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 3_1 is computed via a single summation. Indeed,

The symbols in the above formula require a definition:

More precisely, 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 qBinomial[n, k, q] is

${\binom {n}{k}}_{q}={\begin{cases}{\frac {(q^{n-k+1};q)_{k}}{(q;q)_{k}}}&k\geq 0\\0&k<0.\end{cases}}$

The function qExpand replaces every occurence of a qPochhammer[a, q, k] symbol or a qBinomial[n, k, q] symbol by its definition:

Hence,

Finally,

[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.