On Character varieties of two-bridge knot groups

Abstract

We find explicit models for the PSL(2,C)- and SL(2,C)-character varieties of the fundamental groups of complements in S^3 of an infinite family of two-bridge knots that contains the twist knots. We compute the genus of the components of these character varieties, and deduce upper bounds on the degree of the associated trace fields. We also show that these knot complements are fibered if and only if they are commensurable to a fibered knot complement in a Z/2Z-homology sphere, resolving a conjecture of Hoste and Shanahan.

36 pages, 4 figures

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• The $\mathrm{SL}(2,\mathbb{C})$-character variety of a Montesinos knot
• A vanishing identity on adjoint Reidemeister torsions of twist knots
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• Some non-abelian covers of knots with non-trivial Alexander polynomial
• On the volume of double twist link cone-manifolds
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• Intercusp geodesics and the invariant trace field of hyperbolic 3-manifolds
• A conjecture of Chinburg-Reid-Stover for surgeries on twist knots
• Genera of two-bridge knots and epimorphisms of their knot groups
• Hasse-Weil zeta functions of SL_2-character varieties of arithmetic two bridge link complements
• Symmetries and Detection of Surfaces by the Character Variety
• Character varieties of some families of links
• $PS{L}_2(\mathbb{C})$-character varieties and Seifert fibered cosmetic surgeries