The Algebraic Geometry of Character Varieties of Double Twist Links

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May 2023

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Abstract

The thesis focuses on the study of SL2(C) character varieties for hyperbolic 3-manifolds, which encode topological information about the manifold such as the number of cusps and the genus. Specifically, the thesis explores the canonical component of the character variety of the double twist link J(2m+1, 2m+1), which contains a character corresponding to a discrete faithful representation. Even though character varieties have been used to study the topology of hyperbolic 3-manifolds, there is still a lot we don’t know about their properties. The thesis investigates the canonical component of character variety of two-component double twist links, which form an infinite family of two-bridge links, and fills a gap in previous research (Petersen and Tran, 2016) by proving that the singularities obtained in that paper are of order one. The main result of the thesis is Theorem 1.0.1, which states that the desingularization of the projective model for the canonical component of SL2(C) character variety of the double twist link J(2m + 1, 2m + 1) is a rational surface isomorphic to P1 × P1 blown up at 6m + 1, for m ≥ 1 and (−6m − 3) times if m ≤ −2.

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