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{{Use American English|date=January 2019}}{{Short description|Polynomials arising in knot theory
}}
In the [[mathematics|mathematical]] field of [[knot theory]], the '''HOMFLY polynomial''', sometimes called the '''HOMFLY-PT''' polynomial or the generalized [[Jones polynomial]], is a 2-variable [[knot polynomial]], i.e. a [[knot invariant]] in the form of a [[polynomial]] of variables ''m'' and ''l''.   

A central question in the [[knot theory|mathematical theory of knots]] is whether two [[knot diagram]]s represent the same knot. One tool used to answer such questions is a knot polynomial, which is computed from a diagram of the knot and can be shown to be an [[knot invariant|invariant of the knot]], i.e. diagrams representing the same knot have the same [[polynomial]]. The converse may not be true. The HOMFLY polynomial is one such invariant and it generalizes two polynomials previously discovered, the [[Alexander polynomial]] and the [[Jones polynomial]], both of which can be obtained by appropriate substitutions from HOMFLY. The HOMFLY polynomial is also a [[quantum invariant]].

The name  ''HOMFLY'' combines the initials of its co-discoverers: [[Jim Hoste]], [[Adrian Ocneanu]], [[Kenneth Millett]], [[Peter J. Freyd]], [[W. B. R. Lickorish]], and David N. Yetter.<ref>{{cite journal|authors= Freyd, P., Yetter, D., Hoste, J., Lickorish, W.B.R., Millett, K., and Ocneanu, A.|title = A New Polynomial Invariant of Knots and Links|journal = Bulletin of the American Mathematical Society|volume = 12|issue = 2|year = 1985|pages = 239–246|doi = 10.1090/S0273-0979-1985-15361-3}}</ref> The addition of ''PT'' recognizes independent work  carried out by [[Józef H. Przytycki]] and Paweł Traczyk.

==Definition==
The polynomial is defined using [[skein relation]]s:

: <math>P( \mathrm{unknot} ) = 1,\,</math>

: <math>\ell P(L_+) + \ell^{-1}P(L_-) + mP(L_0)=0,\,</math>

where <math>L_+, L_-, L_0</math> are links formed by crossing and smoothing changes on a local region of a link diagram, as indicated in the figure.  [[Image:Skein (HOMFLY).svg|200px|center]]

The HOMFLY polynomial of a link ''L'' that is a split union of two links <math>L_1</math> and <math>L_2</math> is given by

: <math>P(L) = \frac{-(\ell+\ell^{-1})}{m} P(L_1)P(L_2).</math>

See the page on [[skein relation]] for an example of a computation using such relations.

==Other HOMFLY skein relations==
This polynomial can be obtained also using other skein relations:
: <math>\alpha P(L_+) - \alpha^{-1}P(L_-) = zP(L_0),\,</math>
: <math>xP(L_+) + yP(L_-) + zP(L_0)=0,\,</math>

==Main properties==
: <math>P(L_1 \# L_2)=P(L_1)P(L_2),\,</math>, where # denotes the [[knot sum]]; thus the HOMFLY polynomial of a [[composite knot]] is the product of the HOMFLY polynomials of its components.

: <math>P_K(\ell,m)=P_{\text{Mirror Image}(K)}(\ell^{-1},m),\,</math>, so the HOMFLY polynomial can often be used to distinguish between two knots of different [[chirality]]. However there exist chiral pairs of knots that have the same HOMFLY polynomial, e.g. knots 9<sub>42</sub> and 10<sub>71</sub><ref>{{Cite journal |arxiv = hep-th/9401095|doi = 10.1142/S0217732394003026|title = Chirality of Knots 942 and 1071 and Chern-Simons Theory|journal = Modern Physics Letters A|volume = 09|issue = 34|pages = 3205–3217|year = 1994|last1 = Ramadevi|first1 = P.|last2 = Govindarajan|first2 = T.R.|last3 = Kaul|first3 = R.K.}}</ref>

The Jones polynomial, ''V''(''t''), and the Alexander polynomial, <math>\Delta(t)\,</math> can be computed in terms of the HOMFLY polynomial (the version in <math>\alpha</math> and <math>z</math> variables) as follows:
: <math>V(t)=P(\alpha=t^{-1},z=t^{1/2}-t^{-1/2}),\,</math>

: <math>\Delta(t)=P(\alpha=1,z=t^{1/2}-t^{-1/2}),\,</math>

==References==
{{reflist}}

==Further reading==
* [[Louis Kauffman|Kauffman, L.H.]], "Formal knot theory", Princeton University Press, 1983.
* [[W. B. R. Lickorish|Lickorish, W.B.R.]] "An Introduction to Knot Theory". Springer. {{ISBN|0-387-98254-X}}.

==External links==
* {{springer|title=Jones-Conway polynomial|id=p/j130040}}
* {{MathWorld|HOMFLYPolynomial|HOMFLY Polynomial}}
* {{Knot Atlas|The HOMFLY-PT Polynomial}}

{{Knot theory}}

{{DEFAULTSORT:Homfly Polynomial}}
[[Category:Knot theory]]
[[Category:Polynomials]]