Cut open null-bordisms and derivatives of slice knots

TL;DR

This paper introduces a novel method of analyzing null-bordisms to derive new obstructions for slice knots and their derivatives, providing insights into knot sliceness and related algebraic properties.

Contribution

The authors develop a cutting-open null-bordism technique to produce new sliceness obstructions and re-derive known signature conditions, advancing understanding of slice knots and doubling operators.

Findings

New obstructions to derivatives of slice knots

Re-derivation of Cooper's signature condition

Evidence for weak injectivity of doubling operators

Abstract

In the 60's Levine proved that if $R$ is a slice knot, then on any genus $g$ Seifert surface for $R$ there is a $g$ component link $J$, called a derivative of $R$, on which the Seifert form vanishes. Many subsequent obstructions to $R$ being slice are given in terms of slice obstructions of $J$. Many of these obstructions can be derived from a 4-manifold called a null-bordism. Recently the authors proved that that it is possible for $R$ to be slice without $J$ being slice, disproving a conjecture of Kauffmann from the 80's. In this paper we cut open these null-bordisms in order to derive new obstructions to being the derivative of a slice knot. As a proof of the strength of this approach we re-derive a signature condition due to Daryl Cooper. Our results also apply to doubling operators, giving new evidence for their weak injectivity. We close with a new sufficient condition for a genus 1 algebraically slice knot to be $1.5$-solvable.