Character varieties of double twist links

Petersen, Kathleen L; Tran, Anh T

doi:10.2140/agt.2015.15.3569

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Character varieties of double twist links

Kathleen L Petersen and Anh T Tran

Algebraic & Geometric Topology 15 (2015) 3569–3598

DOI: 10.2140/agt.2015.15.3569

Abstract

We compute both natural and smooth models for the ${SL}_{2} (ℂ)$ character varieties of the two-component double twist links, an infinite family of two-bridge links indexed as $J (k, l)$ . For each $J (k, l)$ , the component(s) of the character variety containing characters of irreducible representations are birational to a surface of the form $C \times ℂ$ , 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 ${SL}_{2} (ℂ)$ character variety of the $J (3, 2 m + 1)$ link as the surface obtained from $ℙ^{1} \times ℙ^{1}$ as a series of blow-ups.

Keywords

character variety, canonical component, double twist link

Mathematical Subject Classification 2010

Primary: 57M25

Secondary: 57N10, 14J26

References

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

Volume 15, issue 6 (2015)