I think the best answer I can give is a quote from Louis Kauffman's Knots and Physics:

It is important to come to some practical understanding of how these knots work. The fact that the square knot holds, and that the granny knot does not hold are best observed with actual rope models. It is a tremendous challenge to give a good mathematical analysis of these phenomena. Tension, friction and topology conspire to give the configuration a form - and within that form the events of slippage or interlock can occur.
I raise these direct physical questions about knotting, not because we shall answer them, but rather as an indication of the difficulty in which we stand. A book on knots and physics cannot ignore the physicality of knots of rope in space. Yet all of our successes will be more abstract, more linguistic, patterned and poised between the internal logic of patterns and their external realizations.

(Italics mine.) Based on this, my suspicion - and it's no more than a suspicion; applied math is not my area - is that such studies are ongoing, but that as yet they're in their infancy. (The Kauffman book was published in 1991, and a lot can happen in a decade and a half, though.)