Knots and their Polynomials-9

$e-MATH$

Knots and their polynomials

What happens if we change orientation?

Let us focus on one crossing in an oriented knot diagram:

If we reverse the orientation and rotate the diagram 180 degrees, that crossing appears exactly as it did before.

So reversing the orientation in a diagram and then applying the skein relation to a crossing is the same as rotating the original diagram 180 degrees, applying the skein relation, rotating the products of the relation another 180 degrees, and then reversing the orientation. Since repeated application of the skein relation eventually reduces all the diagrams to unknots, for which the Jones polynomial does not depend on orientation, it follows that the result of the entire calculation does not depend on the orientation of the diagram, and it is appropriate to write

[t]= -t⁴ +t³ + t.

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