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Right-angled polyhedra and alternating links

Abhijit Champanerkar, Ilya Kofman and Jessica S Purcell

Algebraic & Geometric Topology 22 (2022) 739–784

To any prime, alternating link, we associate a collection of hyperbolic right-angled ideal polyhedra by relating geometric, topological and combinatorial methods to decompose the link complement. The sum of the hyperbolic volumes of these polyhedra is a new geometric link invariant, which we call the right-angled volume of the alternating link. We give an explicit procedure to compute the right-angled volume from any alternating link diagram, and prove that it is a new lower bound for the hyperbolic volume of the link.

hyperbolic geometry, alternating knot, guts, circle packing, right-angled polyhedra, ideal polyhedra, checkerboard surface, Conway sphere, Andreev theorem, hyperbolic volume, weaving knot, right-angled knot
Mathematical Subject Classification 2010
Primary: 57M25, 57M27, 57M50
Received: 20 February 2020
Revised: 22 January 2021
Accepted: 22 February 2021
Published: 3 August 2022
Abhijit Champanerkar
Department of Mathematics
College of Staten Island
City University of New York
Staten Island, NY
United States
Ilya Kofman
Department of Mathematics
College of Staten Island
City University of New York
New York, NY
United States
Jessica S Purcell
School of Mathematics
Monash University
Clayton VIC