Development of the Alexander Polynomial of Single Component Knots via Siefert Surfaces

Abstract

This document explains the development of the Alexander Polynomial via Seifert Surfaces. To accomplish this, a few topics are considered. As a pseudo Table of Contents, here is a list of what's included. I. Preliminaries and Definitions II. Construction of Seifert Surfaces Ill. Construction of the Seifert Graph IV. Euler Characteristic and Genus of surfaces V. Homeomorphism between the sphere and the projective plane VI. The Fundamental Group of a Seifert Surface VII. The Seirert Matrix and S-Equivalence of Matrices VIII. The invariance of the Alexander Polynomial. IX. Appendix: A few "cheats" on computing the linking numbers for the Seifert Matrix