2007/04/10 by Sergei Chmutov, S. V. Chmutov, Сергей Константинович Ландо +1 · 57 citations
Mathematics · #Algebraic structures and combinatorial models #Combinatorics #Converse #Discrete mathematics #Finite type invariant #Geometric and Algebraic Topology #Geometry #Graph #Homotopy and Cohomology in Algebraic Topology #Intersection graph #Invariant (physics) #Invariant polynomial #Knot (papermaking) #Knot invariant #Knot theory #Line graph #Mathematical analysis #Mathematics #Polynomial #Pure mathematics #math.CO #math.GT #msc:05C62 #msc:57M15 #msc:57M25
paper · pdf · doi:10.2140/agt.2007.7.1579
published in Algebraic & Geometric Topology 7(3), 1579-1598 (Mathematical Sciences Publishers) · 13 pages, many figures
arxiv created 2007/04/10 · openalex publication_date 2007/12/17 · arxiv updated 2016/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.