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]=- t^1/2 - t^-1/2. (Step 3)

• [t] =t^3/2 - t^1/2 + t² [t] (Step 2)
  = t^3/2 - t^1/2 + t² (- t^1/2 - t^-1/2)

  = -t^5/2 - t^1/2.

• [t] =(t^3/2 - t^1/2)[t] + t² (Step 1)
  = (t^3/2 - t^1/2)(- t^5/2 - t^1/2) +t²

  = -t⁴ + t³ + t.

This completes the calculation.

On to the next knot page.

Back to the previous step.

Back to the first knot page.