Applications of 2-braids via chart diagrams

The classical notion of a geometric braid has a natural generalization to four-dimensional space, called a 2-braid. In analogy with Alexander's theorem, for example, any closed, orientable surface in 4-space may be described as the closure of a 2-braid. One way to view a 2-braid is as a “movie” of classical braids, but this can be cumbersome for deciding if two such 2-braids are equivalent. To remedy this, Kamada introduced chart diagrams to describe 2-braids, which (roughly speaking) are to classical braids what Cerf diagrams are to Morse functions. In this talk we describe chart diagrams, and discuss their use in defining Vassiliev invariants and approaching problems in link homotopy in the four-dimensional setting. In particular, we give a new proof that an embedded link of two 2-spheres in the 4-sphere is link homotopic to the trivial link.