Approaches to GI inspired by knot problem
One connection is that graph isomorphism and knot isomorphism are both special cases of 3-manifold homeomorphism. In the knot case, two knots are isomorphic if their complements (manifolds formed by deleting the points of the knot from 3-space) are homeomorphic.

And in the graph case it's possible to transform graphs into manifolds in such a way that the graphs are isomorphic if and only if the manifolds are homeomorphic. I wrote a comment about this on a Google+ post last December, but unfortunately not on a post I can share. The construction is to start with a manifold for each vertex v, in the form of the complement in a 3-sphere of a bouquet of degree(v) loops (connected together at a common vertex). For each edge uv, connect the manifolds for u and v together by a surgery, and link one loop from u and one loop from v across the surgery ball. Then every isomorphism of graphs lifts to a homeomorphism of the resulting manifold (this would be true even if we just used surgery on 3-spheres without the bouquets) and the bouquets prevent the manifold from having extra homeomorphisms that don't come from the graph.