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The Volume Density Conjecture for biperiodic alternating links

From papers

Let L\mathcal L be a biperiodic alternating link with alternating quotient link LL, and let KnK_n be alternating hyperbolic links satisfying Kn⟶FLK_n\stackrel{\rm F}{\longrightarrow}\mathcal L. The crossing number c(Kn)c(K_n) is that of KnK_n, while c(L)c(L) is the crossing number of the reduced alternating projection of LL on the torus. Volume Density Conjecture.

lim⁡n→∞vol⁡(Kn)c(Kn)=vol⁡((T2×I)−L)c(L).\lim_{n\to\infty}\frac{\operatorname{vol}(K_n)}{c(K_n)}=\frac{\operatorname{vol}((T^2\times I)-L)}{c(L)}.

This conjecture extends the known volume-density convergence for sequences converging to the infinite square weave to arbitrary biperiodic alternating links. Its status is not resolved in the supplied source.

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The volume density conjecture for biperiodic alternating links

    Let L\mathcal{L} be a biperiodic alternating link with alternating quotient link LL. Let {Kn}\{K_n\} be a sequence of alternating hyperbolic links that F\o lner converges to L\mathcal{L}. Volume density conjecture.

    lim⁡n→∞vol⁡(Kn)c(Kn)=vol⁡((T2×I)−L)c(L).\displaystyle {\lim_{n \to \infty}} \frac{\operatorname{vol}(K_n)}{c(K_n)} = \frac{\operatorname{vol}((T^2 \times I) - L)}{c(L)}.

    The conjecture predicts that the volume density of a Følner-convergent sequence is determined by the volume density of its quotient link. The paper states that it proves this conjecture for fully augmented links; its general status is not specified here.

    source: Alice Kwon, “Fully Augmented Links in the Thickened Torus”, arXiv:2007.12773 (2023).

Sources & referencesView supporting material

Primary source

Abhijit Champanerkar, Ilya Kofman and Jessica S. Purcell, “Geometry of biperiodic alternating links”, arXiv:1802.05343 (2018).

alternating-linkshyperbolic-geometryvolume-density

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