Non-integer characterizing slopes and knot Floer homology

TL;DR

This paper investigates the conjecture that non-integer non-characterizing slopes are finite for knots in the 3-sphere, verifying it for knots with certain Floer homology simplicity conditions and exploring properties of L-space knots.

Contribution

It proves the conjecture for a broad class of knots including alternating and L-space knots, and shows that most slopes are characterizing for arbitrary knots.

Findings

Finiteness of non-integer non-characterizing slopes for certain knots.

Almost all slopes with |q|≥3 are characterizing for arbitrary knots.

L-space and almost L-space knots have infinitely many integer characterizing slopes.

Abstract

Conjecturally, a knot in the 3-sphere has only finitely many non-integer non-characterizing slopes. We verify this conjecture for all knots with knot Floer homology satisfying certain simplicity conditions. The class of knots satisfying our notion of simplicity includes alternating knots, $L$-space knots and the vast majority of knots with at most 12 crossings. For arbitrary knots in the 3-sphere we show that almost all slopes $p/q$ with $|q|\geq 3$ are characterizing. In addition, we show that all $L$-space knots and almost $L$-space knots have infinitely many integer characterizing slopes.