#### Volume 15, issue 6 (2015)

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Character varieties of double twist links

### Kathleen L Petersen and Anh T Tran

Algebraic & Geometric Topology 15 (2015) 3569–3598
##### Abstract

We compute both natural and smooth models for the ${SL}_{2}\left(ℂ\right)$ character varieties of the two-component double twist links, an infinite family of two-bridge links indexed as $J\left(k,l\right)$. For each $J\left(k,l\right)$, the component(s) of the character variety containing characters of irreducible representations are birational to a surface of the form $C×ℂ$, where $C$ is a curve. The same is true of the canonical component. We compute the genus of this curve, and the degree of irrationality of the canonical component. We realize the natural model of the canonical component of the ${SL}_{2}\left(ℂ\right)$ character variety of the $J\left(3,2m+1\right)$ link as the surface obtained from ${ℙ}^{1}×{ℙ}^{1}$ as a series of blow-ups.

##### Keywords
character variety, canonical component, double twist link
##### Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 57N10, 14J26
##### Publication
Received: 13 November 2014
Revised: 16 April 2015
Accepted: 26 April 2015
Published: 12 January 2016
##### Authors
 Kathleen L Petersen Department of Mathematics Florida State University 208 Love Building 1017 Academic Way Tallahassee, FL 32306-4510 USA Anh T Tran Department of Mathematical Sciences The University of Texas at Dallas 800 W Campbell Rd FO 35 Richardson, TX 75080 USA