HOMFLY polynomial

HOMFLY polynomial
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "HOMFLY_polynomial".

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 name HOMFLY combines the initials of its co-discoverers: Hoste, Ocneanu, Millett, Freyd, Lickorish, and Yetter¹. The addition of PT recognizes independent work carried out by Przytycki and Traczyk.

The polynomial is defined using skein relations:

$P( \mathrm{unknot} ) = 1,\,$

$\ell P(L_+) + \ell^{-1}P(L_-) + mP(L_0)=0,\,$

where $L +, L -, L 0$ are 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 $L 1$ and $L 2$ is given by $P(L) = \frac{-(l+l^{-1})}{m} P(L_1)*P(L_2)$ .

See the page on skein relation for an example of a computation using these relations.

Other HOMFLY skein relations

This polynomial can be obtained also using other skein relations:

$\alpha P(L_+) - \alpha^{-1}P(L_-) = zP(L_0),\,$

$xP(L_+) + yP(L_-) + zP(L_0)=0,\,$

Main properties

$V(t)=P(\alpha=t,z=t^{1/2}-t^{-1/2}),\,$

where V(t) is the Jones polynomial.

$\Delta(t)=P(\alpha=1,z=t^{1/2}-t^{-1/2}),\,$

where $\Delta(t)\,$ is the Alexander polynomial.

$P(L_1 \# L_2)=P(L_1)P(L_2),\,$

$P_K(\ell,m)=P_{Mirror Image(K)}(\ell^{-1},m),\,$

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/S0273-0979-1985-15361-3.

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

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