Twist Number and the Alternating Volume of Knots

TL;DR

This paper introduces the concept of alternating volume for knots, relating it to the twist number, and demonstrates their coarse equivalence, providing new insights into knot invariants and hyperbolic geometry.

Contribution

It defines the alternating volume of a knot and proves its coarse equivalence to the twist number, linking geometric and combinatorial knot invariants.

Findings

Alternating volume is coarsely equivalent to the twist number.

The paper establishes a new relationship between knot volume and twist number.

It extends understanding of hyperbolic link invariants.

Abstract

It was previously shown by the second author that every knot in $S^3$ is ambient isotopic to one component of a two-component, alternating, hyperbolic link. In this paper, we define the alternating volume of a knot $K$ to be the minimum volume of any link $L$ in a natural class of alternating, hyperbolic links such that $K$ is ambient isotopic to a component of $L$. Our main result shows that the alternating volume of a knot is coarsely equivalent to the twist number of a knot.