# Rob Gaebler

Harvey Mudd College Mathematics 2004

Thesis Proposal: | Alexander Polynomials of Tunnel Number One Knots |
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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.