The A2 Invariant: Difference between revisions
From Knot Atlas
Jump to navigationJump to search
No edit summary |
No edit summary |
||
| (9 intermediate revisions by 5 users not shown) | |||
| Line 6: | Line 6: | ||
<!--$$?A2Invariant$$--> |
<!--$$?A2Invariant$$--> |
||
<!--Robot Land, no human edits to "END"--> |
|||
{{HelpLine| |
|||
n = 2 | |
|||
in = <nowiki>A2Invariant</nowiki> | |
|||
out= <nowiki>A2Invariant[L][q] computes the A2 (sl(3)) invariant of a knot or link L as a function of the variable q.</nowiki>}} |
|||
<!--END--> |
<!--END--> |
||
{| align=center |
{| align=center |
||
|[[Image: |
|[[Image:10_22.gif|thumb|180px|<center>[[10_22]]</center>]] |
||
|[[Image: |
|[[Image:10_35.gif|thumb|none|<center>[[10_35]]</center>|180px]] |
||
|} |
|} |
||
| Line 16: | Line 21: | ||
<!--$$Jones[Knot[10, 22]][q] == Jones[Knot[10, 35]][q]$$--> |
<!--$$Jones[Knot[10, 22]][q] == Jones[Knot[10, 35]][q]$$--> |
||
<!--Robot Land, no human edits to "END"--> |
|||
{{InOut| |
|||
n = 3 | |
|||
in = <nowiki>Jones[Knot[10, 22]][q] == Jones[Knot[10, 35]][q]</nowiki> | |
|||
out= <nowiki>True</nowiki>}} |
|||
<!--END--> |
<!--END--> |
||
<!--$$A2Invariant[Knot[10, 22]][q]$$--> |
<!--$$A2Invariant[Knot[10, 22]][q]$$--> |
||
<!--Robot Land, no human edits to "END"--> |
|||
{{InOut| |
|||
n = 4 | |
|||
in = <nowiki>A2Invariant[Knot[10, 22]][q]</nowiki> | |
|||
out= <nowiki> -12 -8 -6 -4 2 4 6 8 10 12 14 |
|||
-1 + q + q + q - q + -- - q - 2 q + q - q + q + q + |
|||
2 |
|||
q |
|||
18 |
|||
q</nowiki>}} |
|||
<!--END--> |
<!--END--> |
||
<!--$$A2Invariant[Knot[10, 35]][q]$$--> |
<!--$$A2Invariant[Knot[10, 35]][q]$$--> |
||
<!--Robot Land, no human edits to "END"--> |
|||
{{InOut| |
|||
n = 5 | |
|||
in = <nowiki>A2Invariant[Knot[10, 35]][q]</nowiki> | |
|||
out= <nowiki> -14 -12 -10 -8 2 2 2 6 8 10 14 16 |
|||
q + q - q + q - -- + -- + q - q + q - 2 q + q - q + |
|||
4 2 |
|||
q q |
|||
18 20 |
|||
q + q</nowiki>}} |
|||
<!--END--> |
<!--END--> |
||
The <math>A2</math> invariant attains <!--$all=Join[AllKnots[], AllLinks[]]; Length[Union[A2Invariant[#][q]& /@ all]]$--><!--END--> values on the <!--$Length[all]$--><!--END--> knots and links known to <code>KnotTheory</code>: |
The <math>A2</math> invariant attains <!--$all=Join[AllKnots[], AllLinks[]]; Length[Union[A2Invariant[#][q]& /@ all]]$--><!--Robot Land, no human edits to "END"-->2163<!--END--> values on the <!--$Length[all]$--><!--Robot Land, no human edits to "END"-->2226<!--END--> knots and links known to <code>KnotTheory</code>: |
||
<!--$$all = Join[AllKnots[], AllLinks[]];$$--> |
<!--$$all = Join[AllKnots[], AllLinks[]];$$--> |
||
<!--Robot Land, no human edits to "END"--> |
|||
{{In| |
|||
n = 6 | |
|||
in = <nowiki>all = Join[AllKnots[], AllLinks[]];</nowiki>}} |
|||
<!--END--> |
<!--END--> |
||
<!--$$Length /@ {Union[A2Invariant[#][q]& /@ all], all}$$--> |
<!--$$Length /@ {Union[A2Invariant[#][q]& /@ all], all}$$--> |
||
<!--Robot Land, no human edits to "END"--> |
|||
{{InOut| |
|||
n = 7 | |
|||
in = <nowiki>Length /@ {Union[A2Invariant[#][q]& /@ all], all}</nowiki> | |
|||
out= <nowiki>{2163, 2226}</nowiki>}} |
|||
<!--END--> |
<!--END--> |
||
Latest revision as of 17:22, 21 February 2013
We compute the (or quantum ) invariant using the normalization and formulas of [Khovanov], which in itself follows [Kuperberg]:
(For In[1] see Setup)
| ||||
As an example, let us check that the knots 10_22 and 10_35 have the same Jones polynomial but different invariants:
In[3]:=
|
Jones[Knot[10, 22]][q] == Jones[Knot[10, 35]][q]
|
Out[3]=
|
True
|
In[4]:=
|
A2Invariant[Knot[10, 22]][q]
|
Out[4]=
|
-12 -8 -6 -4 2 4 6 8 10 12 14
-1 + q + q + q - q + -- - q - 2 q + q - q + q + q +
2
q
18
q
|
In[5]:=
|
A2Invariant[Knot[10, 35]][q]
|
Out[5]=
|
-14 -12 -10 -8 2 2 2 6 8 10 14 16
q + q - q + q - -- + -- + q - q + q - 2 q + q - q +
4 2
q q
18 20
q + q
|
The invariant attains 2163 values on the 2226 knots and links known to KnotTheory:
In[6]:=
|
all = Join[AllKnots[], AllLinks[]];
|
In[7]:=
|
Length /@ {Union[A2Invariant[#][q]& /@ all], all}
|
Out[7]=
|
{2163, 2226}
|
[Khovanov] ^ M. Khovanov, link homology I, arXiv:math.QA/0304375.
[Kuperberg] ^ G. Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996) 109-151, arXiv:q-alg/9712003.
