Knots and Their Polynomials-10 ## Knots and Their Polynomials

## What about the left trefoil?

Left- and right-handed trefoilsA 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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