Character Varieties of Knots and Links with Symmetries

• Other Authors: Sparaco, Leona H. (authoraut)
• Format: Others
• Language: English; English
• Published: Florida State University
• Subjects: Mathematics
• Online Access: http://purl.flvc.org/fsu/fd/FSU_SUMMER2017_Sparaco_fsu_0071E_13851

: Let M be a hyperbolic manifold. The SL2(C) character variety of M is essentially the set of all representations ρ : π1(M) → SL2(C) up to trace equivalence. This algebraic set is connected to many geometric properties of the manifold M. We examine the effect of symmetries of M on its character variety. We compute the SL2(C) and PSL2(C) character varieties for an infinite family of two-bridge hyperbolic knots with symmetry. We explore the effect the symmetry has on the character variety and exploit this symmetry to factor the character variety. We then find the geometric genus of both components of the character variety. We compute the SL2(C) character variety for the Borromean ring complement in S^3. Further, we explore how the symmetries effect this character variety. Finally, we prove some general results about the structure of character varieties of links with symmetries. === A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. === Summer Semester 2017. === July 6, 2017. === Includes bibliographical references. === Kathleen Petersen, Professor Directing Dissertation; Kristine Harper, University Representative; Sam Ballas, Committee Member; Philip Bowers, Committee Member; Eriko Hironaka, Committee Member.