2-Tangles

Just as links may be algebraically described as certain morphisms in the category of tangles, compact surfaces smoothly embedded in R 4 may be described as certain 2-morphisms in the 2-category of `2-tangles in 4 dimensions'. In this announcement we give a purely algebraic characterization of the 2-category of unframed unoriented 2-tangles in 4 dimensions as the `free semistrict braided monoidal 2-category with duals on one unframed self-dual object'. A forthcoming paper will contain a proof of this result using the movie moves of Carter, Rieger and Saito. We comment on how one might use this result to construct invariants of 2-tangles. 1 Introduction Recent work on `quantum invariants' of knots, links, tangles, and 3-manifolds depends crucially on a purely algebraic characterization of tangles in 3-dimensional space. It follows from work of Freyd and Yetter, Turaev, and Shum [13, 18, 19, 20] that isotopy classes of framed oriented tangles in 3 dimensions are the morphisms of a cert..