Commensurability classes of twist knots

Jim Hoste; Patrick D. Shanahan

Commensurability classes of twist knots

Jim Hoste, Patrick D. Shanahan

Mathematics, Statistics and Data Science

Pitzer College

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we prove that if M _K is the complement of a non-fibered twist knot K in S ³ , then M _K is not commensurable to a fibered knot complement in a Z/2Z-homology sphere. To prove this result we derive a recursive description of the character variety of twist knots and then prove that a commensurability criterion developed by D. Calegari and N. Dunfield is satisfied for these varieties. In addition, we partially extend our results to a second infinite family of 2-bridge knots.

Original language	American English
Pages (from-to)	1-10
Journal	Journal of Knot Theory and its Ramifications
Volume	14
Issue number	1
State	Published - 2005

Keywords

Geometric topology
knot theory
hyperbolic geometry

Disciplines

Geometry and Topology

Access to Document

https://digitalcommons.lmu.edu/math_fac/50

Cite this

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title = "Commensurability classes of twist knots",

abstract = " In this paper we prove that if M K is the complement of a non-fibered twist knot K in S 3 , then M K is not commensurable to a fibered knot complement in a Z/2Z-homology sphere. To prove this result we derive a recursive description of the character variety of twist knots and then prove that a commensurability criterion developed by D. Calegari and N. Dunfield is satisfied for these varieties. In addition, we partially extend our results to a second infinite family of 2-bridge knots.",

keywords = "Geometric topology, knot theory, hyperbolic geometry",

author = "Jim Hoste and Shanahan, \{Patrick D.\}",

note = "Hoste, Jim and Patrick D. Shanahan. {"}Commensurability classes of twist knots.{"} Journal of Knot Theory and its Ramifications, Vol. 14, No. 1 (2005) 1-10. arXiv:math/0311051",

year = "2005",

language = "American English",

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journal = "Journal of Knot Theory and its Ramifications",

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T1 - Commensurability classes of twist knots

AU - Hoste, Jim

AU - Shanahan, Patrick D.

N1 - Hoste, Jim and Patrick D. Shanahan. "Commensurability classes of twist knots." Journal of Knot Theory and its Ramifications, Vol. 14, No. 1 (2005) 1-10. arXiv:math/0311051

PY - 2005

Y1 - 2005

N2 - In this paper we prove that if M K is the complement of a non-fibered twist knot K in S 3 , then M K is not commensurable to a fibered knot complement in a Z/2Z-homology sphere. To prove this result we derive a recursive description of the character variety of twist knots and then prove that a commensurability criterion developed by D. Calegari and N. Dunfield is satisfied for these varieties. In addition, we partially extend our results to a second infinite family of 2-bridge knots.

AB - In this paper we prove that if M K is the complement of a non-fibered twist knot K in S 3 , then M K is not commensurable to a fibered knot complement in a Z/2Z-homology sphere. To prove this result we derive a recursive description of the character variety of twist knots and then prove that a commensurability criterion developed by D. Calegari and N. Dunfield is satisfied for these varieties. In addition, we partially extend our results to a second infinite family of 2-bridge knots.

KW - Geometric topology

KW - knot theory

KW - hyperbolic geometry

M3 - Article

VL - 14

SP - 1

EP - 10

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

IS - 1

ER -