Are there knot theoretic formulations of NP complete problems? Are there NP complete(or even NP-hard, or NP) problems that have good topological properties to study. Do NP problems have knot theoretic formulations? We know about #$P$ results about the Jones polynomial. Graph problems(embeddings?), specifically colourings of graphs can be seen to have nice knot theoretic properties. It is an open ended question, and any references for this topic appreciated.
You can take a look to:

Peter Golbus, Robert W. McGrail, Tomasz Przytycki, Mary Sharac, and Aleksandar Chakarov. 2009. Tricolorable torus knots are NP-complete. In Proceedings of the 47th Annual Southeast Regional Conference (ACM-SE 47). ACM, New York, NY, USA, , Article 42 , 6 pages. 

Abstract: This work presents a method for associating a class of constraint satisfaction problems to a three-dimensional knot. Given a knot, one can build a knot quandle, which is generally an infinite free algebra. The desired collection of problems is derived from the set of invariant relations over the knot quandle, applying theory that relates finite algebras to constraint satisfaction problems. This allows us to develop notions of tractable and NP-complete quandles and knots. In particular, we show that all tricolorable torus knots and all but at most 2 non-trivial knots with 10 or fewer crossings are NP-complete.

and also to its seminal report: 

P. Golbus, R. W. McGrail, M. Merling, K. Ober, M. Sharac, and J. Wood. The class of constraint satisfaction problems over a knot. Technical Report number
BARD-CMSC-2008-01, Bard College, 2008.