A number theoretic result for Berge's conjecture
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∈
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
.