Non-integer characterizing slopes for torus knots

TL;DR

This paper proves that for each torus knot, almost all non-integer slopes are characterizing, meaning they uniquely determine the knot from the resulting surgery, extending previous results and analyzing Heegaard Floer homology.

Contribution

It establishes that all but finitely many non-integer slopes are characterizing for torus knots, generalizing prior work and linking surgery properties to knot invariants.

Findings

Almost all non-integer slopes are characterizing for torus knots.

Homeomorphic surgeries imply equal genera and Alexander polynomials for large slopes.

Heegaard Floer homology grading helps distinguish knots from surgery data.

Abstract

A slope $p/q$ is a characterizing slope for a knot $K$ in $S^3$ if the oriented homeomorphism type of $p/q$-surgery on $K$ determines $K$ uniquely. We show that for each torus knot its set of characterizing slopes contains all but finitely many non-integer slopes. This generalizes work of Ni and Zhang who established such a result for $T_{5,2}$. Along the way we show that if two knots $K$ and $K'$ in $S^3$ have homeomorphic $p/q$-surgeries, then for $q\geq 3$ and $p$ sufficiently large we can conclude that $K$ and $K'$ have the same genera and Alexander polynomials. This is achieved by consideration of the absolute grading on Heegaard Floer homology.