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(EDIT: Powers of $q$ in the formula corrected.)

I've been doing some computations with skein modules, and I found the following formula for the N-th colored Jones polynomial of the trefoil:

$\frac{1}{q^2-q^6}\sum_{i=1}^{N+1} (-1)^i q^{6(N^2-i^2)}(q^{10i-4}-q^{2i})$

I'm interested in comparing this to formulas that appear in the literature. I'm pretty sure I've seen similar formulas before, but I haven't been able to find them again.

Where can I find explicit formulas for colored Jones polynomials of the trefoil (or even better, for some family of "small" knots that includes the trefoil)?

(To be precise I'll explain my normalization. Identify the Kauffman bracket skein module of the tubular neighborhood of the (0-framed) trefoil with $\mathbb C[x]$, where $x^n$ is identified with $n$ parallel copies of the trefoil. When I say "N-th colored Jones polynomial," I mean the Chebyshev polynomial $S_N(x)$ viewed as an element of the skein module of $S^3$. In particular, the first Jones polynomial of the unknot is $-q^2-q^{-2}$.) (Again, to be precise, $S_0=1, S_1=x$, and $S_{n+1}(x) = xS_n(x) - S_n(x)$.)

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  • $\begingroup$ Sorry about the bump to the front page, I forgot that happened... $\endgroup$
    – Peter Samuelson
    Commented Mar 6, 2012 at 19:46

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This appears to be done by K. Habiro in "On the colored jones polynomials of some simple links" (the trefoil is the last section of the paper).

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  • $\begingroup$ Thanks, this (and its references) have been very helpful. $\endgroup$
    – Peter Samuelson
    Commented Mar 8, 2012 at 17:31

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