**Knots and their polynomials**

## 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 over-crossing into an under-crossing.

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 under-crossing and the over-crossing, 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 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

[*t*] = - *t*^{-4} + *t*^{-3} +
*t*^{-1}
different from the value for the Right Trefoil!

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