Slicing degree of knots

TL;DR

This paper investigates the slicing degree of knots, establishing bounds using advanced invariants, and computes these degrees for small knots and certain torus knots, enhancing understanding of knot sliceness properties.

Contribution

The paper introduces new bounds for the slicing degree of knots utilizing Rasmussen's s-invariant, knot Floer homology, and instanton homology, with explicit computations for small and torus knots.

Findings

Bounds for slicing degrees established using multiple invariants

Computed slicing degrees for knots with up to 9 crossings

Determined slicing degrees for some families of torus knots

Abstract

The slicing degree of a knot $K$ is defined as the smallest integer $k$ such that $K$ is $k$-slice in $\#^n \overline{\mathbb{CP}^2}$ for some $n$. In this paper, we establish bounds for the slicing degrees of knots using Rasmussen's $s$-invariant, knot Floer homology and singular instanton homology. We compute the slicing degrees for many small knots (with crossing numbers up to $9$) and for some families of torus knots.