Braids and Knots

Barnard College, New York NY

Investigators

Abstract

The main focus of this proposal is the conjugacy problem in braid groups, and ultimately in Garside and Artin groups and also in surface mapping class groups. To begin, we propose to investigate the interface between two known approaches to the conjugacy problem in the braid groups: (a) the combinatorial solution that originated with the work of Frank Garside in 1968; and (b) the dynamic approach which was first studied by Jakob Nielsen in 1932 and later given new meaning, in 1982, by William Thurston. We note that (a) gives a definitive test for when two braids are conjugate, whereas (b) gives less than that. On the other hand (a) is exponential in both word length and braid index, whereas (b) suggests the possible existence of a polynomial solution. The moment seems to be ripe for an investigation which combines aspects of both. The new work of Gebhardt on (a) shows that the `Super Summit Set', a complete class invariant, can be replaced by the smaller `Ultra Summit Set' (USS). Our hope is to use the known dynamic classification to study the USS. We will begin with a study of reducing curves for reducible braids, and go on to study finite order (FO) and pseudo-Anosov (PA) braids. We conjecture that for braids in the USS the reducing curves can be chosen to be round circles which meet a line containing the punctures in 2 points. We conjecture further that PA braids have `rigid powers whose combinatorics is particularly simple. Looking at the braid group as a special case of mapping class groups, one then expects further that the associated train tracks will be special and interesting. If all this goes well, there will be further problems to investigate, including related combinatorics in more general mapping class groups, and related dynamics in more general Garside and Artin groups. In recent years there has been great interest in certain codes, in `Public Key Cryptography, which are based upon the use of braid groups. The underlying idea has been that the so-called `word problem in the braid groups is known to have a solution which is fast. That is, the time required to check whether two words represent the same element in the braid group is known to be polynomial as a function of word length L and braid index N. On the other hand, the conjugacy problem, which is more difficult, has been thought to be fundamentally exponential in L and N. Like most such problems in complexity theory (e.g. the factorization of numbers into primes) the possibility always exists that a particular problem which was thought to be exponential will turn out to be polynomial if a more ingenious solution can be found. The PI has been a leading expert in the study of braid groups for many years. She now proposes to investigate the conjugacy problem in the braid groups from a new point of view which she hopes will show it to be polynomial, like the word problem, in both L and N. The underlying idea is to look at two very different approaches to the conjugacy problem, one based on `combinatorics and the other on `dynamics, and to apply ideas from the second approach to simplify the first. If successful, this research would have implications in Public Key Cryptography as regards the security of codes based upon braid groups. It should also have many applications in mathematics because of the central role played by the braid groups.