Detection of knots and a cabling formula for $A$–polynomials

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 $T\left(p,q\right)$ 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.