Let L be a biperiodic alternating link with alternating quotient link L, and let K_n be alternating hyperbolic links satisfying K_n stackrelrm Flongrightarrow L. The crossing number c(K_n) is that of K_n, while c(L) is the crossing number of the reduced alternating projection of L on the torus. Volume Density Conjecture. lim_n→∞fracoperatornamevol(K_n)c(K_n)=fracoperatornamevol((T^2× 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.

The Volume Density Conjecture for biperiodic alternating links

About 8 years old · traced to 2018

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

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

The volume density conjecture for biperiodic alternating links
Let $\mathcal{L}$ be a biperiodic alternating link with alternating quotient link $L$ . Let $\{K_n\}$ be a sequence of alternating hyperbolic links that F\o lner converges to $\mathcal{L}$ . Volume density conjecture.
$\displaystyle {\lim_{n \to \infty}} \frac{\operatorname{vol}(K_n)}{c(K_n)} = \frac{\operatorname{vol}((T^2 \times I) - L)}{c(L)}.$
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).

References

Primary source

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

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The volume density conjecture for biperiodic alternating links

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