Topology of n-gonal Curve Complements
This thesis has two parts. The first part concerns topological invariants of the n-gonal plane curves. Our first result is an application of Krammer representations to the Libgober invariant for plane curve complements. This gives a multivariable invariant that depends only on the fundamental group....
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| Format: | Others |
| Language: | English English |
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Florida State University
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| Online Access: | http://purl.flvc.org/fsu/fd/FSU_2017SP_Aktas_fsu_0071E_13779 |
| Summary: | This thesis has two parts. The first part concerns topological invariants of the n-gonal plane curves. Our first result is an application of Krammer representations to the Libgober invariant for plane curve complements. This gives a multivariable invariant that depends only on the fundamental group. Our second results is an algorithm to compute the braid monodromy and Libgober polynomial invariant of n-gonal curves. We show that the algorithm improves on existing algorithms. We compare the information one gets from Alexander and Krammer polynomials. The second and main part of our thesis focuses on properties of dessins d'enfants associated to trigonal curves. Degtyarev first studied dessins d'enfants in this context giving a new method for computing braid monodromies and fundamental groups. Our first result is a classification of all possible combinatorial data that can occur for trigonal curves of low degree, as well as bounds on the number of possibilities for all degree. We also study deformations of trigonal curves and corresponding deformations of their dessins. Of special interest to Degtyarev was the case when the dessins are maximal. Our second result gives a sufficient condition for a trigonal curve to be deformable to one that is maximal. === A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. === Spring Semester 2017. === April 10, 2017. === Alexander Polynomial, Braid Monodromy, Dessin D'enfant, Krammer Polynomial, Krammer Representation === Includes bibliographical references. === Eriko Hironaka, Professor Co-Directing Dissertation; Washington Mio, Professor Co-Directing Dissertation; Piyush Kumar, University Representative; Wolfgang Heil, Committee Member; Paolo Aluffi, Committee Member. |
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