Knots and their polynomials

What about the Left Trefoil?

Left trefoil Right 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 over-crossing into an under-crossing.

If we rewrite the skein relation

t-1J(undercrossing)[t] - t J(overcrossing)[t] = (t1/2 - t-1/2)J(no crossing)[t]

so as to interchange the position of the under-crossing and the over-crossing, we obtain

- t J(overcrossing)[t] + t-1J(undercrossing)[t] = (t1/2 - t-1/2)J(no crossing)[t].

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

t J(overcrossing)[t] - t-1J(undercrossing)[t] = (t-1/2 - t1/2)J(no crossing)[t].

This manipulation shows that the skein relation holds if under-crossing and over-crossing 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

Left trefoil[t] = - t-4 + t-3 + t-1

different from the value for the Right Trefoil!

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