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Knots and Their Polynomials-10

 Knots and Their Polynomials

What about the left trefoil?

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

so as to interchange the positionof the undercrossing and the overcrossing, 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 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

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

which is different from the value for the right trefoil!

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