The Head and Tail of the Colored Jones Polynomial

Cody Armond

Knots in Washington XXXIII; Categorification of Knots, Algebras, and Quandles; Quantum Computing
December 2-4, 2011
George Washington University
Washington, DC, USA

Organizers
Valentina Harizanov (GWU),Mark Kidwell (U.S. Naval Academy and GWU), Jozef H. Przytycki (GWU), Yongwu Rong (GWU), Radmila Sazdanovic (U.Penn), Alexander Shumakovitch (GWU), Hao Wu (GWU)

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The Head and Tail of the Colored Jones Polynomial
by
Cody Armond
Louisiana State University
Coauthors: Oliver Dasbach

The colored Jones polynomial is a sequence of Laurent-polynomial knot invariants beginning with the Jones polynomial. We use techniques from skein theory to show that for alternating knots (and more generally, adequate knots) the j-th coefficient of the N-th colored Jones polynomial viewed as a sequence in N, will stabalize at N=j. For a knot with this property, we can define a pair of power series called the head and tail of the colored Jones polynomial.

In a joint work with Oliver Dasbach, we show many examples including torus knots and twist knots. Also, by considering the reduced A and B-graphs of an adequate diagram, we develop methods for calculating the head and tail of complicated knot from simpler knots. This allows us to determine the head and tail of a large family of knots, including, for example, all 2-bridge knots.

Date received: October 31, 2011