Knots and Their Polynomials-7

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] =(t1/2 - 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]= - t1/2 - t-1/2.

Since the two knots

and

are topologically the same, it follows that

[t]=[t]=- t1/2 - t-1/2.

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