The Coloured Jones Polynomials: Difference between revisions
No edit summary |
DrorsRobot (talk | contribs) No edit summary |
||
Line 7: | Line 7: | ||
<!--Robot Land, no human edits to "END"--> |
<!--Robot Land, no human edits to "END"--> |
||
{{HelpAndAbout| |
{{HelpAndAbout| |
||
n = |
n = 2 | |
||
n1 = |
n1 = 3 | |
||
in = <nowiki>ColouredJones</nowiki> | |
in = <nowiki>ColouredJones</nowiki> | |
||
out= <nowiki>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.</nowiki> | |
out= <nowiki>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.</nowiki> | |
||
Line 20: | Line 20: | ||
<!--Robot Land, no human edits to "END"--> |
<!--Robot Land, no human edits to "END"--> |
||
{{InOut| |
{{InOut| |
||
n = |
n = 4 | |
||
in = <nowiki>ColouredJones[Knot[4, 1], 3][q]</nowiki> | |
in = <nowiki>ColouredJones[Knot[4, 1], 3][q]</nowiki> | |
||
out= <nowiki> -12 -11 -10 2 2 3 3 2 4 6 |
out= <nowiki> -12 -11 -10 2 2 3 3 2 4 6 |
||
Line 38: | Line 38: | ||
<!--Robot Land, no human edits to "END"--> |
<!--Robot Land, no human edits to "END"--> |
||
{{InOut| |
{{InOut| |
||
n = |
n = 5 | |
||
in = <nowiki>ColouredJones[Knot[4, 1], 1][q]</nowiki> | |
in = <nowiki>ColouredJones[Knot[4, 1], 1][q]</nowiki> | |
||
out= <nowiki> -2 1 2 |
out= <nowiki> -2 1 2 |
||
Line 48: | Line 48: | ||
<!--Robot Land, no human edits to "END"--> |
<!--Robot Land, no human edits to "END"--> |
||
{{InOut| |
{{InOut| |
||
n = |
n = 6 | |
||
in = <nowiki>Jones[Knot[4, 1]][q]</nowiki> | |
in = <nowiki>Jones[Knot[4, 1]][q]</nowiki> | |
||
out= <nowiki> -2 1 2 |
out= <nowiki> -2 1 2 |
||
Line 60: | Line 60: | ||
<!--Robot Land, no human edits to "END"--> |
<!--Robot Land, no human edits to "END"--> |
||
{{HelpLine| |
{{HelpLine| |
||
n = |
n = 7 | |
||
in = <nowiki>CJ`Summand</nowiki> | |
in = <nowiki>CJ`Summand</nowiki> | |
||
out= <nowiki>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.</nowiki>}} |
out= <nowiki>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.</nowiki>}} |
||
Line 70: | Line 70: | ||
<!--Robot Land, no human edits to "END"--> |
<!--Robot Land, no human edits to "END"--> |
||
{{InOut| |
{{InOut| |
||
n = |
n = 8 | |
||
in = <nowiki>s = CJ`Summand[Mirror[Knot[3, 1]], n]</nowiki> | |
in = <nowiki>s = CJ`Summand[Mirror[Knot[3, 1]], n]</nowiki> | |
||
out= <nowiki> (3 n)/2 + n CJ`k[1] + (-n + 2 CJ`k[1])/2 1 |
out= <nowiki> (3 n)/2 + n CJ`k[1] + (-n + 2 CJ`k[1])/2 1 |
||
Line 94: | Line 94: | ||
<!--Robot Land, no human edits to "END"--> |
<!--Robot Land, no human edits to "END"--> |
||
{{HelpLine| |
{{HelpLine| |
||
n = |
n = 9 | |
||
in = <nowiki>qPochhammer</nowiki> | |
in = <nowiki>qPochhammer</nowiki> | |
||
out= <nowiki>qPochhammer[a, q, k] represents the q-shifted factorial of a in base q with index k. See Eric Weisstein's |
out= <nowiki>qPochhammer[a, q, k] represents the q-shifted factorial of a in base q with index k. See Eric Weisstein's |
||
Line 104: | Line 104: | ||
<!--Robot Land, no human edits to "END"--> |
<!--Robot Land, no human edits to "END"--> |
||
{{HelpLine| |
{{HelpLine| |
||
n = |
n = 10 | |
||
in = <nowiki>qBinomial</nowiki> | |
in = <nowiki>qBinomial</nowiki> | |
||
out= <nowiki>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 |
out= <nowiki>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 |
||
Line 136: | Line 136: | ||
<!--Robot Land, no human edits to "END"--> |
<!--Robot Land, no human edits to "END"--> |
||
{{HelpLine| |
{{HelpLine| |
||
n = |
n = 11 | |
||
in = <nowiki>qExpand</nowiki> | |
in = <nowiki>qExpand</nowiki> | |
||
out= <nowiki>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.</nowiki>}} |
out= <nowiki>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.</nowiki>}} |
||
Line 146: | Line 146: | ||
<!--Robot Land, no human edits to "END"--> |
<!--Robot Land, no human edits to "END"--> |
||
{{InOut| |
{{InOut| |
||
n = |
n = 12 | |
||
in = <nowiki>qPochhammer[a, q, 6] // qExpand</nowiki> | |
in = <nowiki>qPochhammer[a, q, 6] // qExpand</nowiki> | |
||
out= <nowiki> 2 3 4 5 |
out= <nowiki> 2 3 4 5 |
||
Line 155: | Line 155: | ||
<!--Robot Land, no human edits to "END"--> |
<!--Robot Land, no human edits to "END"--> |
||
{{InOut| |
{{InOut| |
||
n = |
n = 13 | |
||
in = <nowiki>First[s] /. {n -> 3, CJ`k[1] -> 2} // qExpand</nowiki> | |
in = <nowiki>First[s] /. {n -> 3, CJ`k[1] -> 2} // qExpand</nowiki> | |
||
out= <nowiki> 11 2 3 |
out= <nowiki> 11 2 3 |
||
Line 168: | Line 168: | ||
<!--Robot Land, no human edits to "END"--> |
<!--Robot Land, no human edits to "END"--> |
||
{{HelpLine| |
{{HelpLine| |
||
n = |
n = 14 | |
||
in = <nowiki>ColoredJones</nowiki> | |
in = <nowiki>ColoredJones</nowiki> | |
||
out= <nowiki>Type ColoredJones and see for yourself.</nowiki>}} |
out= <nowiki>Type ColoredJones and see for yourself.</nowiki>}} |
Revision as of 13:01, 14 September 2005
KnotTheory`
can compute the coloured Jones polynomial of knots and links, using the formulas in [Garoufalidis Le]:
(For In[1] see Setup)
|
|
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
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 ; 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 |
|
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:
|
|
More precisely, qPochhammer[a, q, k]
is
and qBinomial[n, k, q]
is
The function qExpand
replaces every occurence of a qPochhammer[a, q, k]
symbol or a qBinomial[n, k, q]
symbol by its definition:
|
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,
|
[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.