Rob Gaebler

Harvey Mudd College Mathematics 2004

mugshot
Thesis Proposal: Alexander Polynomials of Tunnel Number One Knots
Thesis Advisor: Prof. Jim Hoste
Second Reader: Prof. Weiqing Gu
E-Mail: rgaebler@hmc.edu
Final Draft: Alexander Polynomials of Two-Bridge Knots and Links

Alexander Polynomials of Two-Bridge Knots and Links

Every two-bridge knot or link is characterized by a rational number p/q, and has a fundamental group which has a simple presentation with only two generators and one relator. The relator has a form that gives rise to a formula for the Alexander polynomial of the knot or link in terms of p and q. Every two-bridge knot or link also has a corresponding ``up-down'' graph in terms of p and q. This graph is analyzed combinatorially to prove several properties of the Alexander polynomial. The number of two-bridge knots and links of a given crossing number are also counted.