Cusp areas of Farey manifolds and applications to knot theory

TL;DR

This paper provides explicit bounds on cusp areas of certain hyperbolic 3-manifolds using Farey tessellation data, leading to new volume estimates and insights into knot invariants like the Jones polynomial.

Contribution

It introduces the first explicit estimates on cusp areas of specific hyperbolic manifolds based on combinatorial Farey data, with applications to volume bounds and knot theory.

Findings

Explicit cusp area bounds for specific manifolds

Sharp volume bounds for Dehn fillings

Counterexamples to a conjecture relating Jones polynomial and volume

Abstract

This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey tesselation of the hyperbolic plane. The bounds on cusp area lead to explicit bounds on the volume of Dehn fillings of these manifolds, for example sharp bounds on volumes of hyperbolic closed 3-braids in terms of the Schreier normal form of the associated braid word. Finally, these results are applied to derive relations between the Jones polynomial and the volume of hyperbolic knots, and to disprove a related conjecture.