Grid diagrams and the Ozsvath-Szabo tau-invariant

TL;DR

This paper employs grid diagrams to analyze the Ozsvath-Szabo tau-invariant, establishing bounds related to knot cobordisms and providing a grid diagram-based proof of the Milnor conjecture.

Contribution

It introduces a grid diagram approach to study tau-invariant and offers a new proof of the Milnor conjecture using this method.

Findings

Proves |tau(K_1)-tau(K_2)|<=g for genus g cobordisms.

Provides a grid diagram-based proof of the Milnor conjecture.

Establishes bounds on tau-invariant differences via knot cobordisms.

Abstract

We use grid diagrams to investigate the Ozsvath-Szabo concordance invariant tau, and to prove that |tau(K_1)-tau(K_2)|<=g, whenever there is a genus g knot cobordism joining K_1 to K_2. This leads to an entirely grid diagram-based proof of Kronheimer-Mrowka's theorem, formerly known as the Milnor conjecture.