Abstract
Knits (or BMW tangles) are tangles in a cylinder generated by generators of the BMW (Birman-Murakami-Wenzl) algebras, consisting of standard generators of the braid group and their inverses, and splices of crossings called pairs of hooks. We give a new construction of surfaces in D2 B2, called knitted surfaces (or BMW surfaces), that are described as the trace of deformations of knits, and we give the notion of charts for knitted surfaces, that are graphs in B2. We show that a knitted surface has a chart description. Knitted surfaces and their chart description include 2-dimensional braids and their chart description. This is joint work with Jumpei Yasuda (Osaka University).
Gist
We can define braided surfaces by taking multiple surfaces which lie in a 4-dimensional cylinder and which does not backtrack (the projection is a homeo, restricted on the connected components?)
How can we think of that? Imagine two planes cutting across each other in 3D-space. This is reminicent of two segments crossing in 2D space. By adding a dimension two both we can control whether a plane/segment crosses in front of behind the other. Applying a generator puts a crossing into two parallel planes. (If I understood correctly the boundary of a plane is one-dimensional, just like the boundary of a braid is zerodimensional. We imagine that the boundary can be braided too).
We can encode the information about this surface braid using a structure called a chart (allso BMW diagram/chart) which is a chart with black vertices, white vertices and crossings.
The main idea is that black directed edges describe pane croossings and red edges indicate places where the planes return instead of crossing. The last operation is usually descibed by a generator .