Infinitely many knots admitting the same integer surgery and a 4-dimensional extension

TL;DR

This paper demonstrates the existence of infinitely many knots in the 3-sphere that produce the same 3-manifold after surgery with a fixed integer, and extends this to a 4-dimensional context, solving longstanding problems.

Contribution

It constructs infinite families of knots with identical surgery outcomes and extends the results to 4-manifolds, addressing open problems in knot theory and 4-manifold topology.

Findings

Existence of infinitely many knots with the same integer surgery in $S^3$.

Construction of two families of such knots via twisting methods.

Extension of results to 4-manifolds via 2-handle additions.

Abstract

We prove that for any integer $n$ there exist infinitely many different knots in $S^3$ such that $n$-surgery on those knots yields the same 3-manifold. In particular, when $|n|=1$ homology spheres arise from these surgeries. This answers Problem 3.6(D) on the Kirby problem list. We construct two families of examples, the first by a method of twisting along an annulus and the second by a generalization of this procedure. The latter family also solves a stronger version of Problem 3.6(D), that for any integer $n$, there exist infinitely many mutually distinct knots such that 2-handle addition along each with framing $n$ yields the same 4-manifold.