Mutant knots and intersection graphs

Chmutov, Sergei¹ ; Lando, Sergei²

¹The Ohio State University - Mansfield, 1680 University Drive, Mansfield OH 44906, USA
²Institute for System Research RAS and the Poncelet Laboratory, Independent University of Moscow, Bolshoy Vlasyevskiy Pereulok 11, Moscow 119002, Russia

Algebraic and Geometric Topology, Tome 7 (2007) no. 3, pp. 1579-1598

Résumé

We prove that if a finite order knot invariant does not distinguish mutant knots, then the corresponding weight system depends on the intersection graph of a chord diagram rather than on the diagram itself. Conversely, if we have a weight system depending only on the intersection graphs of chord diagrams, then the composition of such a weight system with the Kontsevich invariant determines a knot invariant that does not distinguish mutant knots. Thus, an equivalence between finite order invariants not distinguishing mutants and weight systems depending only on intersections graphs is established. We discuss the relationship between our results and certain Lie algebra weight systems.

Détail

DOI : 10.2140/agt.2007.7.1579

Keywords: mutant knots, Vassiliev invariants, intersection graphs, Lie algebra weight systems

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