The Coloured Jones Polynomials
KnotTheory`
can compute the coloured Jones polynomial of braid
closures, using the same formulas as in [Garoufalidis Le]:
(For In[1] see Setup)
Thus, for example, here's the coloured Jones polynomial of the knot 4_1 in the 4-dimensional representation of :
And here's the coloured Jones polynomial of the same knot in the two
dimensional representation of ; this better be equal to the ordinary
Jones polynomial of 4_1!
The coloured Jones polynomial of \hlink{../Knots/3.1.html}{$3_1$} is
computed via a single summation. Indeed,
<*InOut@"s = CJ`Summand[Mirror[Knot[3, 1]], n]"*> \vskip 6pt
The symbols in the above formula require a definition:
\index{Riese, Axel} \index{Weisstein, Eric} <* HelpBox[{qPochhammer, qBinomial}] *>
More precisely, {\tt qPochhammer[a, q, k]} is \[
(a;q)_k=\begin{cases} (1-a)(1-aq)\dots(1-aq^{k-1}) & k>0 \\ 1 & k=0 \\ \left((1-aq^{-1})(1-aq^{-2})\dots(1-aq^{k})\right)^{-1} & k<0 \end{cases}
\] and {\tt qBinomial[n, k, q]} is \[
\binom{n}{k}_q = \begin{cases} \frac {\displaystyle (q^{n-k+1};q)_k} {\displaystyle (q;q)_k } & k\geq 0 \\ 0 & k<0. \end{cases}
\]
The function {\tt qExpand} replaces every occurence of a {\tt qPochhammer} symbol or a {\tt qBinomial} symbol by its definition:
<* HelpBox[qExpand] *>
Hence,
<*InOut@"qPochhammer[a, q, 6] // qExpand"*> <*InOut@"First[s] /. {n -> 3, CJ`k[1] -> 2} // qExpand"*> \vskip 6pt
Finally,
<* ColoredJones=.; HelpBox[ColoredJones] *>
[Garoufalidis Le] ^ S. Garoufalidis and T. Q. T. Le, The Colored Jones Function is -Holonomic, Georgia Institute of Technology preprint, September 2003, arXiv:math.GT/0309214.