Trefoil Knot

$\begin{figure}\begin{center}\BoxedEPSF{Trefoil_Knot.epsf}\end{center}\end{figure}$

The knot 03-001, also called the Threefoil Knot, which is the unique Prime Knot of three crossings. It has Braid Word ${\sigma_1}^3$ . The trefoil and its Mirror Image are not equivalent. The trefoil has Alexander Polynomial and is a (3, 2)-Torus Knot. The Bracket Polynomial can be computed as follows.

$\displaystyle \left\langle{L}\right\rangle{}$	$\textstyle =$	$\displaystyle A^3d^{2-1}+A^2Bd^{1-1}+A^2Bd^{1-1}+AB^2d^{2-1}$
	$\textstyle \phantom{=}$	$\displaystyle +A^2Bd^{1-1}+AB^2d^{2-1}+AB^2d^{2-1}+B^3d^{3-1}$
	$\textstyle =$	$\displaystyle A^3d^1+3A^2Bd^0+3AB^2d^1+B^3d^2.$

Plugging in

$\displaystyle B$	$\textstyle =$	$\displaystyle A^{-1}$
$\displaystyle d$	$\textstyle =$	$\displaystyle -A^2-A^{-2}$

gives

$\begin{displaymath} \left\langle{L}\right\rangle{}=A^{-7}-A^{-3}-A^5. \end{displaymath}$

The normalized one-variable Kauffman Polynomial X is then given by

$\displaystyle X_L$	$\textstyle =$	$\displaystyle (-A^3)^{-w(L)}\left\langle{L}\right\rangle{}=(-A^3)^{-3}(A^{-7}-A^{-3}-A^5)$
	$\textstyle =$	$\displaystyle A^{-4}+A^{-12}-A^{-16},$

where the Writhe

. The Jones Polynomial is therefore

$\begin{displaymath} V(t)=L(A=t^{-1/4})=t+t^3-t^4=t(1+t^2-t^3). \end{displaymath}$

Since $V(t^{-1})\not=V(t)$ , we have shown that the mirror images are not equivalent.

References

Claremont High School. ``Trefoil_Knot Movie.'' Binary encoded QuickTime movie. ftp://chs.cusd.claremont.edu/pub/knot/trefoil.cpt.bin.

Crandall, R. E. Mathematica for the Sciences. Redwood City, CA: Addison-Wesley, 1993.

Kauffman, L. H. Knots and Physics. Singapore: World Scientific, pp. 29-35, 1991.

Pappas, T. ``The Trefoil Knot.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 96, 1989.