Tube-Equivalence of Spanning Surfaces and Seifert Surfaces

Publication Date

4-2008

Type of Culminating Activity

Thesis

Degree Title

Master of Science in Mathematics

Department

Mathematics

Supervisory Committee Chair

Uwe Kaiser

Abstract

Utilizing the tools familiar to the knot theorist, i.e., the Reidemeister moves, the Seifert algorithm, and cut and paste, Bar-Natan, Fulman, and Kauffman have proved that spanning surfaces are tube-equivalent for possibly disconnected spanning surfaces. In this paper, connectivity is added to the assumption and we show: If S1 and S2 are Seifert surfaces for the link L, then S1 and S2 are tube-equivalent. The proof proceeds by examining how changes to a projection of a link affect the corresponding Seifert surfaces. Maintaining connectedness of a surface allows for controlling the first homology and the Seifert pairing by S-equivalence, and thus, is used in proving that the Alexander polynomial of the given link is an invariant.

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