Commensurability classes of (−2<i>,</i>3<i>,n</i>) pretzel knot complements

Macasieb, Melissa L.; Mattman, Thomas W.

doi:10.2140/agt.2008.8.1833

Cited by 16 publications

(18 citation statements)

References 10 publications

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“…It has also been shown in that the

(- 2, 3, n)

pretzel knot is slim provided that it is hyperbolic and that

n

is not divisible by 3. In fact,

(- 2, 3, n)

is hyperbolic precisely when

n \notin {1, 3, 5}

; see [, p. 1834]. We refer the reader to [, p. 2] for a discussion of these facts.…”

Section: Some Computationsmentioning

confidence: 99%

A sheaf‐theoretic SL(2,C) Floer homology for knots

Côté

Manolescu

2019

Proc. London Math. Soc.

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Using the theory of perverse sheaves of vanishing cycles, we define a homological invariant of knots in three‐manifolds, similar to the three‐manifold invariant constructed by Abouzaid and the second author. We use spaces of SL(2,C) flat connections with fixed holonomy around the meridian of the knot. Thus, our invariant is a sheaf‐theoretic SL(2,C) analogue of the singular knot instanton homology of Kronheimer and Mrowka. We prove that for two‐bridge and torus knots, the SL(2,C) invariant is determined by the l‐degree of the trueÂ‐polynomial. However, this is not true in general, as can be shown by considering connected sums of knots.

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“…It has also been shown in that the

(- 2, 3, n)

pretzel knot is slim provided that it is hyperbolic and that

n

is not divisible by 3. In fact,

(- 2, 3, n)

is hyperbolic precisely when

n \notin {1, 3, 5}

; see [, p. 1834]. We refer the reader to [, p. 2] for a discussion of these facts.…”

Section: Some Computationsmentioning

confidence: 99%

A sheaf‐theoretic SL(2,C) Floer homology for knots

Côté

Manolescu

2019

Proc. London Math. Soc.

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“…Their work provides criteria for checking whether or not a hyperbolic knot complement is the only knot complement in its commensurability class. Specifically, we have the following theorem coming from Reid and Walsh's work in [37,Section 5]; this version of the theorem can be found at the beginning of [23]. Theorem 7.1.…”

Section: Commensurability Classes Of Hyperbolic Pretzel Knot Complementsmentioning

confidence: 99%

“…Macasieb and Mattman use this criterion in [23] to show that for any hyperbolic pretzel knot of the form K 1 −2 , 1 3 , 1 n , n ∈ Z \ {7}, its knot complement S 3 \ K 1 −2 , 1 3 , 1 n is the only knot complement in its commensurability class. The main challenge in their work was showing that these knots admit no hidden symmetries.…”

Section: Commensurability Classes Of Hyperbolic Pretzel Knot Complementsmentioning

confidence: 99%

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Mutations and short geodesics in hyperbolic 3-manifolds

Millichap

2017

Communications in Analysis and Geometry

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ABSTRACT. In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurability classes by analyzing their cusp shapes.The knot complements in each class differ by a topological cut-and-paste operation known as mutation. Ruberman has shown that mutations of hyperelliptic surfaces inside hyperbolic 3-manifolds preserve volume. Here, we provide geometric and topological conditions under which such mutations also preserve the initial (complex) length spectrum.This work requires us to analyze when least area surfaces could intersect short geodesics in a hyperbolic 3-manifold.

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“…Aside from the figure-eight, there are no knots with hidden symmetries with at most fifteen crossings [6] and no two-bridge knots with hidden symmetries [11]. Macasieb-Mattman showed that no hyperbolic (−2, 3, n) pretzel knot, n ∈ Z, has hidden symmetries [8]. Hoffman showed the dodecahedral knots are commensurable with no others [7].…”

mentioning

confidence: 99%

Hidden symmetries via hidden extensions

Chesebro

DeBlois

2017

Proc. Amer. Math. Soc.

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Abstract. This paper introduces a new approach to finding knots and links with hidden symmetries using "hidden extensions", a class of hidden symmetries defined here. We exhibit a family of tangle complements in the ball whose boundaries have symmetries with hidden extensions, then we further extend these to hidden symmetries of some hyperbolic link complements.A hidden symmetry of a manifold M is a homeomorphism of finite-degree covers of M that does not descend to an automorphism of M . By deep work of Margulis, hidden symmetries characterize the arithmetic manifolds among all locally symmetric ones: a locally symmetric manifold is arithmetic if and only if it has infinitely many "non-equivalent" hidden symmetries (see [13, Ch. 6]; cf. The partial answers that we know are all negative. Aside from the figure-eight, there are no knots with hidden symmetries with at most fifteen crossings [6] and no two-bridge knots with hidden symmetries [11]. Macasieb-Mattman showed that no hyperbolic (−2, 3, n) pretzel knot, n ∈ Z, has hidden symmetries [8]. Hoffman showed the dodecahedral knots are commensurable with no others [7].Here we offer some positive results with potential relevance to this question. Our first main result exhibits hidden symmetries with the following curious feature. (S).We use a family {L n } of two-component links constructed in previous work [3]. For each n, L n is assembled from a tangle S in B 3 , n copies of a tangle T in S 2 × I, and the mirror image S of S. Figure 1 depicts L 2 , with light gray lines indicating the spheres that divide it into copies of S and T . For n ∈ N and m ≥ 0, we will also use a tangle T n ⊂ L m+n : the connected union of S with n copies of T . For instance, L 2 contains a copy of T 1 (which is pictured in Figure 2 below) and of T 2 .Upon numbering the endpoints of T n as indicated in Figure 2, order-two even permutations determine mutations: mapping classes of ∂(B 3 − T n ) induced by 180-degree rotations of the sphere obtained by filling the punctures. Theorem 1.8. For n ∈ N, the mutation of ∂(B 3 − T n ) determined by (1 3)(2 4) has a hidden extension over a cover of B 3 − T n and for any m ∈ N, taking T n ⊂ L m+n , a hidden extension over a cover of S 3 − L m+n .

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Commensurability classes of (−2,3,n) pretzel knot complements

Cited by 16 publications

References 10 publications

A sheaf‐theoretic SL(2,C) Floer homology for knots

A sheaf‐theoretic SL(2,C) Floer homology for knots

Mutations and short geodesics in hyperbolic 3-manifolds

Hidden symmetries via hidden extensions

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