#### Date of Award

5-2012

#### Degree Type

Thesis

#### Degree Name

Master of Science

#### Major

Mathematics

#### Major Professor

James R. Conant

#### Committee Members

Don B. Hinton, Morwen B. Thistlethwaite

#### Abstract

We disprove the conjecture that if K is amphicheiral and K is concordant to K', then C_{K'}(z)C_{K'}(iz)C_{K\}(z^{2}) is a perfect square inside the ring of power series with integer coefficients. The Alexander polynomial of (p,q)-torus knots are found to be of the form A_{T(p,q)}(t)= (f(t^{q}))/(f(t)) where f(t)=1+t+t^{2}+...+t^{p-1}. Also, for (p^{n},q)-torus knots, the Alexander polynomial factors into the form A_{T(pn },q)=f(t)f(t^{p})f(t^{p2 } )...f(t^{pn-2 } )f(t^{pn-1 } ). A new conversion from the Alexander polynomial to the Conway polynomial is discussed using the Lucas polynomial. This result is used to show that the Conway polynomial of (2^{n},q)-torus knots are of the form C_{T(2n },q)(z)=K_{1}K_{2...}K_{n} where K_{1}=F_{q}(z), F_{q}(z) being the Fibonacci polynomial, and K_{i}(z)=K_{i-1}(√z^{4}+4z^{2}).

#### Recommended Citation

Agle, Katherine Ellen Louise, "Alexander and Conway polynomials of Torus knots. " Master's Thesis, University of Tennessee, 2012.

http://trace.tennessee.edu/utk_gradthes/1127