Classification of Lattices Bounded by Large Surgeries of Knots

TL;DR

This paper classifies certain lattices associated with positive definite four-manifolds bounded by large surgeries on knots, introduces a new concordance invariant, and generalizes Rasmussen's theorem on lens space surgeries.

Contribution

It provides a complete classification of lattices for large surgeries on knots and extends Rasmussen's theorem using this classification.

Findings

Classified all lattices realized as intersection forms for large surgeries on knots.

Defined a new concordance invariant based on the classification.

Generalized Rasmussen's theorem on lens space surgeries.

Abstract

We classify all the lattices realized as the intersection form of a positive definite four manifold with boundary $S_n^3(K)$ for a knot $K$ in the three sphere and a positive integer $n$ greater than $4g_4(K)+3$. We then use this result to define a concordance invariant and generalize a theorem of Rasmussen on lens space surgeries.