# 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.

• from mathworld isomorphic graphs: "n some sense, graph isomorphism is easy in practice except for a set of pathologically difficult graphs that seem to cause all the problems. So, unlike knot theory, there have never been any significant pairs of graphs for which isomorphism was unresolved. ... Unfortunately, there is almost certainly no simple-to-calculate universal graph invariant, whether based on the graph spectrum or any other parameters of a graph (Royle 2004)." – vzn Oct 1 '12 at 15:34
• Apparently knot equivalence is also easy in practice. – Jeffε Oct 2 '12 at 3:49
• I have poster similar question here physics.stackexchange.com/questions/39328/… also – DurgaDatta Oct 8 '12 at 4:54
• To my knowledge, there are no "pathologically difficult" knots that cause all of the problems. It would be very interesting to find a family of unknots that had poor running times on the various unknot recognition programs, either provably or just experimentally. – Sam Nead Feb 24 '14 at 12:00