 
Summary: BRAIDS AND DIAGRAMMATIC ALGEBRAS.
We will investigate various groups, monoids, and other algebraic struc
tures, whose elements are represented by certain kinds of twodimensional
diagrams drawn inside a rectangle. This will continue the research from the
2011 REU which has resulted in a preprint "The Alexander and Jones Poly
nomials through representations of rook algebras" and "A presentation of
the Motzkin algebra" (work in preparation).
A braid is a certain kind of arrangement of pieces of string in space.
The endpoints of the strands are lined up at the top and the bottom of the
braid. The set of braids with a given number of strands forms a group,
where "multiplication" is defined by stacking one braid on top of the other
and joining up the strands.
A braid is a threedimensional object, but we can represent it by a two
dimensional picture that includes "crossings" where one strand goes over or
under another. Recently there has been increasing interest in other algebras
defined using two dimensional pictures. These have applications including
von Neumann algebras, and Feynman diagrams in physics.
Our goal is to define and investigate new algebras of this kind. The
diagrams should not include crossings, but can introduce features such as
coloring or orientations on the strands. We are especially interested in maps
