FAMED by computer: proving the Andersen-Kashaev volume conjecture for 42,000 knots

TL;DR

This paper computationally verifies the Andersen-Kashaev volume conjecture for over 42,000 knot complements using the FAMED property, significantly expanding the set of known cases and providing new insights into the conjecture.

Contribution

It introduces a large-scale computational approach to verify the conjecture for thousands of knots, extending previous results and offering new insights into the FAMED property.

Findings

Verified the conjecture for over 42,000 knots

Identified FAMED triangulations for all knots with 12 or fewer crossings

Discovered new properties of the FAMED condition

Abstract

The FAMED condition is a combinatorial property for ideal triangulations of $3$-manifolds, which was introduced in 2024 by the first and last authors in order to study the Andersen--Kashaev volume conjecture. They notably proved that this conjecture is true for all FAMED geometric triangulations of one-cusped hyperbolic $3$-manifolds with trivial second homology. In this paper, using a straightforward computer implementation in Regina and Snappy, we find FAMED geometric triangulations for more than 42.000 complements of knots in $S^3$, including all knots with $12$ crossings or fewer and all knots whose complement can be triangulated with $23$ tetrahedra or fewer. As a consequence, the Andersen-Kashaev conjecture is now proven to be true for as many new examples. Along the way, we find several new insights about the FAMED property, which have great value in the quest of a general proof of the Andersen-Kashaev volume conjecture for every knot complement.