Crossing Matrices for Braids

Resumen:

Given a geometric braid, one can tabulate the crossings of different strands in a matrix, called the crossing matrix of the braid;  this is invariant under Reidemeister moves and is therefore attached to the braid (isotopy class rel endpoints) represented by the geometric braid.  It is relatively easy, using ideas from Garside and Thurston, to characterize which matrices occur for some braid.  However, when one asks about which of these matrices arise for positive braids (every crossing is left-over-right), the “obvious” modification is false.  My Tufts colleague Mauricio Gutierrez and I have been trying (so far unsuccessfully) to provide such a characterization. We have a working Mathematica program which decides by brute force whether a given matrix (satisfying some relatively easy necessary conditions) does represent a positive braid, but we have been unable to find a conceptual characterization of the matrices which do represent a positive braid.  This special case is particularly interesting because in the general case, some crossings may “cancel” in forming the matrix, while for a positive braid all crossings are exhibited.

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