The Coloured Jones Polynomials

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KnotTheory` can compute the coloured Jones polynomial of braid closures, using the same formulas as in [Garoufalidis Le]:

(For In[1] see Setup)

In[2]:= ?ColouredJones ColouredJones[br, n][q] computes the coloured Jones polynomial of the closure of the braid br in colour n (i.e., in the (n+1)-dimensional representation) and with respect to the variable q. ColouredJones[K, n][q] does the same for knots for which a braid representative is known to this program.

In[3]:= ColouredJones::about The ColouredJones program was written jointly with Stavros Garoufalidis, based on formulas provided to us by Thang Le.

Thus, for example, here's the coloured Jones polynomial of the knot 4_1 in the 4-dimensional representation of $sl(2)$:

In[4]:=	ColouredJones[Knot[4, 1], 3][q]
Out[4]=	-12 -11 -10 2 2 3 3 2 4 6 8 10 11 12 3 + q - q - q + -- - -- + -- - -- - 3 q + 3 q - 2 q + 2 q - q - q + q 8 6 4 2 q q q q

And here's the coloured Jones polynomial of the same knot in the two dimensional representation of $sl(2)$; this better be equal to the ordinary Jones polynomial of 4_1!

In[5]:=	ColouredJones[Knot[4, 1], 1][q]
Out[5]=	-2 1 2 1 + q - - - q + q q

In[6]:=	Jones[Knot[4, 1]][q]
Out[6]=	-2 1 2 1 + q - - - q + q q

In[7]:= ?CJ`Summand CJ`Summand[br, n] returned a pair {s, vars} where s is the summand in the the big sum that makes up ColouredJones[br, n][q] and where vars is the list of variables that need to be summed over (from 0 to n) to get ColouredJones[br, n][q]. CJ`Summand[K, n] is the same for knots for which a braid representative is known to this program.

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 $q$-Holonomic, Georgia Institute of Technology preprint, September 2003, arXiv:math.GT/0309214.