info prev up next book cdrom email home

Kauffman Polynomial X

A 1-variable Knot Polynomial denoted $X$ or ${\mathcal L}$.

{\mathcal L}_L(A)\equiv (-A^3)^{-w(L)}\left\langle{L}\right\rangle{},
\end{displaymath} (1)

where $\left\langle{L}\right\rangle{}$ is the Bracket Polynomial and $w(L)$ is the Writhe of $L$. This Polynomial is invariant under Ambient Isotopy, and relates Mirror Images by
{\mathcal L}_{L^*}={\mathcal L}_L(A^{-1}).
\end{displaymath} (2)

It is identical to the Jones Polynomial with the change of variable
{\mathcal L}(t^{-1/4}) = V(t).
\end{displaymath} (3)

The $X$ Polynomial of the Mirror Image $K^*$ is the same as for $K$ but with $A$ replaced by $A^{-1}$.


Kauffman, L. H. Knots and Physics. Singapore: World Scientific, p. 33, 1991.

© 1996-9 Eric W. Weisstein