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

We apply the skein relation to the right trefoil:

t^-1[t] -t [t] =(t^1/2 - t^-1/2)[t].

The second knot in the equation is topologically an unknot:

=,so [t] = 1,

and we are left with

[t] =(t^3/2 - t^1/2)[t] + t².

The knot on the right is made up of two linked unknots. This isa right-hand link because when we follow the orientation the two circles twist around each other to the right.

The next step is to analyze that link by the same method.

On to the next step.

Back to the previous knot page.

Back to the first knot page.