Dehn surgery functions are never injective

TL;DR

This paper proves that for any fixed rational surgery coefficient, there exist distinct knots with homeomorphic surgeries, confirming a long-standing conjecture in knot theory.

Contribution

It establishes that Dehn surgery functions are not injective, resolving Gordon's 1978 conjecture about the existence of distinct knots with identical surgery outcomes.

Findings

Existence of distinct knots with homeomorphic $p/q$-surgeries for fixed rational $p/q$

Confirmation of Gordon's 1978 conjecture

Advancement in understanding the non-injectivity of Dehn surgery functions

Abstract

We prove that, for each fixed rational number $p/q \in \mathbb{Q}$, there exists a pair of distinct knots whose $p/q$-surgeries are orientation-preservingly homeomorphic. This confirms a 1978 conjecture of Gordon.