The HOMFLY-PT Polynomial: Difference between revisions

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<!--$$?HOMFLYPT$$-->
<!--$$?HOMFLYPT$$-->
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{{HelpAndAbout|
{{HelpAndAbout1|n=1|s=HOMFLYPT}}
n = 1 |
⚫
HOMFLYPT[K][a, z] computes the HOMFLY-PT (Hoste, Ocneanu, Millett, Freyd, Lickorish, Yetter, Przytycki and Traczyk) polynomial of a knot/link K, in the variables a and z.
n1 = 2 |
{{HelpAndAbout2|n=2|s=HOMFLYPT}}
in = <nowiki>HOMFLYPT</nowiki> |
⚫
The HOMFLYPT program was written by Scott Morrison.
⚫
out= <nowiki>HOMFLYPT[K][a, z] computes the HOMFLY-PT (Hoste, Ocneanu, Millett, Freyd, Lickorish, Yetter, Przytycki and Traczyk) polynomial of a knot/link K, in the variables a and z.</nowiki> |
{{HelpAndAbout3}}
⚫
about= <nowiki>The HOMFLYPT program was written by Scott Morrison.</nowiki>}}
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<!--$$K = Knot[8, 1];$$-->
<!--$$K = Knot[8, 1];$$-->
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{{In1|n=3}}
{{In|
n = 3 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>K = Knot[8, 1];</nowiki></pre>
in = <nowiki>K = Knot[8, 1];</nowiki>}}
{{In2}}
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<!--$$HOMFLYPT[Knot[8, 1]][a, z]$$-->
<!--$$HOMFLYPT[Knot[8, 1]][a, z]$$-->
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{{InOut1|n=4}}
{{InOut|
n = 4 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>HOMFLYPT[Knot[8, 1]][a, z]</nowiki></pre>
in = <nowiki>HOMFLYPT[Knot[8, 1]][a, z]</nowiki> |
{{InOut2|n=4}}<pre style="border: 0px; padding: 0em"><nowiki> -2 4 6 2 2 2 4 2
a - a + a - z - a z - a z</nowiki></pre>
out= <nowiki> -2 4 6 2 2 2 4 2
a - a + a - z - a z - a z</nowiki>}}
{{InOut3}}
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<!--$$Expand[HOMFLYPT[K][1/q, Sqrt[q]-1/Sqrt[q]]]$$-->
<!--$$Expand[HOMFLYPT[K][1/q, Sqrt[q]-1/Sqrt[q]]]$$-->
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{{InOut1|n=5}}
{{InOut|
n = 5 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>Expand[HOMFLYPT[K][1/q, Sqrt[q]-1/Sqrt[q]]]</nowiki></pre>
in = <nowiki>Expand[HOMFLYPT[K][1/q, Sqrt[q]-1/Sqrt[q]]]</nowiki> |
{{InOut2|n=5}}<pre style="border: 0px; padding: 0em"><nowiki> -6 -5 -4 2 2 2 2
out= <nowiki> -6 -5 -4 2 2 2 2
2 + q - q + q - -- + -- - - - q + q
2 + q - q + q - -- + -- - - - q + q
3 2 q
3 2 q
q q</nowiki></pre>
q q</nowiki>}}
{{InOut3}}
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<!--$$Jones[K][q]$$-->
<!--$$Jones[K][q]$$-->
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{{InOut1|n=6}}
{{InOut|
n = 6 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>Jones[K][q]</nowiki></pre>
in = <nowiki>Jones[K][q]</nowiki> |
{{InOut2|n=6}}<pre style="border: 0px; padding: 0em"><nowiki> -6 -5 -4 2 2 2 2
out= <nowiki> -6 -5 -4 2 2 2 2
2 + q - q + q - -- + -- - - - q + q
2 + q - q + q - -- + -- - - - q + q
3 2 q
3 2 q
q q</nowiki></pre>
q q</nowiki>}}
{{InOut3}}
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<!--$${HOMFLYPT[K][1, z], Conway[K][z]}$$-->
<!--$${HOMFLYPT[K][1, z], Conway[K][z]}$$-->
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{{InOut1|n=7}}
{{InOut|
n = 7 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>{HOMFLYPT[K][1, z], Conway[K][z]}</nowiki></pre>
in = <nowiki>{HOMFLYPT[K][1, z], Conway[K][z]}</nowiki> |
{{InOut2|n=7}}<pre style="border: 0px; padding: 0em"><nowiki> 2 2
{1 - 3 z , 1 - 3 z }</nowiki></pre>
out= <nowiki> 2 2
{1 - 3 z , 1 - 3 z }</nowiki>}}
{{InOut3}}
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<!--$$L = Link[5, Alternating, 1];$$-->
<!--$$L = Link[5, Alternating, 1];$$-->
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{{In1|n=8}}
{{In|
n = 8 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>L = Link[5, Alternating, 1];</nowiki></pre>
in = <nowiki>L = Link[5, Alternating, 1];</nowiki>}}
{{In2}}
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}]$$-->
}]$$-->
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{{InOut1|n=9}}
{{InOut|
n = 9 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>Simplify[{
in = <nowiki>Simplify[{
(-1)^(Length[Skeleton[L]]-1)(q^2+1+1/q^2)HOMFLYPT[L][1/q^3, q-1/q],
(-1)^(Length[Skeleton[L]]-1)(q^2+1+1/q^2)HOMFLYPT[L][1/q^3, q-1/q],
A2Invariant[L][q]
A2Invariant[L][q]
}]</nowiki></pre>
}]</nowiki> |
{{InOut2|n=9}}<pre style="border: 0px; padding: 0em"><nowiki> -12 -8 -6 2 -2 2 4 6 -12 -8 -6 2 -2 2 4 6
out= <nowiki> -12 -8 -6 2 -2 2 4 6
{2 - q + q + q + -- + q + q + q + q , 2 - q + q + q + -- + q + q + q + q }
{2 - q + q + q + -- + q + q + q + q ,
4 4
4
q q</nowiki></pre>
q
{{InOut3}}
-12 -8 -6 2 -2 2 4 6
2 - q + q + q + -- + q + q + q + q }
4
q</nowiki>}}
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Revision as of 13:10, 30 August 2005


KnotTheory`/Navigation 

The Mathematica Package KnotTheory`

The HOMFLY-PT polynomial (see [HOMFLY] and [PT]) of a knot or link is defined by the skein relation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle aH\left(\{\overcrossing\}\right) -a^{-1}H\left(\{\undercrossing\}\right) = zH\left(\{\smoothing\}\right) }

and by the initial condition =1.

KnotTheory` knows about the HOMFLY-PT polynomial:

(For In[1] see Setup)

In[1]:= ?HOMFLYPT
HOMFLYPT[K][a, z] computes the HOMFLY-PT (Hoste, Ocneanu, Millett, Freyd, Lickorish, Yetter, Przytycki and Traczyk) polynomial of a knot/link K, in the variables a and z.
In[2]:= HOMFLYPT::about
The HOMFLYPT program was written by Scott Morrison.

Thus, for example, here's the HOMFLY-PT polynomial of the knot 8_1:

In[3]:= K = Knot[8, 1];
In[4]:= HOMFLYPT[Knot[8, 1]][a, z]
Out[4]= -2 4 6 2 2 2 4 2 a - a + a - z - a z - a z

It is well known that HOMFLY-PT polynomial specializes to the Jones polynomial at and and to the Conway polynomial at . Indeed,

In[5]:= Expand[HOMFLYPT[K][1/q, Sqrt[q]-1/Sqrt[q]]]
Out[5]= -6 -5 -4 2 2 2 2 2 + q - q + q - -- + -- - - - q + q 3 2 q q q
In[6]:= Jones[K][q]
Out[6]= -6 -5 -4 2 2 2 2 2 + q - q + q - -- + -- - - - q + q 3 2 q q q
In[7]:= {HOMFLYPT[K][1, z], Conway[K][z]}
Out[7]= 2 2 {1 - 3 z , 1 - 3 z }

In our parametirzation of the link invariant, it satisfies

,

where is some knot or link and where is the number of components of . Let us verify this fact for the Whitehead link, L5a1:

In[8]:= L = Link[5, Alternating, 1];
In[9]:= Simplify[{ (-1)^(Length[Skeleton[L]]-1)(q^2+1+1/q^2)HOMFLYPT[L][1/q^3, q-1/q], A2Invariant[L][q] }]
Out[9]= -12 -8 -6 2 -2 2 4 6 {2 - q + q + q + -- + q + q + q + q , 4 q -12 -8 -6 2 -2 2 4 6 2 - q + q + q + -- + q + q + q + q } 4 q

[HOMFLY] ^  J. Hoste, A. Ocneanu, K. Millett, P. Freyd, W. B. R. Lickorish and D. Yetter, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 12 (1985) 239-246.

[PT] ^  J. Przytycki and P. Traczyk, , Proc. Amer. Math. Soc. 100 (1987) 744-748.

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