Right-angled volume of Alternating Links

Abstract

In a recent paper Champanerkar, Kofman and Purcell defined right-angled volume for prime alternating links as a sum of volumes of an associated collection of hyperbolic right-angled ideal polyhedra which is an invariant of the alternating link. Around the same time Felsner and Rote gave a graph theoretic algorithm to obtain right-angled circle patterns associated to planar graphs. In this thesis, we extend the Felsner-Rote algorithm to alternating knot and link diagrams by developing graph theoretic analogs of the two moves used to compute right-angled volumes, namely rational reduction and decomposition along prismatic 4-circuits. Using this technique we compute right-angled volume for knots in the alternating knot census up to 17 crossings, and links in the alternating link census up to 14 crossings. In addition, using our methods we extend computations of right-angled volume of weaving knots and links, verify their conjecture on the existence of right-angled knots for alternating knots up to 17 crossings, give a new method to generate volumes of right-angled polyhedra recreating volumes computed by Vesnin and Egorov, and explore volumes of fully augmented link complements.