Computations of the Ozsvath-Szabo knot concordance invariant

TL;DR

This paper presents methods to efficiently compute the Ozsvath-Szabo tau invariant, demonstrating its nontriviality for certain knots and providing explicit calculations, along with a new proof of the Slice-Bennequin Inequality.

Contribution

It introduces computational shortcuts for the tau invariant and applies them to various knots, including new examples and a novel proof of an important inequality.

Findings

Tau invariant can be computed more efficiently using the proposed methods.

The invariant is nontrivial for all iterated untwisted positive doubles of certain knots.

A new proof of the Slice-Bennequin Inequality is derived from these techniques.

Abstract

Ozsvath and Szabo have defined a knot concordance invariant tau that bounds the 4-ball genus of a knot. Here we discuss shortcuts to its computation. We include examples of Alexander polynomial one knots for which the invariant is nontrivial, including all iterated untwisted positive doubles of knots with nonnegative Thurston-Bennequin number, such as the trefoil, and explicit computations for several 10 crossing knots. We also note that a new proof of the Slice-Bennequin Inequality quickly follows from these techniques.