In [A minus sign that used to annoy me but now I know why it is there][1], 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 $S_n$][2]) [form a Hopf algebra][3]. 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? <hr> More examples of [Hopf algebras][4] in combinatorics. [1]: http://www-math.mit.edu/~ptingley/publications/minus-sign.pdf [2]: https://arxiv.org/abs/math/0503040 [3]: http://math.columbia.edu/~ellis/ugrad_seminar_spring_2011/geissinger.pdf [4]: http://garsia.math.yorku.ca/~zabrocki/talks/Hopfzoo.pdf