Geometry of biperiodic alternating links

Abstract

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.

25 pages, 15 figures. V2: Minor changes. Added reference, fixed typos, and clarified proof of Theorem 5.1

References in corpus (5)

• Volume bounds for weaving knots
• Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds
• Mahler Measure and the Vol-Det Conjecture
• Determinant density and biperiodic alternating links
• From 3-moves to Lagrangian tangles and cubic skein modules

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• On the geometry of rod packings in the 3-torus
• Universality in minor-closed graph classes
• TG-Hyperbolicity of Virtual Links
• Alternating links with totally geodesic checkerboard surfaces
• Alternating links on surfaces and volume bounds
• A volumish theorem for alternating virtual links
• A geometric classification of rod complements in the 3-torus
• Cusp Volumes of Alternating Knots on Surfaces
• Infinitely many virtual geometric triangulations
• Degeneration of 3-dimensional hyperbolic cone structures with decreasing cone angles
• A Quantum Invariant of Links in $T^2\times I$ with Volume Conjecture Behavior
• Examples of homological torsion and volume growth