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 name HOMFLY combines the initials of its co-discoverers: Hoste, Ocneanu, Millett, Freyd, Lickorish, and Yetter. 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 − ,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 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.

[edit] 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,\,

[edit] main properties

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

where V(t) is Jones Polynomial

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

where \Delta(t)\, is Alexander Polynomial

P(L_1 \# L_2)=P(L_1)P(L_2),\,
P_K(\ell,m)=P_{Mirror Image(K)}(\ell^{-1},m),\,
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