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On the head and the tail of the colored Jones polynomial

Published online by Cambridge University Press:  25 September 2006

Oliver T. Dasbach
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USAkasten@math.lsu.edu
Xiao-Song Lin
Affiliation:
Department of Mathematics, University of California, Riverside, CA 92521, USAxl@math.ucr.edu
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Abstract

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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.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2006