Knots that share four surgeries

TL;DR

This paper disproves two conjectures by constructing examples of distinct knots sharing multiple Dehn surgeries and non-isotopic Legendrian knots with contactomorphic surgeries, challenging previous assumptions in knot theory and contact geometry.

Contribution

The authors construct explicit examples of knots sharing multiple surgeries and Legendrian knots with contactomorphic surgeries, providing counterexamples to longstanding conjectures.

Findings

Constructed pairs of knots sharing four distinct Dehn surgeries.

Disproved folk conjecture on uniqueness of shared surgeries for fixed knot pairs.

Presented non-isotopic Legendrian knots with contactomorphic contact (+1) and (-1) surgeries.

Abstract

Distinct knots K, K' can sometimes share a common p/q-framed Dehn surgery. A folk conjecture held that for a fixed pair of knots, this can occur for at most one value of p/q. We disprove this conjecture by constructing pairs of distinct knots K,K' that have common Dehn surgeries for four distinct slopes. We also construct non-isotopic Legendrian knots K,K' that have contactomorphic contact (+1)-and (-1)-surgeries, disproving an analogous conjecture in contact geometry.