The Set of Minimal Braids is co-NP-Complete

Braids ñàï Üå represented as two-dimenSional diagrams showing the crossings of strings or as words over the generators of à braid group. À minimal braid is îïå with the fewest crossings (or the shortest words) among all possible repre~entations topologically equivalent to that braid. ÒÜå main result of this paper is that the set of minimal braids is co-NP-complete. Algorithmic problems in braid groups have received much attention since [À1]. An algorithm for the word problem was given Üó Artin [À1, A2],and Garside solved the conjugacy problem [G]. More recently, with the increasing interest in complexity, these problems have Üååï reexamined with regard to efficiency. Artin's algorithm involves generating à canonical form of length exponential in the length of the original word. Apparently the first polynomial-time algorithm for the word problem results from recent work of Thurston [Th]. Whether there exists à polynomial time algorithm for the conjugacy problem seems to Üå unknown. In his polynomial-time algorithm for the word problem Thurston produces à canonical form for braid group elements whose length is quadratic .This work was carried out during à visit to Warwick University Üó the second author, supported Üó à Visiting Fellowship from the SERC of the UK. ÒÜå first author was partially supported Üó à Senior Fellowship from the SERC and Üó the ESPRIT II BRA Programme of the ÅÑ under Contract 3075 (ALCOM).