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Trefoil Knot


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



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 $w(L)=3$. The Jones Polynomial is therefore


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


Claremont High School. ``Trefoil_Knot Movie.'' Binary encoded QuickTime movie.

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.

Nordstrand, T. ``Threefoil Knot.''

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

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© 1996-9 Eric W. Weisstein