The HOMFLY-PT Polynomial: Difference between revisions
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{{Manual TOC Sidebar}} |
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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 |
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Line 36: | Line 36: | ||
in = <nowiki>HOMFLYPT[Knot[8, 1]][a, z]</nowiki> | |
in = <nowiki>HOMFLYPT[Knot[8, 1]][a, z]</nowiki> | |
||
out= <nowiki> -2 4 6 2 2 2 4 2 |
out= <nowiki> -2 4 6 2 2 2 4 2 |
||
a - a |
a - a a - z - a z - a z</nowiki>}} |
||
<!--END--> |
<!--END--> |
||
Line 47: | Line 47: | ||
in = <nowiki>Expand[HOMFLYPT[K][1/q, Sqrt[q]-1/Sqrt[q]]]</nowiki> | |
in = <nowiki>Expand[HOMFLYPT[K][1/q, Sqrt[q]-1/Sqrt[q]]]</nowiki> | |
||
out= <nowiki> -6 -5 -4 2 2 2 2 |
out= <nowiki> -6 -5 -4 2 2 2 2 |
||
2 |
2 q - q q - -- -- - - - q q |
||
3 2 q |
3 2 q |
||
q q</nowiki>}} |
q q</nowiki>}} |
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Line 58: | Line 58: | ||
in = <nowiki>Jones[K][q]</nowiki> | |
in = <nowiki>Jones[K][q]</nowiki> | |
||
out= <nowiki> -6 -5 -4 2 2 2 2 |
out= <nowiki> -6 -5 -4 2 2 2 2 |
||
2 |
2 q - q q - -- -- - - - q q |
||
3 2 q |
3 2 q |
||
q q</nowiki>}} |
q q</nowiki>}} |
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Line 76: | Line 76: | ||
In our parametrization of the <math>A_2</math> link invariant, it satisfies |
In our parametrization of the <math>A_2</math> link invariant, it satisfies |
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<center><math>A_2(L)(q) = (-1)^c(q^2 |
<center><math>A_2(L)(q) = (-1)^c(q^2 1 q^{-2})H(L)(q^{-3},\,q-q^{-1})</math>,</center> |
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where <math>L</math> is some knot or link and where <math>c</math> is the number of components of <math>L</math>. Let us verify this fact for the Whitehead link, [[L5a1]]: |
where <math>L</math> is some knot or link and where <math>c</math> is the number of components of <math>L</math>. Let us verify this fact for the Whitehead link, [[L5a1]]: |
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Line 88: | Line 88: | ||
<!--$$Simplify[{ |
<!--$$Simplify[{ |
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(-1)^(Length[Skeleton[L]]-1)(q^2 |
(-1)^(Length[Skeleton[L]]-1)(q^2 1 1/q^2)HOMFLYPT[L][1/q^3, q-1/q], |
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A2Invariant[L][q] |
A2Invariant[L][q] |
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}]$$--> |
}]$$--> |
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Line 95: | Line 95: | ||
n = 9 | |
n = 9 | |
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in = <nowiki>Simplify[{ |
in = <nowiki>Simplify[{ |
||
(-1)^(Length[Skeleton[L]]-1)(q^2 |
(-1)^(Length[Skeleton[L]]-1)(q^2 1 1/q^2)HOMFLYPT[L][1/q^3, q-1/q], |
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A2Invariant[L][q] |
A2Invariant[L][q] |
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}]</nowiki> | |
}]</nowiki> | |
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out= <nowiki> -12 -8 -6 2 -2 2 4 6 |
out= <nowiki> -12 -8 -6 2 -2 2 4 6 |
||
{2 - q |
{2 - q q q -- q q q q , |
||
4 |
4 |
||
q |
q |
||
-12 -8 -6 2 -2 2 4 6 |
-12 -8 -6 2 -2 2 4 6 |
||
2 - q |
2 - q q q -- q q q q } |
||
4 |
4 |
||
q</nowiki>}} |
q</nowiki>}} |
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The HOMFLY-PT polynomial (see [HOMFLY] and [PT]) of a knot or link is defined by the skein relation
and by the initial condition =1.
KnotTheory`
knows about the HOMFLY-PT polynomial:
(For In[1] see Setup)
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Thus, for example, here's the HOMFLY-PT polynomial of the knot 8_1:
In[3]:=
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K = Knot[8, 1];
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In[4]:=
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HOMFLYPT[Knot[8, 1]][a, z]
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Out[4]=
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-2 4 6 2 2 2 4 2
a - a a - z - a z - a z
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It is well known that HOMFLY-PT polynomial specializes to the Jones polynomial at and and to the Conway polynomial at . Indeed,
In[5]:=
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Expand[HOMFLYPT[K][1/q, Sqrt[q]-1/Sqrt[q]]]
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Out[5]=
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-6 -5 -4 2 2 2 2
2 q - q q - -- -- - - - q q
3 2 q
q q
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In[6]:=
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Jones[K][q]
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Out[6]=
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-6 -5 -4 2 2 2 2
2 q - q q - -- -- - - - q q
3 2 q
q q
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In[7]:=
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{HOMFLYPT[K][1, z], Conway[K][z]}
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Out[7]=
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2 2
{1 - 3 z , 1 - 3 z }
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8_1 |
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[8]:=
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L = Link[5, Alternating, 1];
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In[9]:=
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Simplify[{
(-1)^(Length[Skeleton[L]]-1)(q^2 1 1/q^2)HOMFLYPT[L][1/q^3, q-1/q],
A2Invariant[L][q]
}]
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Out[9]=
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-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
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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, , Proc. Amer. Math. Soc. 100 (1987) 744-748.