The HOMFLY-PT Polynomial: Difference between revisions

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{{Manual TOC Sidebar}}
{{Manual TOC Sidebar}}


The ''HOMFLY-PT polynomial'' <math>H(L)(a,z)</math> (see {{ref|HOMFLY}} and {{ref|PT}} of a knot or link <math>L</math> is defined by the skein relation
The ''HOMFLY-PT polynomial'' <math>H(L)(a,z)</math> (see {{ref|HOMFLY}} and {{ref|PT}}) of a knot or link <math>L</math> is defined by the skein relation


<center><math>
<center><math>
aH\left(\{overcrossing\}\right)
aH\left(\overcrossing\right)-a^{-1}H\left(\undercrossing\right)=zH\left(\smoothing\right)
-a^{-1}H\left(\{undercrossing\}\right)
= zH\left(\{smoothing\}\right)
</math></center>
</math></center>


and by the initial condition <math>H(\{bigcirc\})</math>=1.
and by the initial condition <math>H(\bigcirc)</math>=1.


<code>KnotTheory`</code> knows about the HOMFLY-PT polynomial:
<code>KnotTheory`</code> knows about the HOMFLY-PT polynomial:
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{{Startup Note}}
{{Startup Note}}
<!--$$?HOMFLYPT$$-->
<!--$$?HOMFLYPT$$-->
<!--Robot Land, no human edits to "END"-->
<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
{{HelpAndAbout|
{{HelpAndAbout1|n=2|s=HOMFLYPT}}
n = 2 |
⚫
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 = 3 |
{{HelpAndAbout2|n=3|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>}}
<!--END-->
<!--END-->


Thus, for example, here's the HOMFLY-PT polynomial of the knot [[8_1]]:
Thus, for example, here's the HOMFLY-PT polynomial of the knot [[8_1]]:
<!--$$K = Knot[8, 1];$$-->
<!--$$K = Knot[8, 1];$$-->
<!--Robot Land, no human edits to "END"-->
<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
{{In1|n=4}}
{{In|
n = 4 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>K = Knot[8, 1];</nowiki></pre>
in = <nowiki>K = Knot[8, 1];</nowiki>}}
{{In2}}
<!--END-->
<!--END-->


<!--$$HOMFLYPT[Knot[8, 1]][a, z]$$-->
<!--$$HOMFLYPT[Knot[8, 1]][a, z]$$-->
<!--Robot Land, no human edits to "END"-->
<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
{{InOut1|n=5}}
{{InOut|
n = 5 |
<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=5}}<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}}
<!--END-->
<!--END-->


It is well known that HOMFLY-PT polynomial specializes to the Jones polynomial at <math>a=q^{-1}</math> and <math>z=q^{1/2}-q^{-1/2}</math> and to the Conway polynomial at <math>a=1</math>. Indeed,
It is well known that HOMFLY-PT polynomial specializes to the Jones polynomial at <math>a=q^{-1}</math> and <math>z=q^{1/2}-q^{-1/2}</math> and to the Conway polynomial at <math>a=1</math>. Indeed,


<!--$${Expand[HOMFLYPT[K][1/q, Sqrt[q]-1/Sqrt[q]]], Jones[K][q]}$$-->
<!--$$Expand[HOMFLYPT[K][1/q, Sqrt[q]-1/Sqrt[q]]]$$-->
<!--Robot Land, no human edits to "END"-->
<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
{{InOut1|n=6}}
{{InOut|
n = 6 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>{Expand[HOMFLYPT[K][1/q, Sqrt[q]-1/Sqrt[q]]], Jones[K][q]}</nowiki></pre>
in = <nowiki>Expand[HOMFLYPT[K][1/q, Sqrt[q]-1/Sqrt[q]]]</nowiki> |
{{InOut2|n=6}}<pre style="border: 0px; padding: 0em"><nowiki> -6 -5 -4 2 2 2 2 -6 -5 -4 2 2 2 2
{2 + q - q + q - -- + -- - - - q + q , 2 + q - q + q - -- + -- - - - q + q }
out= <nowiki> -6 -5 -4 2 2 2 2
3 2 q 3 2 q
2 + q - q + q - -- + -- - - - q + q
q q q q</nowiki></pre>
3 2 q
q q</nowiki>}}
{{InOut3}}
<!--END-->

<!--$$Jones[K][q]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 7 |
in = <nowiki>Jones[K][q]</nowiki> |
out= <nowiki> -6 -5 -4 2 2 2 2
2 + q - q + q - -- + -- - - - q + q
3 2 q
q q</nowiki>}}
<!--END-->
<!--END-->


<!--$${HOMFLYPT[K][1, z], Conway[K][z]}$$-->
<!--$${HOMFLYPT[K][1, z], Conway[K][z]}$$-->
<!--Robot Land, no human edits to "END"-->
<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
{{InOut1|n=7}}
{{InOut|
n = 8 |
<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}}
<!--END-->
<!--END-->


{{Knot Image Pair|8_1|gif|L5a1|gif}}
⚫
In our parametirzation of the <math>A_2</math> link invariant, it satisfies

⚫
In our parametrization of the <math>A_2</math> link invariant, it satisfies


<center><math>A_2(L)(q) = (-1)^c(q^2+1+q^{-2})H(L)(q^{-3},\,q-q^{-1})</math>,</center>
<center><math>A_2(L)(q) = (-1)^c(q^2+1+q^{-2})H(L)(q^{-3},\,q-q^{-1})</math>,</center>
Line 69: Line 81:


<!--$$L = Link[5, Alternating, 1];$$-->
<!--$$L = Link[5, Alternating, 1];$$-->
<!--Robot Land, no human edits to "END"-->
<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
{{In1|n=8}}
{{In|
n = 9 |
<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}}
<!--END-->
<!--END-->


<!--$$Simplify[{\n
<!--$$Simplify[{
(-1)^(Length[Skeleton[L]]-1)(q^2+1+1/q^2)HOMFLYPT[L][1/q^3, q-1/q],\n
(-1)^(Length[Skeleton[L]]-1)(q^2+1+1/q^2)HOMFLYPT[L][1/q^3, q-1/q],
A2Invariant[L][q]\n
A2Invariant[L][q]
}]$$-->
}]$$-->
<!--Robot Land, no human edits to "END"-->
<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
{{InOut1|n=9}}
{{InOut|
n = 10 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>Simplify[{\n
in = <nowiki>Simplify[{
(-1)^(Length[Skeleton[L]]-1)(q^2+1+1/q^2)HOMFLYPT[L][1/q^3, q-1/q],\n
(-1)^(Length[Skeleton[L]]-1)(q^2+1+1/q^2)HOMFLYPT[L][1/q^3, q-1/q],
A2Invariant[L][q]\n
A2Invariant[L][q]
}]</nowiki></pre>
}]</nowiki> |
{{InOut2|n=9}}<pre style="border: 0px; padding: 0em"><nowiki>$Failed</nowiki></pre>
out= <nowiki> -12 -8 -6 2 -2 2 4 6
{{InOut3}}
{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</nowiki>}}
<!--END-->
<!--END-->

====Other Software to Compute the HOMFLY-PT Polynomial====
A C-based program running under windows by M. Ochiai can compute the HOMFLY-PT polynomial of certain knots and links with up to hundreds of crossings using "base tangle decompositions". His program, bTd, is available at [http://amadeus.ics.nara-wu.ac.jp/~ochiai/freesoft.html].

====References====


{{note|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.
{{note|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.


{{note|PT}} J. Przytycki and P. Traczyk, <math>Conway Algebras and Skein Equivalence of Links</math>, Proc. Amer. Math. Soc. '''100''' (1987) 744-748.
{{note|PT}} J. Przytycki and P. Traczyk, ''Conway Algebras and Skein Equivalence of Links'', Proc. Amer. Math. Soc. '''100''' (1987) 744-748.

Latest revision as of 20:56, 8 August 2013


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[2]:= ?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[3]:= 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[4]:= K = Knot[8, 1];
In[5]:= HOMFLYPT[Knot[8, 1]][a, z]
Out[5]= -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[6]:= Expand[HOMFLYPT[K][1/q, Sqrt[q]-1/Sqrt[q]]]
Out[6]= -6 -5 -4 2 2 2 2 2 + q - q + q - -- + -- - - - q + q 3 2 q q q
In[7]:= Jones[K][q]
Out[7]= -6 -5 -4 2 2 2 2 2 + q - q + q - -- + -- - - - q + q 3 2 q q q
In[8]:= {HOMFLYPT[K][1, z], Conway[K][z]}
Out[8]= 2 2 {1 - 3 z , 1 - 3 z }
8 1.gif
8_1 		L5a1.gif
L5a1

In our parametrization 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[9]:= L = Link[5, Alternating, 1];
In[10]:= Simplify[{ (-1)^(Length[Skeleton[L]]-1)(q^2+1+1/q^2)HOMFLYPT[L][1/q^3, q-1/q], A2Invariant[L][q] }]
Out[10]= -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

Other Software to Compute the HOMFLY-PT Polynomial

A C-based program running under windows by M. Ochiai can compute the HOMFLY-PT polynomial of certain knots and links with up to hundreds of crossings using "base tangle decompositions". His program, bTd, is available at [1].

References

[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, Conway Algebras and Skein Equivalence of Links, Proc. Amer. Math. Soc. 100 (1987) 744-748.

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