Topology Atlas | Conferences

How Big is the Kauffman Bracket ?
by
Charles Frohman
Department of Mathematics, The University of Iowa

Jozef Przytycki introduced skein modules as a way of constructing a Jones polynomial for links in an arbitrary three manifold. In this lecture we achieve his goal for links in circle bundles over a surface. Amazingly the polynomial is no longer a polynomial, but a holomorphic function on the unit disk. We also see that our answer extends Turaev's shadow world evaluations at roots of unity.

1. The Kauffman bracket of a link in S³.

   Here I will recall the definition of the Kauffman bracket, with variable t that will be a complex number. Throughout the lecture we will let s=t², and q=t⁴=s². Finally, we let h be a number with e^h=q. We let [n]=[(sⁿ-s^-n)/(s-s^-1)].

2. The Colored Jones polynomial

   • Jones-Wenzl idempotents We review the iterative formula for the nth Jones-Wenzl idempotent, and what it means to color a framed link with an idempotent.
   • Kauffman Triads We review what it means to color a framed trivalent graph admissibly.
3. Example: Colored Jones polynomial of the figure eight knot at t=.1 We present a table of numerical values of the Colored Jones polynomial of the figure eight knot. The numbers are astoundingly large.
4. The Skein Algebra of a cylinder over a surface.
5. The basic estimates. We develop estimates of the size of the basic evaluations that are needed to estimate the size of a Kauffman bracket.

   • \prod_i=1^\infty(1-qⁱ)
   • Relation to the dilogarithm
   • Quantized Factorials
   • Theta evaluations
   • Tetrahedral evaluations
6. The Yang-Mills meausure

   • The Yang-Mills measure on a surface with boundary
   • The Yang-Mills measure on a closed surface
   • Relation to the Shadow world

Date received: November 23, 2000

Copyright © 2000 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 # cafy-04.