Detection of knots and a cabling formula for A–polynomials

Ni, Yi; Zhang, Xingru

doi:10.2140/agt.2017.17.65

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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

DOI: 10.2140/agt.2017.17.65

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.

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/

Volume 17, issue 1 (2017)