Unique Surgery Descriptions along Knots

TL;DR

This paper proves that many 3-manifolds obtained by surgery on knots have a unique surgery description, and constructs examples where multiple descriptions exist, advancing understanding of characterising slopes and knot surgeries.

Contribution

It establishes conditions for unique surgery descriptions for hyperbolic L-space knots and constructs families of manifolds with multiple descriptions, generalizing characterising slopes.

Findings

Infinitely many r-surgeries on non-trivial knots have unique descriptions.

Hyperbolic L-space knots have manifolds with unique surgery descriptions for infinitely many slopes.

Constructed families of manifolds with multiple distinct surgery descriptions.

Abstract

We prove that for any non-trivial knot K, infinitely many r-surgeries K(r) along K have a unique surgery description along a knot. Moreover, we show that for any hyperbolic L-space knot K and infinitely many integer slopes n, the manifold K(n) has a unique surgery description. Here we say a 3-manifold M has a unique surgery description along a knot in S^3 if there is a unique pair (K,r) of a knot K and a slope r such that M is orientation-preservingly diffeomorphic to K(r). This generalises the notion of characterising slopes. Conversely, we provide new families of manifolds with several distinct surgery descriptions along knots. More precisely, we construct for every non-zero integer m a knot K_m such that for any integer n, the manifold K_m(m+1/n) can also be obtained by surgery on another knot.