Geometry of biperiodic alternating links

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

doi:10.1112/jlms.12195

Cited by 33 publications

(69 citation statements)

References 44 publications

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“…The characteristic polynomial of the dimer model is defined as

p (z, w) = det κ (z, w)

, where

κ (z, w)

is the weighted, signed adjacency matrix with rows indexed by black vertices and columns by white vertices, and matrix entries determined by a certain choice of signs on edges, and a choice of homology basis for the

normalΛ

‐action. See Section 2 and for details and examples.…”

Section: Introductionmentioning

confidence: 99%

“…The link

L

is often hyperbolic in

T^{2} \times I

; that is,

(T^{2} \times I) - L

is a complete finite‐volume hyperbolic 3‐manifold . In , it was proved that

vol false((T^{2} \times I) - L false) ⩽ {vol}^{◊} false(L false),

with equality for semi‐regular links. Thus, Conjecture would imply that

vol false((T^{2} \times I) - L false) ⩽ {vol}^{◊} false(L false) ⩽ 2 π m false(p (z, w) false) .

…”

Section: Introductionmentioning

confidence: 99%

“…In , biperiodic alternating links were considered as limits of sequences of finite hyperbolic links. In Section 2, we define a natural notion of convergence for a sequence of alternating links to a biperiodic alternating link

scriptL

, called Følner convergence almost everywhere , denoted by

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

.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2, we define a natural notion of convergence for a sequence of alternating links to a biperiodic alternating link

scriptL

, called Følner convergence almost everywhere , denoted by

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

. It was proved in that for any sequence of alternating links

K_{n}

that converge to a biperiodic alternating link

scriptL

in this sense, the determinant densities of

K_{n}

converge to the density of the Mahler measure of the characteristic polynomial

p (z, w)

of the associated toroidal dimer model:

K_{n} \overset{normalF}{\to} L ⟹ lim_{n \to \infty} \frac{prefixlog trueprefixdet false(K_{n} false)}{c false(K_{n} false)} = \frac{normalm false(p false(z, w false) false)}{c false(L false)} .

…”

Section: Introductionmentioning

confidence: 99%

“…In , the first two authors defined the following notion of convergence of links in

S^{3}

to a biperiodic alternating link. Definition We will say that a sequence of alternating links

K_{n}

Følner converges almost everywhere to the biperiodic alternating link

scriptL

, denoted by

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

, if the respective projection graphs

{G false(K_{n} false)}

and

G (L)

satisfy the following: There are subgraphs

G_{n} \subset G false(K_{n} false)

such that (i)

G_{n} \subset G_{n + 1}

, and

⋃ G_{n} = G false(scriptL false)

, (ii)

{trueprefixlim}_{n \to \infty} false| \partial G_{n} false| / false| G_{n} false| = 0

, where

| \cdot |

denotes number of vertices, and

\partial G_{n} \subset G false(scriptL false)

consists of the vertices of

G_{n}

that share an edge in

G (L)

with a vertex not in

G_{n}

, (iii)

G_{n} \subset G false(scriptL false) \cap false(n normalΛ false)

, where

n Λ

represents

n^{2}

copies of the

normalΛ

‐fundamental domain for the lattice

…”

Section: Introductionmentioning

confidence: 99%

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Mahler measure and the Vol‐Det Conjecture

Champanerkar

Kofman

Laĺın

2018

J. London Math. Soc.

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The Vol‐Det Conjecture relates the volume and the determinant of a hyperbolic alternating link in S3. We use exact computations of Mahler measures of two‐variable polynomials to prove the Vol‐Det Conjecture for many infinite families of alternating links. We conjecture a new lower bound for the Mahler measure of certain two‐variable polynomials in terms of volumes of hyperbolic regular ideal bipyramids. Associating each polynomial to a toroidal link using the toroidal dimer model, we show that every polynomial which satisfies this conjecture with a strict inequality gives rise to many infinite families of alternating links satisfying the Vol‐Det Conjecture. We prove this new conjecture for six toroidal links by rigorously computing the Mahler measures of their two‐variable polynomials.

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“…The characteristic polynomial of the dimer model is defined as

p (z, w) = det κ (z, w)

, where

κ (z, w)

normalΛ

‐action. See Section 2 and for details and examples.…”

Section: Introductionmentioning

confidence: 99%

“…The link

L

is often hyperbolic in

T^{2} \times I

; that is,

(T^{2} \times I) - L

is a complete finite‐volume hyperbolic 3‐manifold . In , it was proved that

vol false((T^{2} \times I) - L false) ⩽ {vol}^{◊} false(L false),

with equality for semi‐regular links. Thus, Conjecture would imply that

vol false((T^{2} \times I) - L false) ⩽ {vol}^{◊} false(L false) ⩽ 2 π m false(p (z, w) false) .

…”

Section: Introductionmentioning

confidence: 99%

scriptL

, called Følner convergence almost everywhere , denoted by

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

.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2, we define a natural notion of convergence for a sequence of alternating links to a biperiodic alternating link

scriptL

, called Følner convergence almost everywhere , denoted by

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

. It was proved in that for any sequence of alternating links

K_{n}

that converge to a biperiodic alternating link

scriptL

in this sense, the determinant densities of

K_{n}

converge to the density of the Mahler measure of the characteristic polynomial

p (z, w)

of the associated toroidal dimer model:

K_{n} \overset{normalF}{\to} L ⟹ lim_{n \to \infty} \frac{prefixlog trueprefixdet false(K_{n} false)}{c false(K_{n} false)} = \frac{normalm false(p false(z, w false) false)}{c false(L false)} .

…”

Section: Introductionmentioning

confidence: 99%

“…In , the first two authors defined the following notion of convergence of links in

S^{3}

to a biperiodic alternating link. Definition We will say that a sequence of alternating links

K_{n}

Følner converges almost everywhere to the biperiodic alternating link

scriptL

, denoted by

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

, if the respective projection graphs

{G false(K_{n} false)}

and

G (L)

satisfy the following: There are subgraphs

G_{n} \subset G false(K_{n} false)

such that (i)

G_{n} \subset G_{n + 1}

, and

⋃ G_{n} = G false(scriptL false)

, (ii)

{trueprefixlim}_{n \to \infty} false| \partial G_{n} false| / false| G_{n} false| = 0

, where

| \cdot |

denotes number of vertices, and

\partial G_{n} \subset G false(scriptL false)

consists of the vertices of

G_{n}

that share an edge in

G (L)

with a vertex not in

G_{n}

, (iii)

G_{n} \subset G false(scriptL false) \cap false(n normalΛ false)

, where

n Λ

represents

n^{2}

copies of the

normalΛ

‐fundamental domain for the lattice

…”

Section: Introductionmentioning

confidence: 99%

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Mahler measure and the Vol‐Det Conjecture

Champanerkar

Kofman

Laĺın

2018

J. London Math. Soc.

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Concordances to prime hyperbolic virtual knots

Chrisman

2020

Geom Dedicata

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Right-angled links in thickened surfaces

Kaplan-Kelly

2024

Geom Dedicata

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Traditionally, alternating links are studied with alternating diagrams on $$S^2$$ S 2 in $$S^3$$ S 3 . In this paper, we consider links which are alternating on higher genus surfaces F in thickened surface $$F \times I$$ F × I . We define what it means for such a link to be right-angled generalized completely realizable (RGCR) and show that this property is equivalent to the link having two totally geodesic checkerboard surfaces, equivalent to each checkerboard surface consisting of one type of polygon, and equivalent to a set of restrictions on the link’s alternating projection diagram. We then use these diagram restrictions to classify RGCR links according to the polygons in their checkerboard surfaces, provide a bound on the number of RGCR links for a given surface of genus g, and find a right-angled knot in a thickened surface. Along the way, we answer a question posed by Champanerkar, Kofman, and Purcell about links with alternating projections on the torus.

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Geometry of biperiodic alternating links

Cited by 33 publications

References 44 publications

Mahler measure and the Vol‐Det Conjecture

Mahler measure and the Vol‐Det Conjecture

Concordances to prime hyperbolic virtual knots

Right-angled links in thickened surfaces

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