HOMFLY polynomial
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In the 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. It generalizes both the Alexander polynomial and the Jones polynomial both of which can be obtained by appropriate substitutions from HOMFLY. The HOMFLY polynomial is a quantum invariant.[1]
The name HOMFLY combines the initials of its co-discoverers: Hoste, Ocneanu, Millett, Freyd, Lickorish, and Yetter[2]. The addition of PT recognizes independent work carried out by Przytycki and Traczyk.
The polynomial is defined using skein relations:
where L + ,L − ,L0 are crossing and smoothing changes on a local region of a link diagram, as indicated in the figure.
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The HOMFLY polynomial of a link L that is a split union of two links L1 and L2 is given by 
.
See the page on skein relation for an example of a computation using these relations.
[edit] Other HOMFLY skein relations
This polynomial can be obtained also using other skein relations:
[edit] Main properties
where V(t) is the Jones polynomial.
where 
is the Alexander polynomial.
[edit] References
- ^ http://www.hausdorff-research-institute.uni-bonn.de/geometry-and-physics-seminars
- ^ 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:.
- Kauffman, L.H., "Formal knot theory", Princeton University Press, 1983.
- Lickorish, W.B.R.. "An Introduction to Knot Theory". Springer. ISBN 038798254X.
- Weisstein, Eric W. "HOMFLY Polynomial." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HOMFLYPolynomial.html
