## Algebraic & Geometric Topology

### Character varieties of double twist links

#### Abstract

We compute both natural and smooth models for the $SL2(ℂ)$ 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 $C × ℂ$, 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 $SL2(ℂ)$ character variety of the $J(3,2m + 1)$ link as the surface obtained from $ℙ1 × ℙ1$ as a series of blow-ups.

#### Article information

Source
Algebr. Geom. Topol., Volume 15, Number 6 (2015), 3569-3598.

Dates
Received: 13 November 2014
Revised: 16 April 2015
Accepted: 26 April 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841077

Digital Object Identifier
doi:10.2140/agt.2015.15.3569

Mathematical Reviews number (MathSciNet)
MR3450771

Zentralblatt MATH identifier
1360.57021

#### Citation

Petersen, Kathleen L; Tran, Anh T. Character varieties of double twist links. Algebr. Geom. Topol. 15 (2015), no. 6, 3569--3598. doi:10.2140/agt.2015.15.3569. https://projecteuclid.org/euclid.agt/1510841077

#### References

• K L Baker, K L Petersen, Character varieties of once-punctured torus bundles with tunnel number one, Internat. J. Math. 24 (2013)
• S Boyer, E Luft, X Zhang, On the algebraic components of the ${\rm SL}(2,\mathbb C)$ character varieties of knot exteriors, Topology 41 (2002) 667–694
• M Culler, P B Shalen, Varieties of group representations and splittings of $3$–manifolds, Ann. of Math. 117 (1983) 109–146
• R Fricke, Ueber die Theorie der automorphen Modulgruppen, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1896 (1896) 91–101
• S Harada, Canonical components of character varieties of arithmetic two bridge link complements, preprint (2012)
• E Landes, Identifying the canonical component for the Whitehead link, Math. Res. Lett. 18 (2011) 715–731
• E Landes, On the canonical components of character varieties of hyperbolic $2$–bridge link complements, PhD thesis, University of Texas at Austin (2011) Available at \setbox0\makeatletter\@url http://hdl.handle.net/2152/ETD-UT-2011-08-2877 {\unhbox0
• M L Macasieb, K L Petersen, R M van Luijk, On character varieties of two-bridge knot groups, Proc. Lond. Math. Soc. 103 (2011) 473–507
• T Ohtsuki, Ideal points and incompressible surfaces in two-bridge knot complements, J. Math. Soc. Japan 46 (1994) 51–87
• T Ohtsuki, R Riley, M Sakuma, Epimorphisms between $2$–bridge link groups, from: “The Zieschang Gedenkschrift”, (M Boileau, M Scharlemann, R Weidmann, editors), Geom. Topol. Monogr. 14 (2008) 417–450
• K L Petersen, A W Reid, Gonality and genus of canonical components of character varieties, preprint (2014)
• K Qazaqzeh, The character variety of a family of one-relator groups, Internat. J. Math. 23 (2012)
• P B Shalen, Representations of $3$–manifold groups, from: “Handbook of geometric topology”, (R J Daverman, R B Sher, editors), North-Holland, Amsterdam (2002) 955–1044
• W P Thurston, The geometry and topology of three-manifolds, lecture notes, Princeton University (1979) Available at \setbox0\makeatletter\@url http://msri.org/publications/books/gt3m {\unhbox0
• H-o Tokunaga, H Yoshihara, Degree of irrationality of abelian surfaces, J. Algebra 174 (1995) 1111–1121
• A T Tran, The universal character ring of the $(-2,2m+1,2n)$–pretzel link, Internat. J. Math. 24 (2013)
• H Vogt, Sur les invariants fondamentaux des équations différentielles linéaires du second ordre, Ann. Sci. École Norm. Sup. 6 (1889) 3–71
• H Yoshihara, Degree of irrationality of a product of two elliptic curves, Proc. Amer. Math. Soc. 124 (1996) 1371–1375