HOMFLY Polynomial

A 2-variable oriented Knot Polynomial $P_L(a,z)$ motivated by the Jones Polynomial (Freyd et al. 1985). Its name is an acronym for the last names of its co-discoverers: Hoste, Ocneanu, Millett, Freyd, Lickorish, and Yetter (Freyd et al. 1985). Independent work related to the HOMFLY polynomial was also carried out by Prztycki and Traczyk (1987). HOMFLY polynomial is defined by the Skein Relationship

$$a^{-1}P_{L_+}(a,z)-aP_{L_-}(a,z)=zP_{L_0}(a,z)$$ (1)

(Doll and Hoste 1991), where $v$ is sometimes written instead of $a$ (Kanenobu and Sumi 1993) or, with a slightly different relationship, as

$$\alpha P_{L_+}(\alpha,z)-\alpha^{-1} P_{L_-}(\alpha,z)=zP_{L_0}(\alpha,z)$$ (2)

(Kauffman 1991). It is also defined as $P_L(\ell,m)$ in terms of Skein Relationship

$$\ell P_{L_+}+\ell^{-1}P_{L_-}+mP_{L_0}=0$$ (3)

(Lickorish and Millett 1988). It can be regarded as a nonhomogeneous Polynomial in two variables or a homogeneous Polynomial in three variables. In three variables the Skein Relationship is written

$$xP_{L_+}(x,y,z)+yP_{L_-}(x,y,z)+zP_{L_0}(x,y,z)=0.$$ (4)

It is normalized so that $P_{\rm unknot}=1$. Also, for $n$ unlinked unknotted components,

$$P_L(x,y,z)=\left({-{x+y\over z}}\right)^{n-1}.$$ (5)

This Polynomial usually detects Chirality but does not detect the distinct Enantiomers of the Knots 09-042, 10-048, 10-071, 10-091, 10-104, and 10-125 (Jones 1987). The HOMFLY polynomial of an oriented Knot is the same if the orientation is reversed. It is a generalization of the Jones Polynomial $V(t)$, satisfying

$$V(t) = P(a=t, z=t^{1/2}-t^{-1/2})$$ (6)

$$V(t) = P(\ell=it^{-1}, m=i(t^{-1/2}-t^{1/2})).$$ (7)

It is also a generalization of the Alexander Polynomial $\nabla(z)$, satisfying

$$\Delta(z)=P(a=1,z=t^{1/2}-t^{-1/2}).$$ (8)

The HOMFLY Polynomial of the Mirror Image $K^*$ of a Knot $K$ is given by

$$P_{K^*}(\ell, m) = P_K(\ell^{-1}, m),$$ (9)

so $P$ usually but not always detects Chirality.

A split union of two links (i.e., bringing two links together without intertwining them) has HOMFLY polynomial

$$P(L_1\cup L_2)=-(\ell+\ell^{-1})m^{-1}P(L_1)P(L_2).$$ (10)

Also, the composition of two links

$$P(L_1\char93 L_2)=P(L_1)P(L_2),$$ (11)

so the Polynomial of a Composite Knot factors into Polynomials of its constituent knots (Adams 1994).

Mutants have the same HOMFLY polynomials. In fact, there are infinitely many distinct Knots with the same HOMFLY Polynomial (Kanenobu 1986). Examples include (05-001, 10-132), (08-008, 10-129) (08-016, 10-156), and (10-025, 10-056) (Jones 1987). Incidentally, these also have the same Jones Polynomial.

M. B. Thistlethwaite has tabulated the HOMFLY polynomial for Knots up to 13 crossings.

See also Alexander Polynomial, Jones Polynomial, Knot Polynomial

References

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 171-172, 1994.

Doll, H. and Hoste, J. ``A Tabulation of Oriented Links.'' Math. Comput. 57, 747-761, 1991.

Freyd, P.; Yetter, D.; Hoste, J.; Lickorish, W. B. R.; Millett, K.; and Oceanu, A. ``A New Polynomial Invariant of Knots and Links.'' Bull. Amer. Math. Soc. 12, 239-246, 1985.

Jones, V. ``Hecke Algebra Representations of Braid Groups and Link Polynomials.'' Ann. Math. 126, 335-388, 1987.

Kanenobu, T. ``Infinitely Many Knots with the Same Polynomial.'' Proc. Amer. Math. Soc. 97, 158-161, 1986.

Kanenobu, T. and Sumi, T. ``Polynomial Invariants of 2-Bridge Knots through 22 Crossings.'' Math. Comput. 60, 771-778 and S17-S28, 1993.

Kauffman, L. H. Knots and Physics. Singapore: World Scientific, p. 52, 1991.

Lickorish, W. B. R. and Millett, B. R. ``The New Polynomial Invariants of Knots and Links.'' Math. Mag. 61, 1-23, 1988.

Morton, H. R. and Short, H. B. ``Calculating the $2$-Variable Polynomial for Knots Presented as Closed Braids.'' J. Algorithms 11, 117-131, 1990.

Przytycki, J. and Traczyk, P. ``Conway Algebras and Skein Equivalence of Links.'' Proc. Amer. Math. Soc. 100, 744-748, 1987.

Stoimenow, A. ``Jones Polynomials.'' http://www.informatik.hu-berlin.de/~stoimeno/ptab/j10.html.

Weisstein, E. W. ``Knots and Links.'' Mathematica notebook Knots.m.