Knots and Their Polynomials

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What about the left trefoil?

Left- and right-handed trefoils

A diagram for the left trefoil can be obtained by reversing all the crossings in a diagram for the right trefoil: make every overcrossing into an undercrossing.

If we rewrite the skein relation

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

so as to interchange the position of the undercrossing and the overcrossing, we obtain

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

Now let's multiply both sides of the relation by -1:

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

This manipulation shows that the skein relation holds if the undercrossing and the overcrossing are interchanged and at the same time positive and negative powers of t are interchanged.

If we apply this form of the skein relation to the left trefoil, the calculation will proceed exactly as it did for the right trefoil, except that the exponents of t will be exactly opposite from what they were before. At the end we will obtain

[t] = - t^-4 + t^-3 + t^-1

which is different from the value for the right trefoil!

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