Signature and crossing number of links

TL;DR

This paper explores the relationship between the signature and crossing number of links, identifying specific classes of links with particular signature-crossing number properties and their applications in topology and biology.

Contribution

It classifies links where the sum of signature and crossing number equals 2, showing they are closures of positive 3-braids, and discusses implications for band surgeries and biological structures.

Findings

Links with signature + crossing number = 2 are closures of positive 3-braids.

The study refines existing theorems relating signature and crossing number.

Applications to vortex knots and DNA topology are discussed.

Abstract

This paper investigates the relationship between the signature and the crossing number of knots and links. We refine existing theorems and provide a comprehensive classification of links with specific properties, particularly those with signatures that deviate by a fixed amount from their crossing numbers. The main results include the identification of all links for which the sum of the signature and crossing number equals 2, which are shown to be closures of positive 3-braids. Additionally, we explore the implications of these findings in the context of band surgeries and their applications to vortex knots and DNA topology.