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-1J(twisted-to-right)[t] - tJ(twisted-to-left)[t] = (t1/2 - t-1/2) J(concentric unknots)[t].

Since both diagrams on the left come from topological unknots, their Jones polynomials are equal to 1, and the left-hand side reduces to t-1 - t. Solving gives the Jones polynomial of two concentric unknots as

J(concentric unknots)[t] = - t1/2 - t-1/2.

Since the two knots

unlinked unknots and concentric unknots

are topologically the same, it follows that

J(unlinked unknots)[t] = J(concentric unknots)[t] = - t1/2 - t-1/2.

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