Slice obstructions from genus bounds in definite 4-manifolds

TL;DR

This paper introduces a new obstruction to a knot being smoothly slice based on genus bounds in definite 4-manifolds, providing an alternative proof for a specific non-slice knot using gauge theory.

Contribution

It presents a novel obstruction method derived from genus bounds in definite 4-manifolds and applies gauge-theoretic techniques to knot sliceness problems.

Findings

Established an obstruction to smooth sliceness from genus bounds in 4-manifolds.

Provided an alternative proof that the (2,1)-cable of the figure eight knot is not smoothly slice.

Utilized gauge-theoretic obstructions in complex projective plane connected sums.

Abstract

We discuss an obstruction to a knot being smoothly slice that comes from minimum-genus bounds on smoothly embedded surfaces in definite 4-manifolds. As an example, we provide an alternate proof of the fact that the (2,1)-cable of the figure eight knot is not smoothly slice, as shown by Dai--Kang--Mallick--Park--Stoffregen in 2022. The main technical input of our argument consists of gauge-theoretic obstructions to smooth small-genus surfaces representing certain homology classes in $\mathbb{CP}^2\#\mathbb{CP}^2$ proved by Bryan in the 1990s.