**Knots and their polynomials**

## The value of the Jones polynomial does not depend on the orientation chosen.

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.

Symbolically, letting ``-'' represent the orientation change, *Rot*
a 180-degree rotation and *Sk* the effect of applying the
skein relation to the knot diagram *K*, we have

*Sk* (- *K*) = - (*Rot* *Sk* *Rot* *K*)
where we apply operations, as usual, from right to left. Suppose our
analysis involves 2 applications of the skein relation.

*Sk*_{2} *Sk*_{1} (- *K*)
=
*Sk*_{2} (- *Rot* *Sk*_{1} *Rot* *K*)

= - (*Rot* *Sk*_{2} *Rot* *Rot* *Sk*_{1} *Rot* *K*)

= - (*Rot* *Sk*_{2} *Sk*_{1} *Rot* *K*)

since two consecutive 180-degree rotations of the entire figure have
the same effect as no rotation at all,
and similarly for more than two applications.
Suppose that it takes *n*
applications of the skein relation to reduce all the
diagrams to unknots, for which the Jones polynomial (symbolically
*J*{ })
does not depend on orientation. Then

*J*{(- *K*)}
= *J*{*Sk*_{n} ... *Sk*_{1} (- *K*)}
= *J*{ - (*Rot* *Sk*_{n} ... *Sk*_{1} *Rot* *K*)}

= *J*{*Rot* *Sk*_{n} ... *Sk*_{1} *Rot* *K*}

(since the Jones polynomial of an unknot-diagram does not depend on the
orientation)

= *J*{*Sk*_{n} ... *Sk*_{1} *K*} = *J*{*K*}

(since a rotated knot is topologically the same).
So the Jones polynomial of the reversed-orientation diagram is the
same as the Jones polynomial of the original diagram.

So why is an orientation required? Because to apply the skein
relation correctly at a crossing, we need the two strands to
be coherently oriented. Otherwise the calculation derails, as
a little experimentation will show.

On to the next knot page.
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