Generalized chord diagrams and weight systems

TL;DR

This paper extends the theory of weight systems from chord diagrams to arbitrary permutations, providing new relations and analyzing properties of these generalized systems within the context of knot invariants.

Contribution

It introduces generalized relations for functions on permutations that extend Vassiliev's relations, and demonstrates their applicability to $gl$- and $so$-weight systems.

Findings

Weight systems from Lie algebras satisfy the new permutation relations.

Analysis of properties of generalized weight systems.

Study of related Hopf algebras of permutations.

Abstract

Weight systems are functions on chord diagrams satisfying Vassiliev's $4$-term relations. They originate in the theory of finite type knot invariants. Recent developments in understanding weight systems arising from Lie algebras are based on extending these weight systems from chord diagrams (which can be interpreted as involutions without fixed points, considered modulo cyclic shifts) to arbitrary permutations (also modulo cyclic shifts). We suggest relations for functions on permutations, which generalize Vassiliev's relations. We show that the $gl$- and $so$- weight systems satisfy these relations. We also analyze certain properties of these weight systems and study realted Hopf algebras of permutations.