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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 S3 is detected by its knot Floer homology and A–polynomial if whenever a knot K S3 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 S3 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 S3, which is of independent interest. In particular we give explicitly the A–polynomials of iterated torus knots.

Keywords
knot Floer homology, A-polynomial, cabling formula, Eudave-Muñoz knots
Mathematical Subject Classification 2010
Primary: 57M25
References
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/