Celebratio Mathematica — Birman

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J. Birman, K. H. Ko, and S. J. Lee: “A new approach to the word and conjugacy problems in the braid groups,” Adv. Math. 139 : 2 (November 1998), pp. 322–353. MR 1654165 Zbl 0937.20016 article

Sang Jin Lee	Related
Ki Hyoung Ko	Related

J. S. Birman, K. H. Ko, and S. J. Lee: “The infimum, supremum, and geodesic length of a braid conjugacy class,” Adv. Math. 164 : 1 (December 2001), pp. 41–56. MR 1870512 Zbl 1063.20039 article

Algorithmic solutions to the conjugacy problem in the braid groups \( B_n \), \( n = 2,\,3,\,4,\dots \) were given in earlier work. This xlinconcerns the computation of two integer class invariants, known as “inf” and “sup.” A key issue in both algorithms is the number \( m \) of times one must “cycle” (resp. “decycle”) in order to either increase inf (resp. decrease sup) or to be sure that it is already maximal (resp. minimal) for the class. Our main result is to prove that \( m \) is bounded above by \[ \frac{n^2-n}{2}-1 \] in the situation stated by E. A. Elrifai and H. R. Morton (1994, Quart. J. Math. Oxford 45, 479–497) and by \( n-2 \) in the situation stated by authors (1998, Adv. Math. 139, 322–353). It follows immediately that the computation of inf and sup is polynomial in both word length and braid index, in both algorithms. The integers inf and sup determine (but are not determined by) the shortest geodesic length for elements in a conjugacy class, and so we also obtain a polynomial-time algorithm for computing this length.

Sang Jin Lee	Related
Ki Hyoung Ko	Related