Search for different links with the same Jones' type polynomials: Ideas from graph theory and statistical mechanics

Abstract

We describe in this talk three methods of constructing different links with the same Jones type invariant. All three can be thought as generalizations of mutation. The first combines the satellite construction with mutation. The second uses the notion of rotant, taken from the graph theory, the third, invented by Jones, transplants into knot theory the idea of the Yang-Baxter equation with the spectral parameter (idea employed by Baxter in the theory of solvable models in statistical mechanics). We extend the Jones result and relate it to Traczyk's work on rotors of links. We also show further applications of the Jones idea, e.g. to 3-string links in the solid torus. We stress the fact that ideas coming from various areas of mathematics (and theoretical physics) has been fruitfully used in knot theory, and vice versa. (This is the detailed version of the talk given at the Banach Center Colloquium, Warsaw, Poland, March 24, 1994: ``W poszukiwaniu nietrywialnego wezla z trywialnym wielomianem Jonesa: grafy i mechanika statystyczna")