Knots and Their Polynomials

The calculation of the Jones polynomial of the right trefoil

• Step 1: from the right trefoil to the right link

• Step 2: from the right link to two unlinked unknots

• Step 3: the Jones polynomial of two unlinked unknots

• Step 4: working backwards to the answer

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.