The Coloured Jones Polynomials

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KnotTheory` can compute the coloured Jones polynomial of knots and links, using the formulas in [Garoufalidis Le]:

(For In[1] see Setup)

In[2]:= ?ColouredJones ColouredJones[K, n][q] returns the coloured Jones polynomial of a knot in colour n (i.e., in the (n+1)-dimensional representation) in the indeterminate q. Some of these polynomials have been precomputed in KnotTheory`. To force computation, use ColouredJones[K,n, Program -> "prog"][q], with "prog" replaced by one of the two available programs, "REngine" or "Braid" (including the quotes). "REngine" (default) computes the invariant for closed knots (as well as links where all components are coloured by the same integer) directly from the MorseLink presentation of the knot, while "Braid" computes the invariant via a presentation of the knot as a braid closure. "REngine" will usually be faster, but it might be better to use "Braid" when (roughly): 1) a "good" braid representative is available for the knot, and 2) the length of this braid is less than the maximum width of the MorseLink presentation of the knot.

In[3]:= ColouredJones::about The "REngine" algorithm was written by Siddarth Sankaran in the summer of 2005, while the "Braid" algorithm was written jointly by Dror Bar-Natan and Stavros Garoufalidis. Both are based on formulas by Thang Le and Stavros Garoufalidis; see [Garoufalidis, S. and Le, T. "The coloured Jones function is q-holonomic." Geom. Top., v9, 2005 (1253-1293)].

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 3 + q - q - q + -- - -- + -- - -- - 3 q + 3 q - 2 q + 8 6 4 2 q q q q 8 10 11 12 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

4_1

3_1

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]=	(3 n)/2 + n CJ`k[1] + (-n + 2 CJ`k[1])/2 1 {CJ`q qBinomial[0, 0, ----] CJ`q 1 1 qBinomial[CJ`k[1], 0, ----] qBinomial[CJ`k[1], CJ`k[1], ----] CJ`q CJ`q n 1 n 1 qPochhammer[CJ`q , ----, 0] qPochhammer[CJ`q , ----, CJ`k[1]] CJ`q CJ`q n - CJ`k[1] 1 qPochhammer[CJ`q , ----, 0], {CJ`k[1]}} 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]=	11 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.