DISTINGUISHING SOME GENUS ONE KNOTS USING FINITE QUOTIENTS

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DISTINGUISHING SOME GENUS ONE KNOTS USING FINITE

QUOTIENTS

TAMUNONYE CHEETHAM-WEST

Abstract. We give a criterion for distinguishing a prime knot Kin S3from every other

knot in S3using the ﬁnite quotients of π1(S3\K). Using recent work of Baldwin-Sivek, we

apply this criterion to the hyperbolic knots 52, 15n43522, and the three-strand pretzel knots

P(−3,3,2n+ 1) for every integer n.

1. Introduction

Finite quotients of the fundamental group are useful for distinguishing 3-manifolds; in

particular, for a 3-manifold M, a ﬁnite quotient of π1(M) corresponds to the deck group

of a ﬁnite-sheeted regular cover of M. If the fundamental groups of two 3-manifolds M

and Nhave diﬀerent ﬁnite quotients, then π1(M)6∼

=π1(N) and the 3-manifolds Mand N

are not homeomorphic. When Mis a compact 3-manifold, its fundamental group π1(M) is

residually ﬁnite [Hem16], and so the set C(π1(M)) of ﬁnite quotients of π1(M) is non-empty

and inﬁnite.

One of the consequences of the residual ﬁniteness of the fundamental groups of compact 3-

manifolds is that the unknot is the only knot in S3whose knot complement has a fundamental

group with only ﬁnite cyclic quotients. Boileau [Boi18] has conjectured that every prime knot

K⊂S3is completely determined by the set of ﬁnite quotients of π1(S3\K), that is, if for

two prime knots Jand K,π1(S3\J) and π1(S3\K) have the same set of ﬁnite quotients,

then Jand Kare isotopic. By work of Boileau-Friedl [BF20] and Bridson-Reid [BR20], it

is known that the ﬁgure-eight knot 41and the trefoil knot 31are completely determined by

the ﬁnite quotients of their knot groups even amongst the fundamental groups of compact

3-manifolds. Furthermore, Wilkes [Wil19] has shown that knots in S3whose complements

are graph manifolds are distinguished by the ﬁnite quotients of their knot groups.

The purpose of this note is to show:

Theorem 1.1. Let Kbe the knot 52(shown in Figure 1A), one of the hyperbolic pretzel

knots P(−3,3,2n+ 1) (n∈Z)(Figure 1B), or the knot 15n43522 (shown in Figure 1C), then

Kis distinguished from every other knot in S3by the ﬁnite quotients of π1(S3\K).

Theorem 1.1 will follow from our next theorem, using recent work of Baldwin-Sivek

[BS22][BS22b], and the work of Wilkes [Wil18] (as described in Section 3). To state the

theorem, we recall the deﬁnition of a characterizing slope α∈Qfor a knot K. For a knot

K⊂S3, let S3

α(K) be the 3-manifold obtained by α−Dehn surgery on K.

arXiv:2210.16426v2 [math.GT] 11 Nov 2022

(a) 52(b) P(−3,3,2n+ 1) (c) 15n43522

Figure 1. Hyperbolic knots in the statement of Theorem 1.1

Deﬁnition 1.2. A slope αis a characterizing slope for a knot K⊂S3if for any knot

J⊂S3,S3

α(J)∼

=S3

α(K)if and only if Jis isotopic to K.

Theorem 1.3. Let Kbe a hyperbolic knot in S3for which

(1) 0 is a characterizing slope for K,

(2) S3

0(K)is distinguished from every other compact, irreducible 3-manifold by the ﬁnite

quotients of π1(S3

0(K)).

then Kis distinguished from other knots in S3by the ﬁnite quotients of π1(S3\K).

We point out that every knot in S3has inﬁnitely many characterizing slopes [Lac19].

On the other hand, it is known that some knots have inﬁnitely many non-characterizing

(integral) slopes [BM18]. Our note relies on recent work of Baldwin-Sivek [BS22][BS22b]

showing that 0 is a characterizing slope for a family of genus 1 knots. Prior to this, the only

knots previously known to have 0 as a characterizing slope were the unknot (Property R),

the trefoil, and the ﬁgure-eight knot [Gab87].

Theorem 1.3 also gives a diﬀerent proof that the ﬁgure-eight knot is distinguished from

every other knot in S3, recovering a result in [BF20] and [BR20] (see Section 3).

Corollary 1.4. The knot 41is distinguished from every other knot in S3by the ﬁnite quo-

tients of π1(S3\41).

Acknowledgements. The author thanks his advisor Alan Reid, and Ryan Spitler for many

helpful conversations and for their support. The author also thanks John Baldwin and Steven

Sivek for helpful conversations, for the proof of Theorem 3.1, and for comments that improved

this note.

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摘要：

DISTINGUISHINGSOMEGENUSONEKNOTSUSINGFINITEQUOTIENTSTAMUNONYECHEETHAM-WESTAbstract.WegiveacriterionfordistinguishingaprimeknotKinS3fromeveryotherknotinS3usingthenitequotientsof1(S3nK).UsingrecentworkofBaldwin-Sivek,weapplythiscriteriontothehyperbolicknots52,15n43522,andthethree-strandpretzelknotsP(...

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