Trefoil Knot
The knot 03-001, also called the Threefoil Knot, which is the unique Prime Knot of threecrossings. It has Braid Word 
. The trefoil and its Mirror Image are not equivalent. Thetrefoil has Alexander Polynomial 
and is a (3, 2)-Torus Knot. The Bracket Polynomial can becomputed as follows.
 |  |  | |
 |  | ||
|  |
Plugging in
 | |  | |
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gives
The normalized one-variable Kauffman Polynomial X is then given by
 | |  | |
|  |
where the Writhe
. The Jones Polynomial is therefore
Since
, 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. Nordstrand, T. ``Threefoil Knot.'' http://www.uib.no/people/nfytn/tknottxt.htm. Pappas, T. ``The Trefoil Knot.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 96, 1989.
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