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# Mathematical Research Letters

## Volume 18 (2011)

### Number 6

### Grid diagrams and the Ozsváth-Szabó tau-invariant

Pages: 1239 – 1257

DOI: https://dx.doi.org/10.4310/MRL.2011.v18.n6.a13

#### Author

#### Abstract

We use grid diagrams to investigate the \Ozsvath-\Szabo{} concordance invariant $\tau$, and to prove that $\left|\tau(K_1)-\tau(K_2)\right|\leq g$, 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.

Published 20 February 2012