-w}=\begin{cases}1,&w\equiv 0\pmod{b}\\\ -1,&w\equiv 1\pmod{b}\\\
0,&\text{otherwise}.\end{cases}$

We let $h=g(C)$, and noting that $\pm\Delta_{C}(t)=t^{h}-t^{h-1}+O(t^{h-2})$
by Corollary 2.3, we examine the $t^{g+hw-w}$-coefficient of

$\displaystyle\pm\Delta_{K}(t)$
$\displaystyle=\pm\Delta_{T_{a,b}}(t)\cdot\Delta_{C}(t^{w})$
$\displaystyle=\left(t^{g}-t^{g-1}+\dots+c_{g-w}t^{g-w}+O(t^{g-w-1})\right)\left(t^{hw}-t^{(h-1)w}+O(t^{(h-2)w})\right).$

By inspection, this coefficient is equal to $c_{g-w}-1$, and if $c_{g-w}=-1$
then $\Delta_{K}(t)$ has a $t^{g+hw-w}$-coefficient of $-2$, contradicting
Theorem 2.2. (It does not matter here whether $K$ or its mirror $\overline{K}$
is an instanton L-space knot, since $\Delta_{K}(t)=\Delta_{\overline{K}}(t)$.)
Thus $w\not\equiv 1\pmod{b}$.

Now if $w\not\equiv 0\pmod{b}$, then the only remaining possibility is that
$c_{g-w}=0$ and so this $t^{g+hw-w}$-coefficient is equal to $-1$. But in this
case, if we write

$w=qb+r,\qquad 2\leq r<b$

then Lemma 3.2 (plus the fact that $g-a\leq g-w-1$) tells us that

$\Delta_{T_{a,b}}(t)=(t^{g}-t^{g-1})+(t^{g-b}-t^{g-b-1})+\dots+(t^{g-qb}-t^{g-qb-1})+O(t^{g-w-1}),$

and we compute that

$\displaystyle\pm\Delta_{K}(t)$
$\displaystyle=\pm\Delta_{T_{a,b}}(t)\cdot\Delta_{C}(t^{w})$
$\displaystyle=\left(t^{g}-\dots+t^{g-qb}-t^{g-qb-1}+O(t^{g-w-1})\right)\left(t^{hw}-t^{(h-1)w}+O(t^{(h-2)w})\right)$
$\displaystyle=t^{g+hw}+\dots+t^{g+hw-qb}-t^{g+hw-
qb-1}-t^{g+hw-w}+O(t^{g+hw-w-1}).$

In this case two consecutive nonzero coefficients of $\Delta_{K}(t)$, namely
those corresponding to $t^{g+hw-qb-1}$ and $t^{g+hw-w}$, have the same sign.
This also contradicts Theorem 2.2, so this case cannot occur either. We
conclude that $w$ must have been a multiple of $b$ after all. ∎

###### Proof of Proposition 3.1.

Suppose that $K=P(C)$ is a satellite knot with $P(U)=T_{a,b}$ for some $a,b$,
with $C$ or $\overline{C}$ an instanton L-space knot, and such that the
winding number $w\geq 1$ of $P$ satisfies

$ab=kw^{2}$

for some integer $k$. If $T_{a,b}$ is a negative torus knot then we can
replace $K$ with $\overline{K}$, and this replaces $T_{a,b}$ with $T_{-a,b}$
but does not change the Alexander polynomial $\Delta_{K}(t)$, so we may as
well assume that $a>b\geq 2$ and then $k\geq 1$.

We must now have $w<a$: assuming otherwise, we would have

$ab=kw^{2}\geq a^{2}>ab,$

which is absurd since $ab>0$. Thus if either $K$ or $\overline{K}$ is an
instanton L-space knot, then Lemma 3.3 says that $w$ is a multiple of $b$. But
now

$a=b\cdot k\left(\frac{w}{b}\right)^{2}$

tells us that $a$ is an integer multiple of $b$, and this is impossible since
$a$ and $b$ must be relatively prime, so neither $K$ nor $\overline{K}$ can be
an instanton L-space knot after all. ∎

##  4\. Thin knots: the proof of Theorem 1.2

In this section, we prove Theorem 1.2. Example 2.13 showed that nontrivial
torus knots are thin. It remains to prove the other direction of Theorem 1.2,
stated as follows:

######  Theorem 4.1.

If $K\subset S^{3}$ is a nontrivial $r$-thin knot, then $K=T_{a,b}$ where
$r=ab$.

###### Proof.

Note that $K$ is not hyperbolic, by Proposition 2.15. If $K$ is a torus knot
then we are done, so it suffices to prove that there are no thin satellite
knots.

Suppose for a contradiction that $K=P(C)$ is an $r$-thin satellite knot with
the smallest genus amongst all thin satellite knots. Proposition 2.16 says
that $P(U)$ and $C$ are nontrivial $r$\- and $(r/w^{2})$-thin knots,
respectively, and that $w\geq 2$. Then from the relation

$g(K)=g(P(U))+w\cdot g(C),$

we see that $P(U)$ and $C$ have genera strictly less than that of $K$, and are
thus torus knots. If we write $P(U)=T_{a,b}$ and $C=T_{c,d}$ then

$ab=r\textrm{ and }cd=r/w^{2}.$

In particular, $ab=cdw^{2}$. Since torus knots are instanton L-space knots,
Proposition 3.1 then implies that neither $K$ nor $\overline{K}$ is an
instanton L-space knot. But this yields a contradiction: since $K$ is thin, it
is $\mathrm{SU}(2)$-averse by Proposition 2.15, which implies by Corollary 2.8
that either $K$ or its mirror must be an instanton L-space knot. ∎

###### Proof of Corollary 1.3.

This follows immediately from Theorem 4.1 and Corollary 2.14. ∎

###### Proof of Corollary 1.4.

Suppose $K\subset S^{3}$ is a knot for which the pair

$\big{(}\tilde{A}_{K}(M,L),\deg\Delta_{K}(t)\big{)}$

agrees with the corresponding pair for the torus knot $T_{a,b}$. If $K$ is the
unknot then the result holds by Corollary 2.10, so we may assume that $K$ is
nontrivial. Then $K$ is a torus knot $T_{p,q}$ with $pq=ab$ by Theorem 4.1, by
comparing the enhanced A-polynomials. Comparing the degrees of the Alexander
polynomials, we also have that $(p-1)(q-1)=(a-1)(b-1).$ These two equations
imply that $\\{p,q\\}=\\{a,b\\}$ and hence that $K=T_{a,b}$. ∎

###  4.1. Boundary slopes

In this section, we use Theorem 1.2 to prove Theorem 1.7. We first provide
some background on boundary slopes, elaborating on the discussion in §1.

Given a knot $K\subset Y$, let $E_{K}=Y\setminus\nu(K)$ denote the exterior of
a tubular neighborhood of $K$. A peripheral slope $\gamma\subset\partial
E_{K}$ is called a _boundary slope_ for $K$ if there is a properly embedded,
incompressible, boundary-incompressible, orientable surface $F\subset E_{K}$
whose boundary $\partial F\subset\partial E_{K}$ is a union of one or more
curves of slope $\gamma$. If $F$ is not a fiber in any fibration $E_{K}\to
S^{1}$, then we say that $\gamma$ is a _strict_ boundary slope.

######  Example 4.2.

Every knot $K\subset S^{3}$ has boundary slope $0$, realized as the boundary
of a Seifert surface of genus $g(K)$.

######  Example 4.3.

A nontrivial torus knot $T_{a,b}$ has boundary slope $ab$: if we view
$T_{a,b}$ as a curve on a standardly embedded torus $\mathbb{T}\subset S^{3}$,
then $\mathbb{T}\setminus N(T_{a,b})$ is an incompressible annulus of boundary
slope $ab$. In fact, $0$ and $ab$ are the only boundary slopes for $T_{a,b}$.

If the knot exterior $E_{K}$ is orientable, compact, and irreducible, then $K$
has only finitely many boundary slopes [Hat82]. Nontrivial knots in $S^{3}$
have at least two boundary slopes [CS84], and conjecturally only torus knots
have exactly two, as discussed in the introduction. The A-polynomial is an
effective source of boundary slopes in the following sense, as proven in
[CCG+94, Theorem 3.4]:

######  Theorem 4.4.

If $r\in\mathbb{Q}\cup\\{\infty\\}$ is the slope of some edge of the Newton
polygon $\mathcal{N}(K)$ of $\tilde{A}_{K}(M,L)$, then $r$ is a strict
boundary slope for $K$.

######  Remark 4.5.

The theorem in [CCG+94] uses $A_{K}(M,L)$ rather than $\tilde{A}_{K}(M,L)$,
and does not claim that $r$ should be strict; indeed, we know that $0$ is
always an edge slope for the Newton polygon of $A_{K}(M,L)$, but it might not
be strict as a boundary slope if $K$ is fibered. But if $0$ is an edge slope
of $\tilde{A}_{K}(M,L)$, then [CCG+94, §3] associates to this edge an ideal
point of the character variety of $E_{K}$, and following [CS83] this gives
rise to an action of $\pi_{1}(E_{K})$ on a tree $T$, from which we extract the
incompressible surface $S$ of boundary slope $0$. By construction $\pi_{1}(S)$
stabilizes some edge of $T$, so by [CGLS87, Proposition 1.2.7] it can’t be a
normal subgroup of $\pi_{1}(E_{K})$, hence $S$ can’t be a fiber in a fibration
$E_{K}\to S^{1}$.

As mentioned above, $0$ is always an edge slope of the Newton polygon of
$A_{K}(M,L)$, corresponding to its $L-1$ factor. Its nonzero edge slopes are
then precisely those of the Newton polygon $\mathcal{N}(K)$ of
$\tilde{A}_{K}(M,L)$. Thus we have the following:

######  Lemma 4.6.

Let $K\subset S^{3}$ be a nontrivial knot with exactly two boundary slopes. If
$K$ is not a torus knot, then $0$ is an edge slope of $\mathcal{N}(K)$ and
thus a strict boundary slope for $K$.

###### Proof.

Since $K$ is not a torus knot, Theorem 1.2 tells us that it is not thin: that
is, the Newton polygon $\mathcal{N}(K)$ is not contained in a line. Then
$\mathcal{N}(K)$ has at least two distinct edge slopes, and Theorem 4.4 says
that each of these is a strict boundary slope for $K$. But $K$ only has two
boundary slopes in the first place, so in fact every boundary slope for $K$ is
both an edge slope for $\mathcal{N}(K)$ and strict. Example 4.2 now says that
$0$ is a boundary slope for $K$, so it must be a strict boundary slope after
all. ∎

We will also need the following fact, which appears to be classically known
but we could not find a reference:

######  Proposition 4.7.

Let $K\subset S^{3}$ be a fibered knot with exterior $E_{K}$. Let $F\subset
E_{K}$ be a fiber surface, and let $S\subset E_{K}$ be a connected,
orientable, nonseparating, incompressible surface whose boundary meets
$\partial E_{K}$ in a union of longitudes. Then $S$ is isotopic to $F$.

###### Proof.

Since $S$ is nonseparating, there is an arc in $E_{K}\setminus N(S)$ from one
side of $S$ to the other; we join the ends of this arc inside $N(S)$ to get a
closed curve $c\subset E_{K}$ that meets $S$ transversely in one point. The
intersection pairing $\langle[S],[c]\rangle$ is then equal to $\pm 1$, so
$[S]$ must be primitive as an element of $H_{2}(E_{K},\partial
E_{K})\cong\mathbb{Z}$. In other words, we have $[S]=\pm[F]$, and so the
boundary homomorphism

$\partial:H_{2}(E_{K},\partial E_{K})\to H_{1}(\partial E_{K})$

sends $[S]$ to $\pm[\partial F]=\pm[\lambda_{K}]$. In other words, $[\partial
S]=\pm[\partial F]$ in $H_{1}(\partial E_{K})$.

We now let $p:\tilde{E}\to E_{K}$ denote the $\mathbb{Z}$-fold cyclic cover
corresponding to the kernel of the abelianization map
$\operatorname{ab}:\pi_{1}(E_{K})\twoheadrightarrow\mathbb{Z}$, and observe
that $\tilde{E}\cong F\times\mathbb{R}$. Given any closed curve $\gamma\subset
S$, we can push $\gamma$ off of $S$ inside $E_{K}$ to conclude that

$\operatorname{lk}(\gamma,K)=\langle[F],[\gamma]\rangle=\pm\langle[S],[\gamma]\rangle=0.$

Thus $\gamma\in\ker(\operatorname{ab})$, and since $\gamma$ was arbitrary we
have $\pi_{1}(S)\subset\ker(\operatorname{ab})=p_{*}(\pi_{1}(\tilde{E}))$. So
$S$ lifts to a surface $\tilde{S}\subset\tilde{E}$, and moreover any
compressing disk for $\tilde{S}$ could be pushed down to a compressing disk
for $S$ itself, so $\tilde{S}$ must be incompressible.

We now let $i:\tilde{S}\hookrightarrow\tilde{E}$ denote inclusion, and
consider the composition

$(\phi,h):\tilde{S}\xrightarrow{i}\tilde{E}\cong F\times\mathbb{R}.$

We note that $i_{*}:\pi_{1}(\tilde{S})\to\pi_{1}(\tilde{E})$ is injective
since $\tilde{S}$ is incompressible, and the projection
$p_{F}:F\times\mathbb{R}\to F$ induces an isomorphism of fundamental groups,
so the map

$\phi_{*}=(p_{F}\circ(\phi,h))_{*}:\pi_{1}(\tilde{S})\to\pi_{1}(F)$

is injective. Since $\phi$ is a map between compact surfaces inducing an
injection of fundamental groups, a theorem of Nielsen [Nie27] says that it is
homotopic to a covering map $\psi:\tilde{S}\to F$. We use $\psi$ to pull back
the orientation of $F$ to $\tilde{S}$, and then the projection $p_{F}$ is an
orientation-preserving homeomorphism from each component of
$\partial\tilde{S}\subset\partial F\times\mathbb{R}$ to $\partial F$. It
follows that

$[\partial\tilde{S}]=(\deg\psi)[\partial F]\in H_{1}(\partial
F\times\mathbb{R}),$

since every component of $\partial\tilde{S}$ is counted with the same sign.
Projecting from $\tilde{E}\cong F\times\mathbb{R}$ to the knot exterior
$E_{K}$, we conclude that

$[\partial S]=(\deg\psi)[\partial F]\in H_{1}(E_{K});$

but we showed above that $[\partial S]=\pm[\partial F]$, so we must have
$\deg\psi=1$. In other words, the covering map $\psi:\tilde{S}\to F$ is a
homeomorphism, and $\tilde{S}$ has connected boundary, which up to translation
we can take to be $\partial F\times\\{0\\}$.

Now the incompressible surfaces $\tilde{S}$ and $F\times\\{0\\}$ in
$\tilde{E}\cong F\times\mathbb{R}$ are homotopic rel boundary, via a homotopy
$(\phi,h)\simeq(\psi,0)$. Composing with the projection $p:\tilde{E}\to
E_{K}$, we see that $S$ is homotopic rel boundary to $F$. A result of
Waldhausen [Wal68, Corollary 5.5] now tells us that $S$ is isotopic rel
boundary to $F$, as desired. ∎

We are now ready to prove Theorem 1.7 from the introduction, restated as
follows:

######  Theorem 4.8.

Let $K\subset S^{3}$ be a nontrivial fibered knot with exactly two boundary
slopes. If $K$ is not a torus knot, then its exterior $E_{K}$ contains a
connected, separating, incompressible surface $S$ which meets $\partial E_{K}$
in curves of slope $0$.

###### Proof.

Let $F\subset E_{K}$ be a fiber of the fibration $E_{K}\to S^{1}$. Lemma 4.6
says that $0$ is a strict boundary slope for $K$; if $S\subset\partial E_{K}$
is the corresponding connected, incompressible surface, then $S$ is not
isotopic to $F$. Now Proposition 4.7 says that if $S$ were nonseparating then
it would have to be isotopic to $F$, so $S$ must separate $E_{K}$ after all. ∎

###  4.2. $\mathrm{PSL}(2,\C)$-averse knots

Proving Theorem 1.1, which states that all $\SL(2,\C)$-averse knots are torus
knots, will take a good deal more work, so we postpone it to §5. Here we show
instead that the analogue of Theorem 1.1 with $\mathrm{PSL}(2,\C)$ in place of
$\SL(2,\C)$ follows readily from Theorem 1.2:

######  Theorem 4.9.

If $K\subset S^{3}$ is a $\mathrm{PSL}(2,\C)$-averse knot, then $K$ is a torus
knot.

Suppose that $K\subset S^{3}$ is a knot and let us recall some facts about the
$\SL(2,\C)$ character variety $X(E_{K})$ associated to the knot exterior
$E_{K}=S^{3}\setminus\nu(K)$. These facts are taken from [BZ98, §5], where
they generalize results of [CGLS87, §1].

Given a peripheral slope $\gamma\subset\partial E_{K}$, the trace function

$\displaystyle f_{\gamma}:X(E_{K})$ $\displaystyle\to\C$ $\displaystyle[\rho]$
$\displaystyle\mapsto\operatorname{tr}(\rho(\gamma))^{2}-4$

is a regular function on $X(E_{K})$. An irreducible curve $X_{0}\subset
X(E_{K})$ defines a _Culler–Shalen seminorm_ $||\cdot||_{X_{0}}$ on
$H_{1}(\partial E_{K};\mathbb{R})$, satisfying exactly one of the following:

  * •

$||\cdot||_{X_{0}}$ is a norm, and every function $f_{\gamma}$ is non-constant
on $X_{0}$.

  * •

$||\cdot||_{X_{0}}$ is a indefinite seminorm but is not identically zero, and
exactly one function $f_{\gamma_{0}}$ is constant on $X_{0}$.

  * •

$||\cdot||_{X_{0}}$ is zero, and every $f_{\gamma}$ is constant on $X_{0}$.

In the first two cases, we call $X_{0}$ a _norm curve_ or a _seminorm curve_ ,
respectively. The following two propositions about such curves are stated in
the form seen here in [NZ17], but essentially appear in [BZ98, §6].

First, if $X_{0}$ is a norm curve, then we let $s_{0}>0$ be the minimal
positive value of $||\alpha||_{X_{0}}$ over all integral classes $\alpha\in
H_{1}(\partial E_{K};\mathbb{Z})$, and then we let $B_{0}\subset
H_{1}(\partial E_{K};\mathbb{R})$ be the convex, finite-sided polygon on which
$||\cdot||_{X_{0}}\leq s_{0}$. Ni and Zhang state the following in [NZ17,
Theorem 2.2(4)]:

######  Proposition 4.10.

If $X_{0}\subset X(E_{K})$ is a norm curve and $S^{3}_{r}(K)$ is
$\mathrm{PSL}(2,\C)$-abelian for some $r=p/q$, then either $r$ is a boundary
slope for $K$, or $(p,q)$ lies on the boundary $\partial B_{0}$ but is not a
vertex of $B_{0}$.

If instead $X_{0}$ is a seminorm curve, and the unique $\gamma$ for which
$f_{\gamma}|_{X_{0}}$ is constant has slope $r$, then we say that $X_{0}$ is
an _$r$ -curve_. For example, the curve of abelian characters of
$\pi_{1}(E_{K})$ is a $0$-curve, since $f_{\lambda}\equiv 0$ on that curve. Ni
and Zhang state the following regarding such curves [NZ17, Theorem 2.5(3)]:

######  Proposition 4.11.

If $X_{0}\subset X(E_{K})$ is an $r$-curve containing the character of an
irreducible representation, and if $S^{3}_{s}(K)$ is
$\mathrm{PSL}(2,\C)$-abelian for some slope $s$, then either $s$ is a boundary
slope for $K$, or $\Delta(r,s)=1$.

Combining these facts with Theorem 4.1 allows us to prove Theorem 4.9:

###### Proof of Theorem 4.9.

Suppose that the character variety $X(E_{K})$ contains a norm curve. Then
$\partial B_{0}$ contains only finitely many integral classes, and a theorem
of Hatcher [Hat82] says that $K$ has only finitely many boundary slopes, so
according to Proposition 4.10 there can only be finitely many
$\mathrm{PSL}(2,\C)$-abelian slopes for $K$.

Now suppose instead that there are $r\neq s$ such that $X(E_{K})$ contains an
$r$-curve and an $s$-curve, both of which contain an irreducible character.
Then Proposition 4.11 says that if $t$ is a $\mathrm{PSL}(2,\C)$-abelian slope
for $K$, then either $t$ is a boundary slope, or $\Delta(r,t)=\Delta(s,t)=1$.
There are only finitely many of the former, and at most two of the latter, so
again $K$ has finitely many $\mathrm{PSL}(2,\C)$-abelian slopes.

In the remaining cases, there is some fixed $r=p/q$ such that every curve
$X_{0}\subset X(E_{K})$ containing an irreducible character must satisfy
either

  * •

$X_{0}$ is an $r$-curve, or

  * •

every function $f_{\gamma}$ is constant on $X_{0}$.

Let $\gamma_{0}\subset\partial E_{K}$ denote the slope corresponding to $r$.
We claim that $f_{\gamma_{0}}$ must be constant on any irreducible component
$C_{0}\subset X(E_{K})$ that contains the character of an irreducible
representation. Suppose otherwise, for a contradiction. Then $f_{\gamma_{0}}$
is not even locally constant since $C_{0}$ is connected. It follows that at
any smooth point $\chi\in C_{0}$ corresponding to an irreducible
representation, we can intersect $C_{0}$ with a suitably chosen affine
subspace through $\chi$ to get a curve $X_{0}$ on which $f_{\gamma_{0}}$ is
still not locally constant, and some irreducible component of $X_{0}$ will not
be an $r$-curve, a contradiction.

The character variety $X(E_{K})$ only has finitely many irreducible
components, so it follows that $f_{\gamma_{0}}([\rho])$ and hence
$\operatorname{tr}(\rho(\mu^{p}\lambda^{q}))$ take only finitely many values
as $\rho$ ranges over all irreducible representations of $\pi_{1}(E_{K})$. But
this means that the plane curve used to define $\tilde{A}_{K}(M,L)$ belongs to
the zero set of $f(M^{p}L^{q})$ for some polynomial $f\in\mathbb{Z}[t]$, so
$\tilde{A}_{K}(M,L)$ is itself a polynomial in $M^{p}L^{q}$. This means that
$K$ is $p/q$-thin, so Theorem 4.1 now guarantees that $K$ is a torus knot. ∎

##  5\. $\SL(2,\C)$-averse knots: the proof of Theorem 1.1

In this section, we prove Theorem 1.1. Note that nontrivial torus knots are
$\SL(2,\C)$-averse, again because $(ab+1/n)$-surgery on $T_{a,b}$ is a lens
space for every $n\in\mathbb{Z}$ and hence $\SL(2,\C)$-abelian. It thus
remains to prove the other direction of Theorem 1.1, stated as:

######  Theorem 5.1.

If $K\subset S^{3}$ is an $\SL(2,\C)$-averse knot, then $K$ is a torus knot.

We first observe the following:

######  Lemma 5.2.

Hyperbolic knots are never $\SL(2,\C)$-averse.

###### Proof.

If $K$ is hyperbolic, then the 6-theorem [Ago00, Lac00] implies that all but
finitely many Dehn surgeries on $K$ are closed hyperbolic manifolds. If
$S^{3}_{r}(K)$ is hyperbolic, then there is a faithful representation

$\pi_{1}(S^{3}_{r}(K))\to\SL(2,\C)$

[CS83, Proposition 3.1.1], which has nonabelian image. Therefore, only the
finitely many non-hyperbolic surgeries can possibly be $\SL(2,\C)$-abelian. ∎

Therefore, to prove Theorem 1.1, we just need to show that there are no
$\SL(2,\C)$-averse satellite knots. This is the aim of the rest of the
section, and we proceed in a manner that is similar in spirit to our proof
that there are no thin satellite knots.

######  Lemma 5.3.

Let $r\in\mathbb{Q}$. If $r$-surgery on a satellite knot $K=P(C)$ is
$\SL(2,\C)$-abelian, then so is $r$-surgery on $P(U)$.

###### Proof.

Our proof is essentially the same as Sivek and Zentner’s proof of [SZ22,
Proposition 10.2], which is the analogue of this lemma with $\mathrm{SU}(2)$
in place of $\SL(2,\C)$. In particular, we use the standard fact that there is
a degree-1 map pinching the exterior $E_{C}=S^{3}\setminus\nu(C)$ onto the
unknot exterior $E_{U}=S^{1}\times D^{2}$, that restricts to a homeomorphism

$\displaystyle\partial E_{C}$ $\displaystyle\to\partial E_{U}$
$\displaystyle\mu_{C},\lambda_{C}$ $\displaystyle\mapsto\mu_{U},\lambda_{U}.$

We can glue the pattern exterior to each of these, and extend this pinching
map across it as the identity map, to get a degree-1 map

$E_{K}=E_{P(C)}\to E_{P(U)}$

sending meridians to meridians and longitudes to longitudes. The induced map
on $\pi_{1}$ is surjective and takes $\mu_{K}^{p}\lambda_{K}^{q}$ to
$\mu_{P(U)}^{p}\lambda_{P(U)}^{q}$ where $r=p/q$, so it descends to a
surjection

$\pi_{1}(S^{3}_{r}(K))\to\pi_{1}(S^{3}_{r}(P(U))).$

Any nonabelian representation

$\pi_{1}(S^{3}_{r}(P(U)))\to\SL(2,\C)$

would therefore induce a nonabelian representation of $\pi_{1}(S^{3}_{r}(K))$,
and since the latter does not exist it follows that $r$-surgery on $P(U)$ must
be $\SL(2,\C)$-abelian as well. ∎

######  Lemma 5.4.

Let $K=P(C)$ be a satellite knot whose pattern $P$ has winding number $w\geq
1$, and let $r\in\mathbb{Q}$. If $r$-surgery on $C$ is not
$\SL(2,\C)$-abelian, then neither is $rw^{2}$-surgery on $K$.

The analogue of this lemma with $\mathrm{SU}(2)$ in place of $\SL(2,\C)$ was
first proved by Sivek and Zentner in [SZ22, Proposition 10.3]. Our proof
follows the same ideas, but is considerably longer: the proof in [SZ22] begins
by assuming that a given representation of $\pi_{1}(E_{C})$ into
$\mathrm{SU}(2)$ is diagonal on the peripheral subgroup (corresponding to Case
1 in our proof below), but here we must consider two additional cases in which
the peripheral values are parabolic elements of $\SL(2,\C)$.

###### Proof of Lemma 5.4.

Write $r=p/q$, with $p$ and $q$ relatively prime and $q>0$. Let
$d=\gcd(q,w^{2})$, and note that

$rw^{2}=\frac{pw^{2}/d}{q/d}$

in lowest terms, since $\gcd(pw^{2}/d,q/d)=1$. Let
$E_{C}=S^{3}\setminus\nu(C)$ denote the exterior of $C$, and fix a nonabelian
representation

$\displaystyle\rho:\pi_{1}(E_{C})$ $\displaystyle\to\SL(2,\C)$
$\displaystyle\mu_{C}^{p}\lambda_{C}^{q}$ $\displaystyle\mapsto 1,$

which exists by hypothesis. We will assume, by replacing $\rho$ with a
conjugate if necessary, that $\rho(\mu_{C})$ is in Jordan normal form. This
means that $\rho(\mu_{C})$ is a diagonal matrix unless its eigenvalues are
equal, in which case these equal eigenvalues are both $+1$ or both $-1$ since
$\det(\rho(\mu_{C}))=1$. We also note that $\rho(\mu_{C})$ cannot be either of
the central elements $\pm I$, since $\rho(\mu_{C})$ normally generates the
nonabelian image of $\rho$.

Letting $V=(S^{1}\times D^{2})\setminus\nu(P)$, we may decompose the exterior
$E_{K}=S^{3}\setminus\nu(K)$ as

$E_{K}\cong E_{C}\cup_{T}V.$

Our goal will be to extend $\rho$ across $\pi_{1}(V)$, by constructing an
_abelian_ representation

$\rho_{V}:\pi_{1}(V)\to\SL(2,\C)$

such that

$\displaystyle\rho_{V}(\mu_{C})=\rho_{V}([\\{\mathrm{pt}\\}\times\partial
D^{2}])$ $\displaystyle=\rho(\mu_{C})$
$\displaystyle\rho_{V}(\lambda_{C})=\rho_{V}([S^{1}\times\\{\mathrm{pt}\\}])$
$\displaystyle=\rho(\lambda_{C})$
$\displaystyle\rho_{V}(\mu_{P}^{pw^{2}/d}\lambda_{P}^{q/d})$
$\displaystyle=1.$

Then $\rho$ and $\rho_{V}$ will glue together to give a representation

$\pi_{1}(E_{K})\to\SL(2,\C)$

that descends to a nonabelian representation of $\pi_{1}(S^{3}_{rw^{2}}(K))$,
as desired.

Observe that any abelian representation $\rho_{V}:\pi_{1}(V)\to\SL(2,\C)$ must
factor through

$H_{1}(V;\mathbb{Z})\cong\mathbb{Z}^{2},$

which is generated by the classes $[\mu_{P}]$ and $[\lambda_{C}]$ subject to
the relations

$[\mu_{C}]=w[\mu_{P}]\,\textrm{ and }\,[\lambda_{P}]=w[\lambda_{C}].$

The desired $\rho_{V}$ is therefore determined by the elements
$\rho_{V}(\mu_{P})$ and $\rho_{V}(\lambda_{C})$, and the value
$\rho_{V}(\lambda_{C})=\rho(\lambda_{C})$ is already fixed by $\rho$ – we note
that this implies that $\rho_{V}(\lambda_{P})=\rho(\lambda_{C})^{w}$ – so we
wish to find $\rho_{V}(\mu_{P})\in\SL(2,\C)$ satisfying

$\displaystyle\rho_{V}(\mu_{P})^{w}$ $\displaystyle=\rho(\mu_{C})$
$\displaystyle\rho_{V}(\mu_{P}^{pw^{2}/d}\lambda_{P}^{q/d})$
$\displaystyle=1.$

We consider several cases separately, recalling that $\rho(\mu_{C})$ is in
Jordan normal form.

Case 1: $\rho(\mu_{C})\in\SL(2,\C)$ is a diagonal matrix
$\left(\begin{smallmatrix}\alpha&0\\\ 0&\alpha^{-1}\end{smallmatrix}\right)$,
where $\alpha\neq\pm 1$.

In this case, $\rho(\mu_{C})$ is diagonal but not central, and
$\rho(\lambda_{C})$ commutes with it, so $\rho(\lambda_{C})$ must be diagonal
as well: we can write

$\displaystyle\rho(\mu_{C})$ $\displaystyle=\begin{pmatrix}\alpha&0\\\
0&\alpha^{-1}\end{pmatrix},$ $\displaystyle\rho(\lambda_{C})$
$\displaystyle=\begin{pmatrix}\beta&0\\\ 0&\beta^{-1}\end{pmatrix}$

where $\alpha^{p}\beta^{q}=1$. We let $\alpha=se^{i\theta}$ and
$\beta=te^{i\phi}$, where $s$ and $t$ are positive real numbers and
$\theta,\phi\in\mathbb{R}$, and then the condition $\alpha^{p}\beta^{q}=1$
means that

$s^{p}t^{q}=1\,\textrm{ and }\,p\theta+q\phi=2\pi m$

for some integer $m$. Let us write

$\eta=s^{1/w}\exp\left(\frac{(\theta+2\pi k)i}{w}\right)$

for some integer $k$, chosen so that

$m+pk\equiv 0\pmod{d};$

we note that this is possible since $d=\gcd(q,w^{2})$ divides $q$ and is thus
coprime to $p$, meaning that $p$ is invertible $\\!\\!\\!\pmod{d}$. Then
$\eta^{w}=\alpha$, so we set

$\displaystyle\rho_{V}(\mu_{P})$ $\displaystyle=\begin{pmatrix}\eta&0\\\
0&\eta^{-1}\end{pmatrix},$ $\displaystyle\rho_{V}(\lambda_{P})$
$\displaystyle=\begin{pmatrix}\beta^{w}&0\\\ 0&\beta^{-w}\end{pmatrix}$

so that $\rho_{V}(\mu_{P})^{w}=\rho(\mu_{C})$ and
$\rho_{V}(\lambda_{P})=\rho(\lambda_{C})^{w}$.

Having done this, we now compute that

$\rho_{V}(\mu_{P}^{pw^{2}/d}\lambda_{P}^{q/d})=\begin{pmatrix}z&0\\\
0&z^{-1}\end{pmatrix},$

where $z=\eta^{pw^{2}/d}\cdot(\beta^{w})^{q/d}$ satisfies

$\displaystyle z$
$\displaystyle=s^{pw/d}t^{qw/d}\cdot\exp\left(\frac{pw(\theta+2\pi
k)i}{d}+\frac{\phi i\cdot qw}{d}\right)$
$\displaystyle=(s^{p}t^{q})^{w/d}\cdot\exp\left(\frac{\big{(}(p\theta+q\phi)+2\pi
pk\big{)}wi}{d}\right)$
$\displaystyle=\exp\left(2\pi\left(\frac{m+pk}{d}\right)\cdot wi\right)$

since $s^{p}t^{q}=1$ and $p\theta+q\phi=2\pi m$. But we chose $k$ so that
$(m+pk)/d$ would be an integer, so $z=1$ and thus
$\rho_{V}(\mu_{P}^{pw^{2}/d}\lambda_{P}^{q/d})=1$, meaning that $\rho_{V}$ is
the desired representation.

Case 2: $\rho(\mu_{C})$ is a $2\times 2$ Jordan block with eigenvalues which
are either both $1$ or both $-1$, where in the latter case we require that $w$
is odd.

Since $\rho(\mu_{C})\neq\pm 1$ commutes with $\rho(\lambda_{C})$, the latter
is also a Jordan block. So, we have

$\displaystyle\rho(\mu_{C})$ $\displaystyle=\epsilon\begin{pmatrix}1&a\\\
0&1\end{pmatrix},$ $\displaystyle\rho(\lambda_{C})$
$\displaystyle=\eta\begin{pmatrix}1&b\\\ 0&1\end{pmatrix}$

for some $a,b\in\C$ with $a\neq 0$ and some $\epsilon,\eta=\pm 1$ with
$\epsilon^{w}=\epsilon$. Then $\rho(\mu_{C}^{p}\lambda_{C}^{q})=1$ is
equivalent to $\epsilon^{p}\eta^{q}=1$ and $ap+bq=0$. We define $\rho_{V}$ by

$\displaystyle\rho_{V}(\mu_{P})$
$\displaystyle=\epsilon\begin{pmatrix}1&a/w\\\ 0&1\end{pmatrix},$
$\displaystyle\rho_{V}(\lambda_{P})$
$\displaystyle=\eta^{w}\begin{pmatrix}1&bw\\\ 0&1\end{pmatrix},$

which satisfies

$\rho_{V}(\mu_{P})^{w}=\rho(\mu_{C})\,\textrm{ and
}\,\rho_{V}(\lambda_{P})=\rho(\lambda_{C})^{w},$

and we compute that

$\displaystyle\rho_{V}(\mu_{P}^{pw^{2}/d}\lambda_{P}^{q/d})$
$\displaystyle=\epsilon^{pw^{2}/d}\begin{pmatrix}1&apw/d\\\
0&1\end{pmatrix}\cdot\eta^{w(q/d)}\begin{pmatrix}1&bqw/d\\\ 0&1\end{pmatrix}$
$\displaystyle=\epsilon^{pw^{2}/d}\eta^{w(q/d)}\begin{pmatrix}1&(ap+bq)w/d\\\
0&1\end{pmatrix}$
$\displaystyle=\epsilon^{pw^{2}/d}\eta^{w(q/d)}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}$

since $ap+bq=0$. If $d$ is odd, then the leading coefficient on the right
satisfies

$\epsilon^{pw^{2}/d}\eta^{w(q/d)}=\left(\epsilon^{pw^{2}/d}\eta^{w(q/d)}\right)^{d}=(\epsilon^{w})^{pw}\eta^{qw}=\epsilon^{pw}\eta^{qw}=(\epsilon^{p}\eta^{q})^{w}=1.$

Otherwise $d=\gcd(q,w^{2})$ is even, hence both $q$ and $w$ must be even; but
then $\eta^{q}=\eta^{w}=1$, and $\epsilon^{p}\eta^{q}=1$ implies that
$\epsilon^{p}=1$, so that

$\epsilon^{pw^{2}/d}\eta^{w(q/d)}=(\epsilon^{p})^{w^{2}/d}(\eta^{w})^{q/d}=1.$

In either case we have $\rho_{V}(\mu_{P}^{pw^{2}/d}\lambda_{P}^{q/d})=1$, so
$\rho_{V}$ is the desired representation.

Case 3: $\rho(\mu_{C})$ is a $2\times 2$ Jordan block with eigenvalues $-1$,
and $w$ is even.

As in the previous case, we can write

$\displaystyle\rho(\mu_{C})$ $\displaystyle=-\begin{pmatrix}1&a\\\
0&1\end{pmatrix},$ $\displaystyle\rho(\lambda_{C})$
$\displaystyle=\eta\begin{pmatrix}1&b\\\ 0&1\end{pmatrix}$

for some $a,b\in\C$ with $a\neq 0$ and some $\eta=\pm 1$, and then
$\rho(\mu_{C}^{p}\lambda_{C}^{q})=1$ is equivalent to $(-1)^{p}\eta^{q}=1$ and
$ap+bq=0$. The problem is that now the matrix $\rho(\mu_{C})$ does not have a
$w$th root. We fix this by considering the nontrivial central character

$\chi:\pi_{1}(E_{C})\twoheadrightarrow H_{1}(E_{C})\to\\{\pm 1\\}$

that sends $\mu_{C}$ to $-1$ and $\lambda_{C}$ to $+1$, and setting
$\rho^{\prime}=\chi\cdot\rho$, so that

$\displaystyle\rho^{\prime}(\mu_{C})$ $\displaystyle=\begin{pmatrix}1&a\\\
0&1\end{pmatrix},$ $\displaystyle\rho^{\prime}(\lambda_{C})$
$\displaystyle=\eta\begin{pmatrix}1&b\\\ 0&1\end{pmatrix}.$

We can extend $\rho^{\prime}$, rather than the original $\rho$, across
$\pi_{1}(V)$ by setting

$\displaystyle\rho_{V}(\mu_{P})$ $\displaystyle=\begin{pmatrix}1&a/w\\\
0&1\end{pmatrix},$ $\displaystyle\rho_{V}(\lambda_{P})$
$\displaystyle=\begin{pmatrix}1&bw\\\ 0&1\end{pmatrix},$

and we have $\rho_{V}(\mu_{P})^{w}=\rho^{\prime}(\mu_{C})$ and
$\rho_{V}(\lambda_{P})=\rho^{\prime}(\lambda_{C})^{w}$ since $w$ is even. Now

$\rho_{V}(\mu_{P}^{pw^{2}/d}\lambda_{P}^{q/d})=\begin{pmatrix}1&(ap+bq)w/d\\\
0&1\end{pmatrix}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},$

so $\rho_{V}$ is the desired extension of $\rho^{\prime}$, and it has
nonabelian image since $\rho^{\prime}$ does.

These three cases cover all possible values of $\rho(\mu_{C})$, so we conclude
that $rw^{2}$-surgery on $K$ cannot be $\SL(2,\C)$-abelian after all. ∎

In preparation for the next result, we note that since $\mathrm{SU}(2)$ is a
subgroup of $\SL(2,\C)$, an $\SL(2,\C)$-abelian 3-manifold is also
$\mathrm{SU}(2)$-abelian, and hence an $\SL(2,\C)$-averse knot is also
$\mathrm{SU}(2)$-averse. In particular, if $K$ is $\SL(2,\C)$-averse, then the
set of slopes $r$ for which $S^{3}_{r}(K)$ is $\SL(2,\C)$-abelian has a unique
accumulation point, which agrees with the limit slope $r(K)$ for
$\mathrm{SU}(2)$-abelian surgeries of Theorem 2.7.

The combination of Lemmas 5.3 and 5.4 then leads to the following, which is an
analogue for $\SL(2,\C)$-averse satellites of Proposition 2.16 on thin
satellites:

######  Corollary 5.5.

Suppose $K=P(C)$ is an $\SL(2,\C)$-averse satellite knot, and let $w$ be the
winding number of the pattern $P$. Then:

  * •

$P(U)$ is nontrivial and $w\geq 1$,

  * •

$P(U)$ is $\SL(2,\C)$-averse, with $r(P(U))=r(K)$,

  * •

the companion $C$ is $\SL(2,\C)$-averse, with $r(K)=w^{2}r(C)$.

###### Proof.

By definition, $K=P(C)$ has infinitely many $\SL(2,\C)$-abelian slopes

$r_{1},r_{2},r_{3},\dots\in\mathbb{Q},$

which are also $\mathrm{SU}(2)$-abelian slopes for $K$ and have a unique
accumulation point $r(K)\in\mathbb{Q}$, as discussed above. Since $K$ is also
$\mathrm{SU}(2)$-averse, Corollary 2.8 says that either $K$ or $\overline{K}$
is an instanton L-space knot. Then $P(U)$ is nontrivial and $w\geq 1$ by Lemma
2.4.

Lemma 5.3 says that each $r_{i}$ is also an $\SL(2,\C)$-abelian slope for
$P(U)$, so $P(U)$ is also $\SL(2,\C)$-averse with limit slope $r(P(U))=r(K)$.

Similarly, Lemma 5.4 says us that each of the slopes $r_{i}/w^{2}$ is also an
$\SL(2,\C)$-abelian slope for $C$, so $C$ is also $\SL(2,\C)$-averse with
limit slope $r(K)/w^{2}$. ∎

We now prove Theorem 5.1, which completes the proof of Theorem 1.1:

###### Proof of Theorem 5.1.

Note that $K$ is not hyperbolic, by Lemma 5.2. If $K$ is a torus knot then we
are done, so it suffices to prove that there are no $\SL(2,\C)$-averse
satellite knots.

Suppose for a contradiction that $K=P(C)$ is an $\SL(2,\C)$-averse satellite
knot with the smallest genus among all $\SL(2,\C)$-averse satellites.
Corollary 5.5 says that $P(U)$ and $C$ are nontrivial and $\SL(2,\C)$-averse,
and that if $P$ has winding number $w$, then $w\geq 1$ and

$r(P(U))=r(K)=w^{2}r(C).$

Then from the relation

$g(K)=g(P(U))+w\cdot g(C),$

we see that $P(U)$ and $C$ have genera strictly less than that of $K$, and are
thus torus knots.

Now we repeat the proof of Theorem 4.1: if we write $P(U)=T_{a,b}$ and
$C=T_{c,d}$, then the above relation between their limit slopes becomes

$ab=r(K)=cdw^{2},$

so $w^{2}$ divides $ab$. Proposition 3.1 then implies that neither $K$ nor
$\overline{K}$ is an instanton L-space knot. But this yields a contradiction:
since $K$ is $\SL(2,\C)$-averse, it is also $\mathrm{SU}(2)$-averse, which
implies by Corollary 2.8 that either $K$ or $\overline{K}$ is an instanton
L-space knot. ∎

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# Adversarial Machine Learning Attacks Against Video Anomaly Detection Systems

Furkan Mumcu, Keval Doshi, Yasin Yilmaz
University of South Florida
4202 E Fowler Ave, Tampa, FL 33620
{furkan, kevaldoshi<EMAIL_ADDRESS>
###### Abstract

Anomaly detection in videos is an important computer vision problem with
various applications including automated video surveillance. Although
adversarial attacks on image understanding models have been heavily
investigated, there is not much work on adversarial machine learning targeting
video understanding models and no previous work which focuses on video anomaly
detection. To this end, we investigate an adversarial machine learning attack
against video anomaly detection systems, that can be implemented via an easy-
to-perform cyber-attack. Since surveillance cameras are usually connected to
the server running the anomaly detection model through a wireless network,
they are prone to cyber-attacks targeting the wireless connection. We
demonstrate how Wi-Fi deauthentication attack, a notoriously easy-to-perform
and effective denial-of-service (DoS) attack, can be utilized to generate
adversarial data for video anomaly detection systems. Specifically, we apply
several effects caused by the Wi-Fi deauthentication attack on video quality
(e.g., slow down, freeze, fast forward, low resolution) to the popular
benchmark datasets for video anomaly detection. Our experiments with several
state-of-the-art anomaly detection models show that the attackers can
significantly undermine the reliability of video anomaly detection systems by
causing frequent false alarms and hiding physical anomalies from the
surveillance system.

##  1 Introduction

In recent years, video anomaly detection has become a popular research topic
with the increasing availability of video-recording hardware and the promising
capabilities of artificial intelligence (AI) algorithms. The fast growing
surveillance video streams prevent manual monitoring by human operators and
necessitate automated monitoring by AI. Deep Neural Networks (DNNs), in
particular Convolutional Neural Networks (CNNs), are known to provide
promising results for video anomaly detection. Since video surveillance is
often used for providing physical safety and security by detecting important
events such as accidents, burglaries, or illegal activities, automated anomaly
detection systems need to be dependable, secure, and robust to attacks.
Because of the sensitive nature of video surveillance, raising frequent false
alarms or not being able to detect the important anomalies could have serious
consequences in real world. In this paper, we show that with an easy-to-
implement adversarial machine learning attack to state-of-the-art video
anomaly detection algorithms, it is possible to cause frequent false alarms or
prevent anomalies from being detected.

Adversarial machine learning attacks damage the liability of machine learning
systems. Although originally adversarial attacks have been widely investigated
for image recognition, lately, attacks targeting video action recognition
models started to gain attention [13, 29, 17, 24]. DNNs, in particular CNNs,
which are commonly used in video understanding models, are highly vulnerable
to adversarial attacks. [28, 10]. The fundamental characteristic of an
adversarial attack is to create input data with perturbations which are
undetectable by humans while they result in errors for the machine learning
algorithms. In the literature, adversarial data are mostly generated by adding
perturbations to the original inputs with an inherent assumption that the
attacker somehow has access to the data input to the machine learning model.
While this assumption easily holds for machine learning models open to public
use or by assuming the attacker is an authorized user (or an intruder), it
needs further justification for video surveillance systems which are typically
private systems with restricted access. In this paper, we demonstrate how an
adversarial machine learning attack can be performed against a video anomaly
detection system through an easy-to-implement cyber-attack.

Wi-Fi deauthentication attack is a type of denial-of-service (DoS) attack,
which targets the communication between clients and a Wi-Fi access point. The
main purpose of this attack is to send deauthentication packets to the Wi-Fi
access point and originate a network outage for the connected devices as a
result. In our experiments, we observed that if surveillance cameras are
connected to a network which is being attacked by Wi-Fi deauthentication, then
the transmitted videos will have malfunctions. These malfunctions include
frame drops, lagging, slowing down, freezing, and drops in the quality of the
frames. We demonstrate that DNN-based anomaly detection methods are prone to
increased false alarms due to such malfunction effects. Furthermore, we show
that an attacker can use the freezing effect to hide a physical anomaly from
the surveillance system by carefully synchronizing the cyber-attack with the
physical anomaly.

In summary, our goal in this paper is to demonstrate that an easy-to-implement
cyber-attack can produce an adversarial machine learning attack to a video
surveillance system. To this end, we followed the following steps: (i) First,
we created a testbed with a security camera connected to Wi-Fi, a router, and
an attacker software to observe the effects of Wi-Fi deauthentication attack
on the surveillance video. (ii) Then, we applied these effects to two popular
benchmark datasets, namely the CUHK Avenue and ShanghaiTech Campus datasets,
to obtain adversarial data for testing benchmark algorithms. (iii) Finally, we
used our new datasets simulating the Wi-Fi deauthentication attack to evaluate
performance of several state-of-the-art DNN-based anomaly detection methods:
Future Frame Prediction [18], Memory-guided Normality for Anomaly Detection
(MNAD) [23], Modular Online Video Anomaly Detector (MOVAD) [7]. Our
contributions in this paper can be summarized as follows:

  * •

We propose the first adversarial machine learning attack that targets video
anomaly detection models.

  * •

We use Wi-Fi deauthentication attack to generate an adversarial machine
learning attack, and to our knowledge, this is the first approach which uses a
cyber-attack for performing adversarial machine learning attack on video
understanding models (e.g., action recognition, anomaly detection, etc.).

Figure 1: Digram of Wi-Fi deauthentication attack, which targets the Wi-Fi
connection between the surveillance camera and the anomaly detection model.

##  2 Related Works

Video anomaly detection is a fast-growing subject in machine learning. The
main goal of video anomaly detection is successfully reporting unexpected
events while controlling false alarms. Although the early methods use hand-
designed features [2, 3], most of the recent works use DNNs [11, 12, 20, 26,
27, 31, 21, 25, 14, 5, 8]. The recent DNN-based methods can be categorized
into two main groups, prediction-based [18, 16, 4, 6] and reconstruction-based
[9, 11, 20, 22, 23]. They, respectively, compare the error in predicting the
next frame and reconstructing the observed frames with a threshold to detect
anomalies.

Although adversarial attacks on images have been investigated for several
years since [10], the vulnerabilities of video recognition models only
recently attracted attention [15, 30, 32, 33, 17, 24]. Adversarial attacks on
videos have been investigated in both white-box and black-box settings. The
difference between these two settings is the accessibility to the knowledge of
target model. While in the white-box setting, an attacker has full access to
the model including its architecture and parameters; in the black-box setting,
the attacker has access only to the output labels.

Adversarial black-box attacks for video classification models are relatively
new compared to the white-box attacks. The most common black-box attack
pattern is sending queries to the target model and estimating gradients to
generate adversarial data. PatchAttack (V-BAD) [15] is the first proposed
black-box video attack. They proposed a method to generate perturbations for
each frame of a video, and then updated their perturbations with queries.
HeuristicAttack [30] is another black-box adversarial attack which uses a
query-based attack strategy to heuristically select the key frames to be
attacked. Similarly, SparseAttack [32] aims to select key frames by using
reinforcement learning. Motion-sampler attack [33] is another attack which
uses query-based strategy, but they aim to take advantage of using motion-
excited sampler to reduce required queries. Geo-Trap [17] uses geometric
transformations to parameterize the temporal space, and as a result they
generate successful perturbations with fewer queries.

The attack we introduce in this paper is also a black-box attack, but it
differs from the existing black-box attacks in two main aspects. The first
difference is that all of these attacks target action recognition models while
our attack targets anomaly detection models. And the other difference is that
while these attacks use query-based strategies to shape their perturbations,
we use a cyber-attack to generate our adversarial data.

Over-the-Air Flickering Attack [24] is a white-box adversarial machine
learning attack. Similar to our work, they demonstrate how to implement their
attack in a real-world scenario. Their idea is attacking the RGB stream of the
data with the help of RGB bulbs. Therefore, in terms of applicability in real-
world, it is the most similar attack to ours. However, their target models are
action recognition models and they use a physical intervention, instead of a
cyber-attack, to generate adversarial data.

##  3 Methodology

Existing video anomaly detection methods typically utilize appearance and/or
motion features. Specifically, appearance features are extracted using object
detection models, and motion features are extracted through optical flow
computation. However, the accuracy of object detection and optical flow
computation are conditional on the video being transmitted without any
disruptions and signal loss. Moreover, it is known that wireless networks are
far more vulnerable to cyber-attacks than wired networks. [1]. Due to the
susceptible nature of wireless networks, we believe that an automated video
surveillance framework would be adversely affected if its associated wireless
network was attacked, especially in a stealthy manner. Instead of a brute-
force DoS attack to completely stop the video streaming, which would make the
attack obvious to the defender, attackers may balance the intensity of their
disruption to make it look like a natural connectivity issue.

To evaluate our hypothesis, we set up a testbed using an Amazon Blink IP
camera transmitting data to a mobile app, emulating a video surveillance
system, as shown in Fig. 1. We targeted our system via a Wi-Fi
deauthentication attack and an ARP spoofing attack using the Evil Limiter
software111https://github.com/bitbrute/evillimiter. By sending
deauthentication packets, an attacker can potentially target the communication
between clients and routers, which can result in clients experiencing
connection outages. On the other hand, Evil Limiter limits the bandwidth of
devices on a local network, effectively disrupting communication. Using an ARP
spoofing tool like Evil Limiter allows the adversary more control over the
attack magnitude since the goal of the adversary is to stealthily attack a
video surveillance framework.

Next, we discuss the threat model posed by the proposed wireless network
attacks, as well as a simulated data injection attack. Our goal in the
following sections is to present the observed effects of the considered
attacks to the video stream and demonstrate the effect of adversarial data
following these effects on the existing state-of-the-art anomaly detection
methods.

###  3.1 Threat Model

Cyber-Attacks to Wi-Fi Network: We begin by exploring the effects of the two
network attacks on the the transmitted video. First, we limited the bandwidth
to mimic a scenario where an adversary attempts to stealthily disrupt the
video stream by affecting the communication quality. Then, we attempted a more
aggressive attack using deauthentication packets, which leads to the security
camera disconnecting from the Wi-Fi node. After empirically analyzing the
received video under both attacks, we broadly categorize the observed effects
into four classes:

  * •

_Slowing effect:_ Once the available bandwidth for the security camera drops
below a certain threshold, we begin to notice the transmitted video being
sluggish or playing in slow motion, which is caused due to a longer delay in
frames being transmitted from the memory buffer.

  * •

_Low-resolution effect:_ During the attack, we also notice that the camera
begins to transmit video at a lower resolution as a way of compensating for
poor network quality.

  * •

_Freezing effect:_ In the case of a deauthentication attack, the camera is
unable to transmit any more frames, and hence the video feed appears to be
frozen. Empirically, the duration of the freeze is related to the intensity of
the attack.

  * •

_Fast-forwarding effect:_ Finally, after the connection gets restored, the
video fast forwards to the current frame, causing a rapid motion. Usually, the
fast forward effect lasts for a fixed duration, irrespective of how long the
attack lasted.

Data Injection Attack: In addition to the proposed cyber-attacks, we also
consider an imaginary scenario in which the attacker has access to the
transmitted data and thus is able to perform a data injection attack. Here,
the adversary infrequently freezes or replicates the frames for an extremely
small duration, without causing any noticeable difference to the human eye.

Theoretically, it is difficult to detect a network attack since it is almost
indistinguishable from video corruption due to ordinary connection problems.
It is important to note that while slowing, freezing, and fast forwarding
occur consequently, low resolution can be observed during the entire attack
duration. On the other hand, the video transmitted during a replicating attack
seems identical to the original video and cannot be differentiated by even a
human operator. To evaluate the effectiveness of existing state-of-the-art
approaches, we imitate these effects on public benchmark datasets and generate
a new set of adversarial videos, as discussed next in Section 3.2.

###  3.2 Generating Adversarial Datasets

We leverage two popular video anomaly detection datasets, the CUHK Avenue
dataset [19] and the ShanghaiTech Campus dataset [20]. We imitated the effects
discussed in Section 3.1 and generated new adversarial datasets. Specifically,
we consider two parameters for generating a new adversarial video: the
duration $D$ of the attack, and the attack onset time $t$. The attack onset
time $t$ is randomly selected for each video. We next describe each generated
dataset in detail.

  * •

Slow-Freeze-Fast Dataset: Similar to what we observe during a network attack,
we first introduce a slowing effect followed by the freezing and fast-
forwarding effects. The lengths of these portions are respectively given by
$D/3$, $D/6$, and $D/2$.

  * •

Low-Resolution Dataset: For the low-resolution dataset, we lower the
resolution of all frames for a duration of $D$.

  * •

Combined Dataset: Finally, we also generate a combined dataset by combining
the effects from the first two datasets.

##  4 Experiments

In this section, we evaluate the performance of existing state-of-the-art
approaches, namely the Future Frame Prediction (FFPN) [18], Memory-guided
Normality for Anomaly Detection (MNAD) [23], and Modular Online Video Anomaly
Detector (MOVAD) [7] on the generated adversarial datasets. We specifically
consider these approaches since their implementations are publicly available,
and can be easily modified to work with the proposed datasets. We evaluate the
performance of all existing approaches using the commonly used Area under
Curve (AUC) metric.

Figure 2: Performance drops of FFPN [18] with adversarial ShanghaiTech dataset
(top), MNAD [23] with adversarial Avenue dataset (middle), MOVAD [7] with
adversarial Avenue dataset (bottom).

###  4.1 Network Attacks

Method | Avenue $\to$ SFF | ShanghaiTech $\to$ SFF
---|---|---
FFPN [18] | 0.85 $\to$ 0.684 | 0.728 $\to$ 0.55
MNAD [23] | 0.885 $\to$ 0.74 | -
MOVAD [7] | 0.887 $\to$ 0.77 | -
Table 1: Performance (AUC) drop in FFPN [18], MNAD [23], and MOVAD [7] due to
the Slow-Fast-Freeze effect. The attack duration is $D=500$.

Figure 3:  The variation in PSNR statistic based on the effect caused by the
adversarial attack. The first figure represents the statistic for the
ShanghaiTech dataset under the slowing, freezing, and fast-forwarding effects.
The second figure represents the Avenue dataset under the low-resolution
effect.

Since all the considered approaches leverage motion features for detecting
anomalies like the vast majority of existing methods, we notice a steady drop
in their performances as the attack intensity increases. In Fig. 2, we compare
the performance of the considered methods on the benchmark datasets by varying
the attack duration $D$. It should be noted that an attack lasting for 500
frames is equivalent to 17 seconds, which is a reasonably short duration. As
summarized in Table 1, the considered state-of-the-art methods are all prone
to the Slow-Freeze-Fast effect. It is interesting to see that FFPN is robust
to the low-resolution effect (Fig. 2). The trained models for MNAD and MOVAD
on ShanghaiTech dataset are not publicly available.

###  4.2 Increased False Alarms

For detecting anomalies, both MNAD and FFPN compute the Peak Signal to Noise
Ratio (PSNR) between the actual frame and predicted frame. Generally, a drop
in the PSNR values constitute an anomaly. Hence, apart from the AUC-based
performance metrics, we also observe the adversarial effect of different
network attacks on the PSNR values in Fig. 3. As discussed in Section 3.1,
there are four distinct types of effects caused by network attacks: slowing,
freezing, fast forwarding, and low resolution. Each of these effects has a
unique impact on the PSNR values. Slow effect results in frequent drops and
rises; fast-forwarding effect results in a large initial drop and maintained
the PSNR value at a low level throughout the effect; low resolution effect
results in drops in PSNR values relative to the original values; and finally,
freezing effect results in stabilized PSNR values. With the exception of
freezing, all effects result in a decrease in PSNR values, leading to false
alarms.

Figure 4: Comparison of the PSNR statistic based on the magnitude of the
replicating attack on the third test video from the Avenue dataset. We show
the PSNR statistic for the original video (left), the PSNR statistic on
Replicated Dataset 1 (middle) and Replicated Dataset 2 (right).

###  4.3 Hiding a Physical Anomaly

While slowing, fast-forwarding, and low-resolution effects result in false
alarms, the freeze effect prevents the PSNR from dropping and thereby
preventing an anomaly from being detected. From the empirical evidence using
our testbed, we observe that the duration of the freeze effect is dependent on
the intensity of the Wi-Fi deauthentication attack. Thus, if an attacker
manages to synchronize a cyber-attack with a physical attack, such as
burglary, it is possible to conceal the physical attack until the freeze
attack is completed.

To validate our hypothesis, we created a new adversarial video using the
Avenue dataset’s second test video. To generate this adversarial video, we
followed the same procedure as described in Section 3.2 for generating the
Slow-Freeze-Fast dataset. However, we increased the duration of the freeze
effect such that it is three times longer than the duration of the slow
effect. We applied these effects to the section that contains the anomaly. We
show the effect of extended freeze attack in Fig. 5, where we notice that
there is no change in the PSNR statistic when the anomaly occurs.

Figure 5: PSNR values of a video with extended freeze effect compared to the
original video. The drop in PSNR, which corresponds to a physical anomaly
successfully detected in the original video, is masked by the freeze effect.

###  4.4 Replicating Attack

Dataset | MNAD [23] | FFPN [18]
---|---|---
Avenue | 0.885 | 0.85
Replicated 1 | 0.831 | 0.81
Replicated 2 | 0.819 | 0.795
Table 2: Performance (AUC) drop in MNAD and FFPN due to Replicating Attack.

We also consider an imaginary scenario in which the adversary has access to
the surveillance system, and hence can directly manipulate the data being
transmitted, similar to the scenario commonly considered in the adversarial
machine learning literature. The goal of the adversary would be to raise
unnecessary false alarms in a stealthy manner. Specifically, we consider a
replicating attack in which the adversary replicates or freezes the frames for
an extremely small duration such that it is unnoticeable to the human eye. To
test the effectiveness of our method, we generate two datasets from the Avenue
dataset. In the first dataset, we replicate one frame every 10 frames.
Similarly, for the second dataset we replicate two frames every 10 frames. We
notice that the AUC drops from 0.855 to 0.795 for the FFPN approach and from
0.885 to 0.819 for the MNAD model. We show the performance drops caused by the
replicating attack in Table 2.

More importantly, we observe many frequent drops in the PSNR values, which
indicate that the system may raise false alarms constantly. In Fig. 4, we show
the PSNR statistic of the third video from the Avenue dataset, for both the
original video and videos from our replicating attack datasets.

##  5 Conclusions

Adversarial machine learning attacks on video recognition models have recently
become a subject of increasing interest. In this work, we, for the first time
in the literature, demonstrated an adversarial attack directed at video
anomaly detection methods. Our adversarial attacks are generated using easy-
to-implement cyber-attacks, in particular the Wi-Fi deauthentication and ARP
spoofing attacks. We observe significant performance deterioration in three
state-of-the-art anomaly detection methods because of our attack.
Additionally, upon examining their decision statistics, we discovered numerous
false alarms caused by our network attacks. We also demonstrate that with the
appropriate synchronization, it is possible to conceal a real-world anomaly
such as a burglary using the explained cyber-attack. For future work, we
intend to conduct additional research on adversarial attacks on video anomaly
detection systems and develop defense mechanisms against them.

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