Only finitely many alternating knots can yield a given manifold by surgery

TL;DR

This paper proves that for any fixed 3-manifold, only finitely many alternating knots in the 3-sphere can produce it via Dehn surgery, establishing a finiteness result in knot theory.

Contribution

It establishes a finiteness theorem for alternating knots that can yield a given 3-manifold through surgery, extending previous partial results.

Findings

Finiteness of alternating knots for a fixed manifold

Extension of prior partial results by Lackenby and Purcell

Contributes to classification of knots via surgery outcomes

Abstract

We show that given a 3-manifold $Y$ there is only a finite number of alternating knots $K \subset S^3$ such that $Y$ can be obtained by surgery on $K$. A very similar but somewhat not complete statement has been obtained in a recent preprint of Lackenby and Purcell.