Abstract
We give necessary and sufficient conditions for a weight system on multiloop chord diagrams to be obtainable from a metrized Lie algebra representation, in terms of a bound on the ranks of associated connection matrices. Here a multiloop chord diagram is a graph with directed and undirected edges so that at each vertex, precisely one directed edge is entering and precisely one directed edge is leaving, and each vertex is incident with precisely one undirected edge. Weight systems on multiloop chord diagrams yield the Vassiliev invariants for knots and links. The k-th connection matrix of a function f on the collection of multiloop chord diagrams is the matrix with rows and columns indexed by k-labeled chord tangles and with entries equal to the f value on the join of the tangles.
Highlights
We describe our results for those familiar with the basic theory of weight systems on chord diagrams
Bar-Natan [1,2] and Kontsevich [10] have shown that any finite-dimensional representation ρ of a metrized Lie algebra g yields a weight system φgρ on chord diagrams—more generally on multiloop chord diagrams
A partial result in this direction was given by Kodiyalam and Raghavan [9]: let g and g be n-dimensional semisimple Lie algebras, with the Killing forms as metrics, and let ρ and ρ be the adjoint representations; if φgρ = φgρ on chord diagrams, g = g
Summary
We describe our results for those familiar with the basic theory of weight systems on chord diagrams (cf. [4]). Bar-Natan [1,2] and Kontsevich [10] have shown that any finite-dimensional representation ρ of a metrized Lie algebra g yields a weight system φgρ on chord diagrams—more generally on multiloop chord diagrams. A k-tangle is a multiloop chord diagram with k directed edges entering it, labeled 1, . Our proof of the reverse implications is based on some basic results of algebraic geometry (Nullstellensatz), invariant theory (first and second fundamental theorem, closed orbit theorem), and (implicitly through [13]) the representation theory of the symmetric group. A partial result in this direction was given by Kodiyalam and Raghavan [9]: let g and g be n-dimensional semisimple Lie algebras, with the Killing forms as metrics, and let ρ and ρ be the adjoint representations; if φgρ = φgρ on (one-loop) chord diagrams, g = g
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