Geometric estimates from spanning surfaces

TL;DR

This paper establishes bounds on meridian length and cusp volume of hyperbolic knots using essential surface topology, providing algorithms and applications to knot theory and Dehn surgery.

Contribution

It introduces a new algorithmic criterion for bounding meridian length and cusp volume based on essential surface topology in hyperbolic knots.

Findings

Derived bounds on meridian length and cusp volume.

Provided an algorithmic criterion for meridian length bounds.

Identified families of knots with meridian length ≤ 4.

Abstract

We derive bounds on the length of the meridian and the cusp volume of hyperbolic knots in terms of the topology of essential surfaces spanned by the knot. We provide an algorithmically checkable criterion that guarantees that the meridian length of a hyperbolic knot is below a given bound. As applications we find knot diagrammatic upper bounds on the meridian length and the cusp volume of hyperbolic adequate knots and we obtain new large families of knots with meridian lengths bounded above by four. We also discuss applications of our results to Dehn surgery.