Polynomial splittings of Ozsvath and Szabo's d-invariant

TL;DR

This paper demonstrates a polynomial splitting property for Ozsvath and Szabo's d-invariants of rational homology 3-spheres, with applications to knot sliceness obstructions, paralleling Casson-Gordon invariants.

Contribution

It establishes a polynomial splitting property for d-invariants, extending structural understanding and enabling new applications in knot theory.

Findings

Polynomial splitting property for d-invariants proven

Application to obstructing smooth sliceness of knots

Structural similarities to Casson-Gordon invariants identified

Abstract

For any rational homology 3-sphere and one of its spin^{c}-structures, Ozsvath and Szabo defined a topological invariant, called d-invariant. Given a knot in the 3-sphere, the d-invariants associated with the prime-power-fold branched covers of the knot, obstruct the smooth sliceness of the knot. These invariants bear some structural resemblances to Casson-Gordon invariants, which obstruct the topological sliceness of a knot. Se-Goo Kim found a polynomial splitting property for Casson-Gordon invariants. In this paper, we show a similar result for Ozsvath and Szabo's d-invariants. We give an application of the result.