Geometrically and diagrammatically maximal knots

Champanerkar, Abhijit; Kofman, Ilya; Purcell, Jessica S.

doi:10.1112/jlms/jdw062

Cited by 24 publications

(89 citation statements)

References 34 publications

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“…We say a torihedron is right‐angled if it admits a hyperbolic structure in which all dihedral angles on edges equal

π / 2

. In this section, we prove that only two semi‐regular biperiodic alternating links admit a decomposition into right‐angled torihedra: the square weave

scriptW

studied in , and the triaxial link

scriptL

. Theorem The square weave

scriptW

and the triaxial link

scriptL

are the only semi‐regular links such that right‐angled torihedra give the complete hyperbolic structure.…”

Section: The Triaxial Linkmentioning

confidence: 92%

“…The volume density of a biperiodic alternating link is defined as

vol false((T^{2} \times I) - L false) / c false(L false)

, where

c (L)

is the crossing number of the reduced alternating projection of

L

on the torus, which is minimal. Hence, as

K_{n} \overset{normalF}{\to} W

, the volume densities of

K_{n}

converge to the volume density of

scriptW

, which is

v_{oct}

; see .…”

Section: Proof Of the Volume Density Conjecture For The Triaxial Linkmentioning

confidence: 98%

“…The checkerboard polyhedral decomposition of the complement of an alternating link in

S^{3}

, as in , can be generalized to that of an alternating link in

T^{2} \times I

. This was done for a single example in , and much more generally in . The following version is appropriate for the links considered here: Theorem Let

L

be a link in

T^{2} \times false(- 1, 1 false)

with a reduced, cellular, alternating diagram on

T^{2} \times false{0 false}

.…”

Section: Torihedramentioning

confidence: 99%

“…In , we considered biperiodic alternating links as limits of sequences of finite hyperbolic links. We focused on the asymptotic behavior of two basic invariants, one geometric and one diagrammatic, for a hyperbolic link

K

: The volume density of

K

is defined as

vol (K) / c (K)

, and the determinant density of

K

is defined as

2 π log det (K) / c (K)

, where

c (K)

denotes crossing number.…”

Section: Proof Of the Volume Density Conjecture For The Triaxial Linkmentioning

confidence: 99%

“…For a biperiodic alternating link, the volume density is that of its quotient link in

T^{2} \times I

. In previous work, the authors showed that if a sequence of knots or links in

S^{3}

converges in an appropriate sense (see Definition ) to the infinite square weave, then their volume densities approaches the volume density of the infinite square weave . The Volume Density Conjecture (Conjecture ) is that the same result holds for any biperiodic alternating link.…”

Section: Introductionmentioning

confidence: 99%

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Geometry of biperiodic alternating links

Champanerkar

Kofman

Purcell

2018

J. London Math. Soc.

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A biperiodic alternating link has an alternating quotient link in the thickened torus. In this paper, we focus on semi‐regular links, a class of biperiodic alternating links whose hyperbolic structure can be immediately determined from a corresponding Euclidean tiling. Consequently, we determine the exact volumes of semi‐regular links. We relate their commensurability and arithmeticity to the corresponding tiling, and assuming a conjecture of Milnor, we show there exist infinitely many pairwise incommensurable semi‐regular links with the same invariant trace field. We show that only two semi‐regular links have totally geodesic checkerboard surfaces; these two links satisfy the Volume Density Conjecture. Finally, we give conditions implying that many additional biperiodic alternating links are hyperbolic and admit a positively oriented, unimodular geometric triangulation. We also provide sharp upper and lower volume bounds for these links.

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“…We say a torihedron is right‐angled if it admits a hyperbolic structure in which all dihedral angles on edges equal

π / 2

. In this section, we prove that only two semi‐regular biperiodic alternating links admit a decomposition into right‐angled torihedra: the square weave

scriptW

studied in , and the triaxial link

scriptL

. Theorem The square weave

scriptW

and the triaxial link

scriptL

are the only semi‐regular links such that right‐angled torihedra give the complete hyperbolic structure.…”

Section: The Triaxial Linkmentioning

confidence: 92%

“…The volume density of a biperiodic alternating link is defined as

vol false((T^{2} \times I) - L false) / c false(L false)

, where

c (L)

is the crossing number of the reduced alternating projection of

L

on the torus, which is minimal. Hence, as

K_{n} \overset{normalF}{\to} W

, the volume densities of

K_{n}

converge to the volume density of

scriptW

, which is

v_{oct}

; see .…”

Section: Proof Of the Volume Density Conjecture For The Triaxial Linkmentioning

confidence: 98%

“…The checkerboard polyhedral decomposition of the complement of an alternating link in

S^{3}

, as in , can be generalized to that of an alternating link in

T^{2} \times I

. This was done for a single example in , and much more generally in . The following version is appropriate for the links considered here: Theorem Let

L

be a link in

T^{2} \times false(- 1, 1 false)

with a reduced, cellular, alternating diagram on

T^{2} \times false{0 false}

.…”

Section: Torihedramentioning

confidence: 99%

K

: The volume density of

K

is defined as

vol (K) / c (K)

, and the determinant density of

K

is defined as

2 π log det (K) / c (K)

, where

c (K)

denotes crossing number.…”

Section: Proof Of the Volume Density Conjecture For The Triaxial Linkmentioning

confidence: 99%

“…For a biperiodic alternating link, the volume density is that of its quotient link in

T^{2} \times I

. In previous work, the authors showed that if a sequence of knots or links in

S^{3}

Section: Introductionmentioning

confidence: 99%

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Geometry of biperiodic alternating links

Champanerkar

Kofman

Purcell

2018

J. London Math. Soc.

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A Survey of Hyperbolic Knot Theory

Futer

Kalfagianni

Purcell

2019

Knots, Low-Dimensional Topology and Applications

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We survey some tools and techniques for determining geometric properties of a link complement from a link diagram. In particular, we survey the tools used to estimate geometric invariants in terms of basic diagrammatic link invariants. We focus on determining when a link is hyperbolic, estimating its volume, and bounding its cusp shape and cusp area. We give sample applications and state some open questions and conjectures.2010 Mathematics Subject Classification. 57M25, 57M27, 57M50.

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Examples of Homological Torsion and Volume Growth

Champanerkar

Kofman

2021

Acta Math Vietnam

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Geometrically and diagrammatically maximal knots

Cited by 24 publications

References 34 publications

Geometry of biperiodic alternating links

Geometry of biperiodic alternating links

A Survey of Hyperbolic Knot Theory

Examples of Homological Torsion and Volume Growth

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