In the mathematical field of knot theory, the HOMFLY polynomial, sometimes called the HOMFLYPT polynomial or the generalized Jones polynomial, is a 2variable knot polynomial, i.e. a knot invariant in the form of a polynomial of variables m and l.
A central question in the mathematical theory of knots is whether two knot diagrams 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 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 codiscoverers: Jim Hoste, Adrian Ocneanu, Kenneth Millett, Peter J. Freyd, W. B. R. Lickorish, and David N. Yetter.^{[1]} The addition of PT recognizes independent work carried out by Józef H. Przytycki and Paweł Traczyk.
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Introduction to the Alexander Polynomial

Introduction to the Jones Polynomial

AlgTop22: Knots and surfaces I
Transcription
Contents
Definition
The polynomial is defined using skein relations:
where are links formed by crossing and smoothing changes on a local region of a link diagram, as indicated in the figure.
The HOMFLY polynomial of a link L that is a split union of two links and is given by
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:
Main properties
 , where # denotes the knot sum; thus the HOMFLY polynomial of a composite knot is the product of the HOMFLY polynomials of its components.
 , 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_{42} and 10_{71}^{[2]}
The Jones polynomial, V(t), and the Alexander polynomial, can be computed in terms of the HOMFLY polynomial (the version in and variables) as follows:
References
 ^ Freyd, P., Yetter, D., Hoste, J., Lickorish, W.B.R., Millett, K., and Ocneanu, A. (1985). "A New Polynomial Invariant of Knots and Links". Bulletin of the American Mathematical Society. 12 (2): 239–246. doi:10.1090/S027309791985153613.
 ^ https://arxiv.org/pdf/hepth/9401095.pdf
Further reading
 Kauffman, L.H., "Formal knot theory", Princeton University Press, 1983.
 Lickorish, W.B.R. "An Introduction to Knot Theory". Springer. ISBN 038798254X.
External links
 Hazewinkel, Michiel, ed. (2001) [1994], "JonesConway polynomial", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 Weisstein, Eric W. "HOMFLY Polynomial". MathWorld.
 "The HOMFLYPT Polynomial", The Knot Atlas.