Heegaard Floer homology of Matsumoto's manifolds

TL;DR

This paper investigates the properties of Matsumoto's manifolds, computes invariants related to Whitehead doubles of torus knots, and distinguishes between sliceness and rational homology 4-ball bounding of their double branched covers.

Contribution

It provides new formulas for Ozsváth-Szabó's τ-invariant, computes δ-invariants and correction terms, and presents the first examples showing a gap between sliceness and rational 4-ball bound-ness.

Findings

No contractible bound exists for certain Matsumoto's manifolds when n<2τ(K2).

Computed δ-invariants and correction terms for Whitehead doubles of torus knots.

Identified examples where double branched covers are not slice but bound rational homology 4-balls.

Abstract

We consider a homology sphere $M_n(K_1,K_2)$ presented by two knots $K_1,K_2$ with linking number 1 and framing $(0,n)$. We call the manifold {\it Matsumoto's manifold}. We show that there exists no contractible bound of $M_n(T_{2,3},K_2)$ if $n<2\tau(K_2)$ holds. We also give a formula of Ozsv\'ath-Szab\'o's $\tau$-invariant as the total sum of the Euler numbers of the reduced filtration. We compute the $\delta$-invariants of the twisted Whitehead doubles of torus knots and correction terms of the branched covers of the Whitehead doubles. By using Owens and Strle's obstruction we show that the $12$-twisted Whitehead double of the $(2,7)$-torus knot and the $20$-twisted Whitehead double of the $(3,7)$-torus knot are not slice but the double branched covers bound rational homology 4-balls. These are the first examples having a gap between sliceness and rational 4-ball bound-ness of the double branched cover.