Categorifying Coloring Numbers
  Coloring numbers are one of the simplest combinatorial invariants of knots and links to describe. And with Joyce's introduction of quandles, we can understand them more algebraically. But can we extend these invariants to tangles -- knots and links with free ends? Indeed we can, once we categorify.   Starting from the definition of coloring numbers, we will categorify them and establish this extension to tangles. Then, decategorifying will leave us with matrix representations of the monoidal category of tangles. 
John Armstrong