Bracket Polynomial

A one-variable Knot Polynomial related to the Jones Polynomial. The bracket polynomial, however, is not a topological invariant, since it is changed by type I Reidemeister Moves. However, the Span of the bracket polynomial is a knot invariant. The bracket polynomial is occasionally given the grandiose name Regular Isotopy Invariant. It is defined by

$$\left\langle{L}\right\rangle{}(A,B,d)\equiv\sum_\sigma \left... ...le{L\vert\sigma}\right\rangle{}d^{\vert\vert\sigma\vert\vert},$$ (1)

where $A$ and $B$ are the ``splitting variables,'' $\sigma$ runs through all ``states'' of $L$ obtained by Splitting the Link, $\left\langle{L\vert\sigma}\right\rangle{}$ is the product of ``splitting labels'' corresponding to $\sigma$, and

$$\vert\vert\sigma\vert\vert \equiv N_L-1,$$ (2)

where $N_L$ is the number of loops in $\sigma$. Letting

$$B = A^{-1}$$ (3)

$$d = -A^2-A^{-2}$$ (4)

gives a Knot Polynomial which is invariant under Regular Isotopy, and normalizing gives the Kauffman Polynomial X which is invariant under Ambient Isotopy. The bracket Polynomial of the Unknot is 1. The bracket Polynomial of the Mirror Image $K^*$ is the same as for $K$ but with $A$ replaced by $A^{-1}$. In terms of the one-variable Kauffman Polynomial X, the two-variable Kauffman Polynomial F and the Jones Polynomial $V$,

$$X(A) = (-A^3)^{-w(L)}\left\langle{L}\right\rangle{},$$ (5)

$$\left\langle{L}\right\rangle{}(A) = F(-A^3, A+A^{-1})$$ (6)

$$\left\langle{L}\right\rangle{}(A) = V(A^{-4}),$$ (7)

where $w(L)$ is the Writhe of $L$.

See also Square Bracket Polynomial

References

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 148-155, 1994.

Kauffman, L. ``New Invariants in the Theory of Knots.'' Amer. Math. Monthly 95, 195-242, 1988.

Kauffman, L. Knots and Physics. Teaneck, NJ: World Scientific, pp. 26-29, 1991.

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