Knots and Their Polynomials

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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.

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On to the next knot page.

Back to the previous step.

Back to the first knot page.