In A minus sign that used to annoy me but now I know why it is there, Peter Tingley shows how to build knot invariants from the representations of the $U_q(\mathfrak{sl}_2) $ quantum group by comparing it to a certain algebra of ribbons. This is to say the representation theory of this quantum group is "tangled" in some way.

It is known that symmetric functions (characters of the symmetric group $S_n$) form a Hopf algebra. Is there a corresponding topological structure?

I'm just wondering if there's a diagrammatic way to look at representations of $S_n$. In general, does Hopf algebra structure of an algebra, imply there exists a diagrammatic way of looking at its representation theory?

More examples of Hopf algebras in combinatorics.

1more comment