Knots and Their Polynomials

What about the left trefoil?

Left- and right-handed trefoils

A diagram for the left trefoil can be obtained by reversingall 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 positionof 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 undercrossingand the overcrossing are interchanged and at the same time positiveand negative powers of t are interchanged.

If we apply this form of the skein relation to the left trefoil, thecalculation will proceed exactly as it did for the right trefoil,except that the exponents of t will be exactly oppositefrom 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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