Braid invariants resource

Question

What are some braid invariants (analogous to the idea of knot invariants) or a resource where I can find them?

Answer 1

For a solid reference on this topic, I would suggest either

J. S. Birman Braids, Links and Mapping Class Groups, Annals of Mathematical Studies 82, Princeton University Press, 1975.

for a classical approach to a solution for the word problem of the braid groups (contained in the first two chapters), or

V. L. Hansen, Braids and Coverings, London Mathematical Society Student Texts 18, Cambridge University Press, 1989.

for a more refined and modern approach, closely based on Birman's text.

To directly answer your question, piecewise linear (non-backtracking) braids on $n$ strands can be entirely classified up to ambient isotopy by 'solving the word problem' in the braid group $B_n$. This means that, given two words in the standard braid generators, we can tell whether these words represent ambient isotopic braids or not in an algorithmic way, as outlined in either of the above references (this amounts to writing braids in a 'standard form' which is shown to exist, and be unique, for every braid).

There are of course many other ways to classify braids. You may want to classify them up to conjugacy, because conjugate braids in $B_n$ will give the same link, up to isotopy, when taking the braid closure (although the converse does not hold. Non-conjugate braids can close to give isotopic links).

You may want to consider classifying braids up to 'homotopy'. This is where you consider two braids to be the same if you allow standard ambient isotopies to get from one to the other, but you also allow single braid strings to 'pass through themselves' (but not each other). A good reference for this concept of homotopic braids is

K. Murasugi & B. Kurpita, A Study of Braids, Kluwer Academic Publishers, 1999.