The HOMFLY-PT Polynomial: Difference between revisions
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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 |
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<center><math> |
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aH\left(\{overcrossing\}\right) |
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-a^{-1}H\left(\{undercrossing\}\right) |
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= zH\left(\{smoothing\}\right) |
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</math></center> |
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and by the initial condition <math>H(\{bigcirc\})</math>=1. |
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<code>KnotTheory`</code> knows about the HOMFLY-PT polynomial: |
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{{Startup Note}} |
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<!--$$?HOMFLYPT$$--> |
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<!--END--> |
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Thus, for example, here's the HOMFLY-PT polynomial of the knot [[8_1]]: |
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<!--$$K = Knot[8, 1];$$--> |
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<!--END--> |
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<!--$$HOMFLYPT[Knot[8, 1]][a, z]$$--> |
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<!--END--> |
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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, |
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<!--$${Expand[HOMFLYPT[K][1/q, Sqrt[q]-1/Sqrt[q]]], Jones[K][q]}$$--> |
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<!--END--> |
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<!--$${HOMFLYPT[K][1, z], Conway[K][z]}$$--> |
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<!--END--> |
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In our parametirzation of the <math>A_2</math> link invariant, it satisfies |
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<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]]: |
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<!--$$L = Link[5, Alternating, 1];$$--> |
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<!--END--> |
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<!--$$Simplify[{\n |
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(-1)^(Length[Skeleton[L]]-1)(q^2+1+1/q^2)HOMFLYPT[L][1/q^3, q-1/q],\n |
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A2Invariant[L][q]\n |
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}]$$--> |
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<!--END--> |
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{{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. |
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{{note|PT}} J. Przytycki and P. Traczyk, <math>Conway Algebras and Skein Equivalence of Links</math>, Proc. Amer. Math. Soc. '''100''' (1987) 744-748. |
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Revision as of 05:56, 27 August 2005
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)
Thus, for example, here's the HOMFLY-PT polynomial of the knot 8_1:
It is well known that HOMFLY-PT polynomial specializes to the Jones polynomial at and and to the Conway polynomial at . Indeed,
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:
[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.
