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] =(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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