A note on the weak splitting number

TL;DR

This paper investigates the weak splitting number of links, providing new lower bounds using link invariants like multivariable signature, and applies these methods to compute the invariant for most small prime links.

Contribution

It introduces conditions linking link invariants to the weak splitting number and combines various invariants to compute this number for numerous small prime links.

Findings

New bounds on weak splitting number using multivariable signature.

Recovery of known bounds via $ au$ and $s$-invariants.

Computed weak splitting number for all but two small prime links.

Abstract

The weak splitting number $wsp(L)$ of a link $L$ is the minimal number of crossing changes needed to turn $L$ into a split union of knots. We describe conditions under which certain $\mathbb{R}$-valued link invariants give lower bounds on $wsp(L)$. This result is used both to obtain new bounds on $wsp(L)$ in terms of the multivariable signature and to recover known lower bounds in terms of the $\tau$ and $s$-invariants. We also establish new obstructions using link Floer homology and apply all these methods to compute $wsp$ for all but two of the $130$ prime links with $9$ or fewer crossings.