Search for different links with the same Jones' type polynomials: Ideas from graph theory and statistical mechanics
Józef H. Przytycki
We describe in this talk three methods of constructing different links with the same Jones type invariant. All can be thought as generalizations mutation. The first combines satellite construction second uses notion rotant, taken from graph theory, third, invented by Jones, transplants into knot theory idea Yang-Baxter equation spectral parameter (idea employed Baxter solvable models statistical mechanics). extend result and relate it to Traczyk’s work on rotors links. also show further applications idea, e.g. 3-string solid torus. stress fact that ideas coming various areas mathematics (and theoretical physics) has been fruitfully used vice versa. 0. Introduction. Exactly ten year ago, at spring 1984, Vaughan introduced his (Laurent) polynomial invariant links, VL(t). He checked immediately distinguishes many knots which were not apart Alexander polynomial, right handed trefoil left knot, square granny knot; Fig. 0.1. 1991 Mathematics Subject Classification: Primary: 57M25. Secondary: 82B.
Banach Center Publications|arXiv (Cornell University)