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 CK'(z)CK'(iz)CK\(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 AT(p,q)(t)= (f(tq))/(f(t)) where f(t)=1+t+t2+...+tp-1. Also, for (pn,q)-torus knots, the Alexander polynomial factors into the form AT(pn ,q)=f(t)f(tp)f(tp2 )...f(tpn-2 )f(tpn-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 CT(2n ,q)(z)=K1K2...Kn where K1=Fq(z), Fq(z) being the Fibonacci polynomial, and Ki(z)=Ki-1(√z4+4z2).

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