Approaches to GI inspired by knot problem GI and Knot Problem both are problem of deciding structural equivalence of mathematical objects. Are there any results establishing connections between them? Nice connections of knot problem to statistical physics have been explored via knot polynomials, are there similar results for $GI$?

It would be particularly helpful to know if there are any standard results/warnings/suggestions/comments before one start looking into $GI$ motivated by knot problem. Actually, I was wondering if its recommended to explore in this direction for my master's thesis. I am interested in quantum/classical approaches to $GI$ and algebraic problems. Any other suggestions are welcome.
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.