Obtaining a braid from a knot/link (Alexander's theorem)

Question

I am following the algorithm for obtaining a braid from a link as explained on pages 23-24 here: https://math.berkeley.edu/~vfr/jonesakl.pdf.

If I let the axis run straight through the center region bounded by the trefoil knot, then it is already braided. If I choose a counter-clockwise orientation, I get that the braid word for the trefoil knot is $\sigma_1^{-1}\sigma_1^{-1}\sigma_1^{-1}$ rather than $\sigma_1\sigma_1\sigma_1$.

Choosing the clockwise orientation, I get the correct $\sigma_1^3$.

Is there a canonical choice of orientation (perhaps clockwise)?

Answer 1

No, there is no canonical choice. Alexander's theorem is an existence theorem, but it is very far from a uniqueness theorem. In fact, there are many, many, many noncanonical choices in this construction:

• the choice of the knot/link within its isotopy class;
• the choice of the projection of the knot/link;
• the choice of the region you run the axis through;
• and the choice of the orientation.

Varying these choices will produce infinitely many possible braids.