A number theoretic result for Berge's conjecture

TL;DR

This paper solves a specific number theoretic case of Berge's conjecture related to lens space surgeries on knots, extending the understanding of which knots admit such surgeries.

Contribution

It provides a solution to the $p > k^2$ case of the lens space realization problem, advancing the classification of knots with lens space surgeries.

Findings

Solved the $p > k^2$ case of Berge's conjecture

Connected number theory with knot surgery classification

Extended the list of known lens space surgeries

Abstract

(Original version of PhD thesis, submitted in Spring 2009 to Harvard University. Provides a solution of the $p > k^2$ case, corresponding to Berge families I-VI, of the "Lens space realization problem" later solved in entirety by Greene.) In the 1980's, Berge proved that a certain collection of knots in $S^3$ admitted lens space surgeries, a list which Gordon conjectured was exhaustive. More recently, J. Rasmussen used techniques from Heegaard Floer homology to translate the related problem of classifying simple knots in lens spaces admitting L-space homology sphere surgeries into a combinatorial number theory question about the data $(p,q,k)$ associated to a knot of homology class $k \in H_1(L(p,q))$ in the lens space $L(p,q)$. In the following paper, we solve this number theoretic problem in the case of $p > k^2$.