We present two classical problems involving braids, classical link invariants, compressed words and trees.
The first problem is about strongly quasipositive links, that is, links which can be seen as closures of positive braids in terms of band generators. We will give a necessary condition for a link with braid index 3 to be strongly quasipositive in terms of its Conway polynomial.
The second problem concerns braid combing, which is a procedure defined by Emil Artin to solve the word problem in braid groups for the first time. Braid combing has exponential complexity. However, we will show how to use the theory of straight line programs (or compressed words) to give a polynomial algorithm which performs braid combing. Applying our procedure to braids on surfaces, we get the first algorithm (to our knowledge) which solves the word problem for braid groups on surfaces with boundary in polynomial time and space.