Detection of knots and a cabling formula for A-polynomials

TL;DR

This paper proves that torus knots and certain hyperbolic knots are uniquely identified by their knot Floer homology and A-polynomial, and provides a cabling formula for A-polynomials of cabled knots.

Contribution

It establishes detection results for torus and hyperbolic knots using knot invariants and derives a new cabling formula for A-polynomials of cabled knots.

Findings

Torus knots are detected by their knot Floer homology and A-polynomial.

A family of hyperbolic knots are similarly detected.

Explicit A-polynomials for iterated torus knots are provided.

Abstract

We say that a given knot $J\subset S^3$ is detected by its knot Floer homology and $A$-polynomial if whenever a knot $K\subset S^3$ has the same knot Floer homology and the same $A$-polynomial as $J$, then $K=J$. In this paper we show that every torus knot $T(p,q)$ is detected by its knot Floer homology and $A$-polynomial. We also give a one-parameter family of infinitely many hyperbolic knots in $S^3$ each of which is detected by its knot Floer homology and $A$-polynomial. In addition we give a cabling formula for the A-polynomials of cabled knots in $S^3$, which is of independent interest. In particular we give explicitly the A-polynomials of iterated torus knots.