The Coloured Jones Polynomials: Difference between revisions
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The coloured Jones polynomial of |
The coloured Jones polynomial of [[3_1]] is computed via a single summation. Indeed, |
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computed via a single summation. Indeed, |
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< |
<!--$$s = CJ`Summand[Mirror[Knot[3, 1]], n]$$--> |
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\vskip 6pt |
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The symbols in the above formula require a definition: |
The symbols in the above formula require a definition: |
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<!--$$?qPochhammer$$--> |
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\index{Riese, Axel} \index{Weisstein, Eric} |
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<* HelpBox[{qPochhammer, qBinomial}] *> |
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<!--$$?qBinomial$$--> |
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\[ |
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(1-a)(1-aq)\dots(1-aq^{k-1}) & k>0 \\ |
(1-a)(1-aq)\dots(1-aq^{k-1}) & k>0 \\ |
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1 & k=0 \\ |
1 & k=0 \\ |
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\left((1-aq^{-1})(1-aq^{-2})\dots(1-aq^{k})\right)^{-1} & k<0 |
\left((1-aq^{-1})(1-aq^{-2})\dots(1-aq^{k})\right)^{-1} & k<0 |
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\end{cases} |
\end{cases} |
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</math</center> |
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\] |
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and |
and <code>qBinomial[n, k, q]</math> is |
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\[ |
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<center><math> |
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\binom{n}{k}_q = \begin{cases} |
\binom{n}{k}_q = \begin{cases} |
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\frac |
\frac |
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0 & k<0. |
0 & k<0. |
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\end{cases} |
\end{cases} |
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</math</center> |
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\] |
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The function |
The function <code>qExpand</code> replaces every occurence of a <code>qPochhammer[a, q, k]</code> |
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symbol or a |
symbol or a <code>qBinomial[n, k, q]</math> symbol by its definition: |
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< |
<!--$$?qExpand$$--> |
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Hence, |
Hence, |
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< |
<!--$$qPochhammer[a, q, 6] // qExpand$$--> |
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Finally, |
Finally, |
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< |
<!--$ColoredJones=.$--><!--END--> |
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<!--$$?ColoredJones$$--> |
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{{note|Garoufalidis Le}} S. Garoufalidis and T. Q. T. Le, ''The Colored Jones Function is <math>q</math>-Holonomic'', Georgia Institute of Technology preprint, September 2003, {{arXiv|math.GT/0309214}}. |
{{note|Garoufalidis Le}} S. Garoufalidis and T. Q. T. Le, ''The Colored Jones Function is <math>q</math>-Holonomic'', Georgia Institute of Technology preprint, September 2003, {{arXiv|math.GT/0309214}}. |
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Revision as of 17:05, 26 August 2005
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 :
| 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 ; 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
![{\displaystyle (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}}</math</center>and<code>qBinomial[n,k,q]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6902a3e3eed732af614a516b04e284b5fef4827a)
![{\displaystyle {\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}}</math</center>Thefunction<code>qExpand</code>replaceseveryoccurenceofa<code>qPochhammer[a,q,k]</code>symbolora<code>qBinomial[n,k,q]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/922a46a20ea7b84964239676de213393e3779cec)
Hence,
Finally,
[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.
