On the head and the tail of the colored Jones polynomial

Abstract

The colored Jones polynomial is a function JK: ℕ → 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 JK(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. © Foundation Compositio Mathematica 2006.