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,

In[8]:=	s = CJ`Summand[Mirror[Knot[3, 1]], n]
Out[8]=	6 + 4 CJ`k[1] + (-4 + 2 CJ`k[1])/2 1 1 {CJ`q qBinomial[0, 0, ----] qBinomial[CJ`k[1], 0, ----] CJ`q CJ`q 1 4 1 qBinomial[CJ`k[1], CJ`k[1], ----] qPochhammer[CJ`q , ----, 0] CJ`q CJ`q 4 1 4 - CJ`k[1] 1 qPochhammer[CJ`q , ----, CJ`k[1]] qPochhammer[CJ`q , ----, 0], {CJ`k[1]}} CJ`q CJ`q

The symbols in the above formula require a definition:

In[9]:= ?qPochhammer qPochhammer[a, q, k] represents the q-shifted factorial of a in base q with index k. See Eric Weisstein's http://mathworld.wolfram.com/q-PochhammerSymbol.html and Axel Riese's www.risc.uni-linz.ac.at/research/combinat/risc/software/qMultiSum/

In[10]:= ?qBinomial qBinomial[n, k, q] represents the q-binomial coefficient of n and k in base q. For k<0 it is 0; otherwise it is qPochhammer[q^(n-k+1), q, k] / qPochhammer[q, q, k].

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:

In[11]:= ?qExpand qExpand[expr_] replaces all occurences of qPochhammer and qBinomial in expr by their definitions as products. See the documentation for qPochhammer and for qBinomial for details.

Hence,

In[12]:=	qPochhammer[a, q, 6] // qExpand
Out[12]=	2 3 4 5 (-1 + a) (-1 + a q) (-1 + a q ) (-1 + a q ) (-1 + a q ) (-1 + a q )

In[13]:=	First[s] /. {n -> 3, CJ`k[1] -> 2} // qExpand
Out[13]=	12 2 3 CJ`q (-1 + CJ`q ) (-1 + CJ`q )

Finally,

In[14]:= ?ColoredJones Type ColoredJones and see for yourself.

[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.