$\tau$-invariants for knots in rational homology spheres

TL;DR

This paper generalizes the $ au$-invariant from knots in the 3-sphere to knots in rational homology spheres, providing new tools for bounding the genus of surfaces in 4-manifolds.

Contribution

It introduces a collection of $ au$-invariants for knots in rational homology spheres, extending the original invariant's applicability.

Findings

Some invariants give lower bounds for surface genus in negative definite 4-manifolds.

The construction broadens the scope of $ au$-invariants beyond $S^3$.

Provides new techniques for studying knot genus in complex 3-manifolds.

Abstract

Ozsv\'ath and Szab\'o used the knot filtration on $\widehat{CF}(S^3)$ to define the $\tau$-invariant for knots in the 3-sphere. In this article, we generalize their construction and define a collection of $\tau$-invariants associated to a knot $K$ in a rational homology sphere $Y$. We then show that some of these invariants provide lower bounds for the genus of a surface with boundary $K$ properly embedded in a negative definite 4-manifold with boundary $Y$..