#### Volume 17, issue 1 (2017)

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

### Yi Ni and Xingru Zhang

Algebraic & Geometric Topology 17 (2017) 65–109
##### 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.

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##### Keywords
knot Floer homology, A-polynomial, cabling formula, Eudave-Muñoz knots
Primary: 57M25
##### Publication
Received: 26 March 2015
Revised: 9 May 2016
Accepted: 19 May 2016
Published: 26 January 2017
##### Authors
 Yi Ni Department of Mathematics Caltech 1200 E California Blvd Pasadena, CA 91125 United States http://www.its.caltech.edu/~yini/ Xingru Zhang Department of Mathematics University at Buffalo 111 Mathematics Building Buffalo, NY 14260-2900 United States http://www.math.buffalo.edu/~xinzhang/