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.

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. if two diagrams have different polynomials, they represent different knots. 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 Nelson Yetter. The addition of PT recognizes independent work carried out by Józef H. Przytycki and Paweł Traczyk.

Definition

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 ₋ ,L₀ 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 L₁ and L₂ 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

• 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