Cusp volumes of alternating knots

TL;DR

This paper establishes bounds on the cusp volume of hyperbolic alternating knots using their diagrammatic twist number, providing new estimates for slopes and implications for Dehn surgery and cusp density.

Contribution

It introduces bounds on cusp volume based on diagrammatic twist number, linking geometric properties to knot diagrams and deriving universal density bounds.

Findings

Cusp volume bounds in terms of twist number

Diagrammatic estimates on slope lengths

Universal lower bound on cusp density

Abstract

We show that the cusp volume of a hyperbolic alternating knot can be bounded above and below in terms of the twist number of an alternating diagram of the knot. This leads to diagrammatic estimates on lengths of slopes, and has some applications to Dehn surgery. Another consequence is that there is a universal lower bound on the cusp density of hyperbolic alternating knots.