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A conjecture on amphicheiral knots
by
Vajira Manathunga
University of Tennessee, Knoxville,TN
Detecting chirality is one of main question in knot theory. For this end, various methods have been developed in the past. The polynomial invariants like Jones, HOMFLYPT, Kauffman are the most prominent methods among them. In 2006, James Conant found that for every natural number n, a certain polynomial in the coefficient of the Conway polynomial is a primitive integer-valued degree n Vassiliev invariant, which named as pcn. It is conjectured that pc4n \mod 2 vanishes on all amphicheiral knots. Another way to formulate this conjecture is, if K is an amphicheiral knot then there is a polynomial F such that C(z)C(iz)C(z2)=F2 Where F ∈ Z4(z2) , C(z) is the Conway polynomial of knot K and i = √{-1}. By work of Kawauchi and Hartley it can be easily shown that this is true for all negative and strong positive amphicheiral knots. However the conjecture still remain unsolved for positive amphicheiral knots (not strong). In this talk we summarize our work on this conjecture.
Date received: February 12, 2015
Copyright © 2015 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbkp-03.