On slicing invariants of knots

TL;DR

This paper investigates the slicing number of knots, establishing bounds using advanced topological invariants and presenting an infinite family of knots with specific slicing properties, enhancing understanding of knot slicing invariants.

Contribution

It demonstrates that existing bounds on unknotting numbers also bound the slicing number and introduces a new invariant with distinct properties, supported by examples and topological methods.

Findings

Bounds on slicing number derived from unknotting number bounds.

Existence of an infinite family of knots with slice genus n and Livingston invariant greater than n.

Application of Donaldson's theorem and Heegaard Floer homology to restrict intersection forms.

Abstract

The slicing number of a knot, $u_s(K)$, is the minimum number of crossing changes required to convert $K$ to a slice knot. This invariant is bounded above by the unknotting number and below by the slice genus $g_s(K)$. We show that for many knots, previous bounds on unknotting number obtained by Ozsvath and Szabo and by the author in fact give bounds on the slicing number. Livingston defined another invariant $U_s(K)$ which takes into account signs of crossings changed to get a slice knot, and which is bounded above by the slicing number and below by the slice genus. We exhibit an infinite family of knots $K_n$ with slice genus $n$ and Livingston invariant greater than $n$. Our bounds are based on restrictions (using Donaldson's diagonalisation theorem or Heegaard Floer homology) on the intersection forms of four-manifolds bounded by the double branched cover of a knot.