Alternating Knots and Links Theory
  The altenating knots, links and twists projected on the S_2 sphere are identified with the phase Space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossing points, the edges, to the stable and unstable manifolds, connecting the saddles. Each facxe is then oriented in one of two different senses determined by the direction of these manifolds. The associated matrix to that connected graph is decomposed in the sum of two permutations. The separation is unique for knots and is not for links. The characteristic polynomial of these graphs was computed for different families of knots in terms of families of Chebyshev polynomials. 
Eduardo Pina