Bridge and pants complexities of knots

TL;DR

This paper introduces a new invariant for knots in 3-manifolds based on bridge splitting distances in dual curve and pants complexes, and explores its relation to hyperbolic volume.

Contribution

It defines a novel complexity measure for knots using dual curve and pants complexes, and investigates its stability and connection to hyperbolic volume.

Findings

The complexity becomes constant after sufficient stabilizations.

The invariant can distinguish different manifold-knot pairs.

Evidence suggests a relationship between pants distance and hyperbolic volume.

Abstract

We modify an approach of Johnson to define the distance of a bridge splitting of a knot in a 3-manifold using the dual curve complex and pants complex of the bridge surface. This distance can be used to determine a complexity, which becomes constant after a sufficient number of stabilizations and perturbations, yielding an invariant of the manifold-knot pair. We also give evidence toward the relationship between the pants distance of a bridge splitting and the hyperbolic volume of the exterior of a knot.