Abstract

Petal diagrams of knots are projections of knots to the plane such that the diagram has exactly one crossing. Petal diagrams offer a convenient and combinatorial way of representing knots via their associated petal permutation. In this thesis we study the fundamental group and Seifert surfaces of knots in petal form. Using the Seifert-Van Kampen theorem, we give a group presentation of the fundamental group of the knot complement of a knot in petal form. We then discuss Seifert surfaces and use decomposition diagrams to represent the Seifert surfaces of knots in petal form. We finally give an algorithm to produce a set of decomposition diagrams for a spanning surface of a knot in petal form and prove that for incompressible surfaces such decomposition diagrams are unique up to perturbation moves.

Degree

MS

College and Department

Physical and Mathematical Sciences; Mathematics

Rights

https://lib.byu.edu/about/copyright/

Date Submitted

2021-08-02

Document Type

Thesis

Handle

http://hdl.lib.byu.edu/1877/etd11891

Keywords

knots, petal diagrams, Seifert surfaces, spanning surfaces, decomposition diagrams, petal decompositions

Language

english

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