Alexander and Conway polynomials of Torus knots

Author

Katherine Ellen Louise Agle, University of Tennessee - KnoxvilleFollow

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\(z²) 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+t²+...+t^p-1. Also, for (pⁿ,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 (2ⁿ,q)-torus knots are of the form C_T(2ⁿ ,q)(z)=K₁K_2...K_n where K₁=F_q(z), F_q(z) being the Fibonacci polynomial, and K_i(z)=K_i-1(√z⁴+4z²).

Recommended Citation

Agle, Katherine Ellen Louise, "Alexander and Conway polynomials of Torus knots. " Master's Thesis, University of Tennessee, 2012.
https://trace.tennessee.edu/utk_gradthes/1127