Kauffman Polynomial X

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

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

$${\mathcal L}_{L^*}={\mathcal L}_L(A^{-1}).$$ (2)

It is identical to the Jones Polynomial with the change of variable

$${\mathcal L}(t^{-1/4}) = V(t).$$ (3)

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

References

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