## Algebraic & Geometric Topology

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

#### 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 17, Number 1 (2017), 65-109.

Dates
Revised: 9 May 2016
Accepted: 19 May 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841309

Digital Object Identifier
doi:10.2140/agt.2017.17.65

Mathematical Reviews number (MathSciNet)
MR3604373

Zentralblatt MATH identifier
1361.57014

#### Citation

Ni, Yi; Zhang, Xingru. Detection of knots and a cabling formula for $A$–polynomials. Algebr. Geom. Topol. 17 (2017), no. 1, 65--109. doi:10.2140/agt.2017.17.65. https://projecteuclid.org/euclid.agt/1510841309

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