Compositio Mathematica



On the head and the tail of the colored Jones polynomial


Oliver T. Dasbach a1 and Xiao-Song Lin a2
a1 Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA kasten@math.lsu.edu
a2 Department of Mathematics, University of California, Riverside, CA 92521, USA xl@math.ucr.edu

Article author query
dasbach o   [Google Scholar] 
lin x   [Google Scholar] 
 

Abstract

The colored Jones polynomial is a function $J_K:{\mathbb{N}}\longrightarrow{\mathbb{Z}}[t,t^{-1}]$ associated with a knot $K$ in 3-space. We will show that for an alternating knot $K$ the absolute values of the first and the last three leading coefficients of $J_K(n)$ are independent of $n$ when $n$ is sufficiently large. Computation of sample knots indicates that this should be true for any fixed leading coefficient of the colored Jones polynomial for alternating knots. As a corollary we get a volume-ish theorem for the colored Jones polynomial.

(Published Online September 25 2006)
(Received December 21 2005)
(Accepted April 4 2006)


Key Words: colored Jones polynomial; volume-ish theorem; volume conjecture.

Maths Classification

57M25.