Knots yielding homeomorphic lens spaces by Dehn surgery

TL;DR

This paper demonstrates the existence of infinitely many knot pairs in the 3-sphere that produce homeomorphic lens spaces via the same Dehn surgery, with various types of knots involved.

Contribution

It establishes the infinite occurrence of such knot pairs and classifies the types of knots that can form these pairs, highlighting an unavoidable exception.

Findings

Infinitely many knot pairs yield homeomorphic lens spaces by the same Dehn surgery.

Pairs can include torus, satellite, or hyperbolic knots, but not two satellites.

The exception involving two satellite knots is explained by classical quadratic form theory.

Abstract

We show that there exist infinitely many pairs of distinct knots in the 3-sphere such that each pair can yield homeomorphic lens spaces by the same Dehn surgery. Moreover, each knot of the pair can be chosen to be a torus knot, a satellite knot or a hyperbolic knot, except that both cannot be satellite knots simultaneously. This exception is shown to be unavoidable by the classical theory of binary quadratic forms.