2018/02/28 by Abhijit Champanerkar, Ilya Kofman, Jessica S. Purcell · 1 citation
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Combinatorics #Conjecture #Euclidean geometry #Geodesic #Geometric and Algebraic Topology #Geometry #Hyperbolic set #Invariant (physics) #Mathematical analysis #Mathematical physics #Mathematics #Physics #Pure mathematics #Quotient #Torus #Unimodular matrix #math.GT #msc:57M25 #msc:57M27 #msc:57M50 #semigroups and automata theory
paper · pdf · doi:10.1112/jlms.12195
published as J. London Math. Soc. (2) 99 (2019), 807-830 · 25 pages, 15 figures. V2: Minor changes. Added reference, fixed typos, and clarified proof of Theorem 5.1
arxiv created 2018/11/08 · openalex publication_date 2018/11/28 · arxiv updated 2019/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.