Algebraic invariants, mutation, and commensurability of link complements

Chesebro, Eric; DeBlois, Jason

doi:10.2140/pjm.2014.267.341

Cited by 14 publications

(32 citation statements)

References 34 publications

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“…The consequence of Propositions 3.10 and 3.12 of [3] below shows that C(∆ n ) is a geometric model for…”

Section: 1mentioning

confidence: 80%

“…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 .…”

Section: (S)mentioning

confidence: 99%

“…In this sub-section we merely summarize geometric details, referring the interested reader to [3] for proofs. In Section 1.2 we prove Theorem 1.8.…”

Section: Existence Of Hidden Extensionsmentioning

confidence: 99%

“…It is equal to H n because their fundamental domains have the same volume. The numbered formulas (8) and (9) above Proposition 6.3 of [3] express the generators of ∆ 0 in terms of a 0 , b 0 , and f 0 and they express Γ T0 in terms of a 0 , a 1 , and f 0 . Therefore, ∆ 0 < H n and Γ T0 < H n .…”

Section: 1mentioning

confidence: 99%

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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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“…The consequence of Propositions 3.10 and 3.12 of [3] below shows that C(∆ n ) is a geometric model for…”

Section: 1mentioning

confidence: 80%

Section: (S)mentioning

confidence: 99%

“…In this sub-section we merely summarize geometric details, referring the interested reader to [3] for proofs. In Section 1.2 we prove Theorem 1.8.…”

Section: Existence Of Hidden Extensionsmentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

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Hidden symmetries via hidden extensions

Chesebro

DeBlois

2017

Proc. Amer. Math. Soc.

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“…Since P consists of two annuli, and since the vertices split into two equivalence classes, ρ 1 determines a one-to-one correspondence between components of P and conjugacy classes of maximal parabolic subgroups ofΓ 1 . This means we are in the setting of [CD,Lem. 2.6.…”

Section: The Example 3-manifoldmentioning

confidence: 99%

A family of non-injective skinning maps with critical points

Gaster

2015

Trans. Amer. Math. Soc.

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Abstract. Certain classes of 3-manifolds, following Thurston, give rise to a 'skinning map', a self-map of the Teichmüller space of the boundary. This paper examines the skinning map of a 3-manifold M , a genus-2 handlebody with two rank-1 cusps. We exploit an orientation-reversing isometry of M to conclude that the skinning map associated to M sends a specified path to itself, and use estimates on extremal length functions to show non-monotonicity and the existence of a critical point. A family of finite covers of M produces examples of non-immersion skinning maps on the Teichmüller spaces of surfaces in each even genus, and with either 4 or 6 punctures.

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A Survey of the Impact of Thurston’s Work on Knot Theory

Sakuma

2020

In the Tradition of Thurston

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Algebraic invariants, mutation, and commensurability of link complements

Cited by 14 publications

References 34 publications

Hidden symmetries via hidden extensions

Hidden symmetries via hidden extensions

A family of non-injective skinning maps with critical points

A Survey of the Impact of Thurston’s Work on Knot Theory

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