A quantitative Birman-Menasco finiteness theorem and its application to crossing number

TL;DR

This paper provides a quantitative version of the Birman-Menasco finiteness theorem, offering estimates for the crossing number of knots based on genus and braid index, with applications to connected sums and satellites.

Contribution

It introduces a new quantitative bound relating crossing number, genus, and braid index, extending the original finiteness theorem.

Findings

Derived explicit crossing number bounds for knots

Applied bounds to connected sums and satellite knots

Enhanced understanding of knot complexity measures

Abstract

Birman-Menasco proved that there are finitely many knots having a given genus and braid index. We give a quantitative version of Birman-Menasco finiteness theorem, an estimate of the crossing number of knots in terms of genus and braid index. This has various applications of crossing numbers, such as, the crossing number of connected sum or satellites.