On the definition of positive and negative crossings

Question

In the definition of the Kauffman bracket, we have to resolve a crossing in two ways.

The way to resolve these crossings involve distinguishing whether the crossings are positive or negative. But without orientation, I do not see why the sign of a crossing can be uniquely defined.

For example, should I apply the rule $$<T> = a <A> + a^{-1}<B>$$ or $$<T> = a <B> + a^{-1}<A>$$ to obtain the Kauffman bracket for T?

The reason I asked is because if you rotate the picture by 90 degrees clockwise you will apply a different rule and get a different answer but I don't think that could be the case for a knot invariant...

Do I miss anything?

Answer 1

Orientation doesn't matter in the Kauffman bracket. In the definition, you can smooth out crossings that have the form:

                         

And then depending on how you resolve you get either a 0-smoothing or a 1-smoothing:

                   and    

If you rotate the figure 90°, you get something like this:

                         

And you cannot apply the rule to crossings like that. If you rotate it 180°, you get the original picture; but then 0- and 1-smoothings are the same as for the original picture. All in all, you can see that orientation doesn't matter. It's not a question of positive and negative crossings.

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In fact if the diagram were oriented, you wouldn't be able to do both smoothings! For example consider the following knot diagram (obviously it's the unknot, but it's just for the example -- you could envision a more complicated diagram):

The crossing is a positive crossing. But you see that if you want to keep an oriented link diagram, only one smoothing is allowed namely . The other one welds the two parts in such a way that the orientations conflict.