On homotopy $K3$ surfaces constructed by two knots and their applications

TL;DR

This paper constructs specific homotopy K3 surfaces using two knots and explores their properties, demonstrating conditions under which certain 3-manifolds do not bound smooth spin rational 4-balls and related knot sliceness results.

Contribution

It introduces a new construction of homotopy K3 surfaces from knots and applies the adjunction inequality to derive novel non-sliceness and bounding results.

Findings

Certain 3-manifolds do not bound smooth spin rational 4-balls under specified conditions.

Negative n-twisted Whitehead doubles of knot connected sums are not slice knots when n exceeds a bound.

New techniques for constructing and analyzing homotopy K3 surfaces from knot data.

Abstract

Let $LHT$ be a left handed trefoil knot and $K$ be any knot. We define $M_n(K)$ to be the homology $3$-sphere which is represented by a simple link of $LHT$ and $LHT \sharp K$ with framings $0$ and $n$ respectively. Starting with this link, we construct homotopy $K3$ and spin rational homology $K3$ surfaces containing $M_n(K)$. Then we apply the adjunction inequality to show that if $n>2g^n_s(K)-2$, $M_n(K)$ does not bound any smooth spin rational $4$-ball, and that under the same assumption the negative $n$-twisted Whitehead double of $LHT \sharp K$ is not a slice knot, where $g^n_s(K)$ is the $n$-shake genus of $K$.