Composition problems for braids: Membership, Identity and Freeness

In this paper we investigate the decidability and complexity of problems related to braid composition. While all known problems for a class of braids with three strands, B3, have polynomial time solutions we prove that a very natural question for braid composition, the membership problem, is NP-complete for braids with only three strands. The membership problem is decidable in NP for B3, but it becomes harder for a class of braids with more strands. In particular we show that fundamental problems about braid compositions are undecidable for braids with at least five strands, but decidability of these problems for B4 remains open. Finally we show that the freeness problem for semigroups of braids from B3 is also decidable in NP. The paper introduces a few challenging algorithmic problems about topological braids opening new connections between braid groups, combinatorics on words, complexity theory and provides solutions for some of these problems by application of several techniques from automata theory, matrix semigroups and algorithms.