Knots and Their Polynomials-7 ## Knots and Their Polynomials

## The calculation of the Jones polynomial of the right trefoil

To calculate the Jones polynomial of the two unlinked unknots, we apply the skein relation to the twisted unknot:

*t*^{-1}[*t*] -*t*[*t*] =(*t*^{1/2} - *t*^{-1/2}) [*t*].Since both diagrams on the left come from topological unknots, their Jones polynomials areequal to 1, and the left-hand side reduces to *t*^{-1} - *t*.Solving gives the Jones polynomial of two concentric unknots as

[*t*]= - *t*^{1/2} - *t*^{-1/2}.Since the two knots

and are topologically the same, it follows that

[*t*]=[*t*]=- *t*^{1/2} - *t*^{-1/2}.

On to the next step.

Back to the previous step.

Back to the first knot page.