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

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 $LHT♯K$ 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 $n>2g_s^n(K)-2$, $M_n(K)$ does not bound any smooth spin rational $4$-ball, and that under the same assumption the negative $n$-twisted Whitehead double of $LHT♯K$ is not a slice knot, where $g_s^n(K)$ is the $n$-shake genus of $K$.

18 pages, 81 figures

References in corpus (2)

• On homology 3-spheres defined by two knots
• Heegaard Floer homology of Matsumoto's manifolds