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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...
\end{displaymath} (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,
\end{displaymath} (2)

where $N_L$ is the number of loops in $\sigma$. Letting
$\displaystyle B$ $\textstyle =$ $\displaystyle A^{-1}$ (3)
$\displaystyle d$ $\textstyle =$ $\displaystyle -A^2-A^{-2}$ (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 $V$,
$\displaystyle X(A)$ $\textstyle =$ $\displaystyle (-A^3)^{-w(L)}\left\langle{L}\right\rangle{},$ (5)
$\displaystyle \left\langle{L}\right\rangle{}(A)$ $\textstyle =$ $\displaystyle F(-A^3, A+A^{-1})$ (6)
$\displaystyle \left\langle{L}\right\rangle{}(A)$ $\textstyle =$ $\displaystyle V(A^{-4}),$ (7)

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

See also Square Bracket Polynomial


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.

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

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© 1996-9 Eric W. Weisstein