Infinitely many knots admitting the same integer surgery

TL;DR

This paper proves that for any integer n, there are infinitely many distinct knots in S^3 that produce the same 3-manifold via n-surgery, expanding understanding of knot surgeries and their resulting manifolds.

Contribution

It demonstrates the existence of infinitely many knots with the same n-surgery manifold for any integer n, including cases leading to homology spheres.

Findings

For any integer n, infinitely many knots yield the same n-surgery manifold.

When |n|=1, the surgeries produce homology spheres.

Bridge numbers of these knots tend to infinity as twists increase.

Abstract

The construction of knots via annular twisting has been used to create families of knots yielding the same manifold via Dehn surgery. Prior examples have all involved Dehn surgery where the surgery slope is an integral multiple of 2. In this note 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 manifold. In particular, when $|n|=1$ homology spheres arise from these surgeries. In addition, when $n \neq 0$ the bridge numbers of the knots constructed tend to infinity as the number of twists along the annulus increases.