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.
College and Department
Physical and Mathematical Sciences; Mathematics
BYU ScholarsArchive Citation
Gardiner, Jason Robert, "Petal Diagrams and Seifert Surfaces" (2021). Theses and Dissertations. 9253.
knots, petal diagrams, Seifert surfaces, spanning surfaces, decomposition diagrams, petal decompositions