Models for Surface Braids, Surface Knots, Singular Surfaces in 4-space, and Beyond

Date started: September 2019
Leads: Scott Carter, Edmund Harriss, Nick Owad


A surface braid is an embedding into a 4-dimensional disk that is thought of as [0,1]\times [0,1] \times [0,1] \times [-\epsilon, \epsilon] such that the projection onto the first two factors is a simple branched covering. Such a braid can be decomposed into pieces of four types: branch points, triple points, type II moves, and expanders. The goal of the project is to produce simple snap-together pieces that can be used to illustrate or construct an arbitrary surface braid. Also we mean to combine these so that braided three and four-dimensional manifolds can be envisioned. In the meantime, other models were created.


Using coaching from Owad and a useful piece of software from Harriss, Carter has been designing and printing prototypes for all the forms. Several photos of the models are illustrated here. The design parameters involve drawing crossings at a particular scale, and interpolating within space, broken surface diagrams of the pieces. Technical issues that are not yet addressed are snapping the pieces together, and providing scaffolds for crossing introduced late in the model.


  • J. Scott Carter and Seiichi Kamada. How to fold a manifold. In New ideas in low dimensional topology, volume 56 of Ser. Knots Everything, pages 31-77. World. Sci. Publ., Hackensack, NJ, 2015.
  • J. Scott Carter and Seiichi Kamada. Three-dimensional braids and their descriptions. Topology Appl., 196(part B):510-521, 2015.
  • Scott Carter, Seiichi Kamada, and Masahico Saito. Surfaces in 4-space, volume 142 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2004. Low-Dimensional Topology, III.
  • Seiichi Kamada. Braid and knot theory in dimension four, volume 95 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2002.