HOMFLY polynomial

The polynomial is defined using skein relations:

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

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

where ${\displaystyle L_{+},L_{-},L_{0}}$ are links formed by 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 ${\displaystyle L_{1}}$ and ${\displaystyle L_{2}}$ is given by

${\displaystyle P(L)={\frac {-(\ell +\ell ^{-1})}{m}}P(L_{1})P(L_{2}).}$

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

This polynomial can be obtained also using other skein relations:

${\displaystyle \alpha P(L_{+})-\alpha ^{-1}P(L_{-})=zP(L_{0}),\,}$

${\displaystyle xP(L_{+})+yP(L_{-})+zP(L_{0})=0,\,}$

${\displaystyle P(L_{1}\#L_{2})=P(L_{1})P(L_{2}),\,}$, where # denotes the knot sum; thus the HOMFLY polynomial of a composite knot is the product of the HOMFLY polynomials of its components.

${\displaystyle P_{K}(\ell ,m)=P_{{\text{Mirror Image}}(K)}(\ell ^{-1},m),\,}$, so the HOMFLY polynomial can often be used to distinguish between two knots of different chirality. However there exist chiral pairs of knots that have the same HOMFLY polynomial, e.g. knots 9₄₂ and 10₇₁[2]

The Jones polynomial, V(t), and the Alexander polynomial, ${\displaystyle \Delta (t)\,}$ can be computed in terms of the HOMFLY polynomial (the version in ${\displaystyle \alpha }$ and ${\displaystyle z}$ variables) as follows:

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

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

• Kauffman, L.H., "Formal knot theory", Princeton University Press, 1983.
• Lickorish, W.B.R. "An Introduction to Knot Theory". Springer. ISBN 0-387-98254-X.

• "Jones-Conway polynomial", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "HOMFLY Polynomial". MathWorld.
• "The HOMFLY-PT Polynomial", The Knot Atlas.