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Mutant knots and intersection graphs

Sergei V Chmutov and Sergei K Lando

Algebraic & Geometric Topology 7 (2007) 1579–1598

arXiv: math.GT/0704.1313


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.

mutant knots, Vassiliev invariants, intersection graphs, Lie algebra weight systems
Mathematical Subject Classification 2000
Primary: 57M25, 57M15
Secondary: 57M27, 05C10
Received: 16 May 2007
Revised: 14 September 2007
Accepted: 17 September 2007
Published: 17 December 2007
Sergei V Chmutov
The Ohio State University - Mansfield
1680 University Drive
Mansfield OH 44906
Sergei K Lando
Institute for System Research RAS and the Poncelet Laboratory
Independent University of Moscow
Bolshoy Vlasyevskiy Pereulok 11
Moscow 119002