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

.

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 are equal 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.

Back to the previous step.

Back to the first knot page.