Splitting numbers and signatures

TL;DR

This paper introduces an elementary yet effective lower bound on the splitting number of links based on multivariable signature and nullity, enabling efficient computation and verification of splitting numbers for many links.

Contribution

It provides a simple, computationally efficient lower bound for the splitting number that matches or improves upon previous methods for various classes of links.

Findings

Successfully computed splitting numbers for 129 of 130 prime links with up to 9 crossings.

Determined splitting numbers for most links studied by prior advanced techniques.

Extended the computation of splitting numbers to a large class of 2-bridge links.

Abstract

The splitting number of a link is the minimal number of crossing changes between different components required to convert it into a split link. We obtain a lower bound on the splitting number in terms of the (multivariable) signature and nullity. Although very elementary and easy to compute, this bound turns out to be suprisingly efficient. In particular, it makes it a routine check to recover the splitting number of 129 out of the 130 prime links with at most 9 crossings. Also, we easily determine 16 of the 17 splitting numbers that were studied by Batson and Seed using Khovanov homology, and later computed by Cha, Friedl and Powell using a variety of techniques. Finally, we determine the splitting number of a large class of 2-bridge links which includes examples recently computed by Borodzik and Gorsky using a Heegaard Floer theoretical criterion.