Related papers: Generalized chord diagrams and weight systems

We describe recent achievements in the theory of weight systems, which are functions on chord diagrams satisfying so-called $4$-term relations. Our main attention is devoted to constructions of weight systems. The two main sources of these…

Weight systems are functions on chord diagrams satisfying so-called Vassiliev's $4$-term relations. They are closely related to finite type knot invariants introduced by Vassiliev. Certain weight systems can be derived from graph…

Each knot invariant can be extended to singular knots according to the skein rule. A Vassiliev invariant of order at most $n$ is defined as a knot invariant that vanishes identically on knots with more than $n$ double points. A chord…

Finite order invariants (Vassiliev invariants) of knots are expressed in terms of weight systems, that is, functions on chord diagrams satisfying the four-term relations. Weight systems have graph analogues, so-called $4$-invariants of…

In the present paper, we discuss a way of generalising Vassiliev knot invariants and weight systems to framed chord diagrams having framing 0 and 1.

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…

A weight system is a function on chord diagrams that satisfies the so-called four-term relations. Vassiliev's theory of finite-order knot invariants describes these invariants in terms of weight systems. In particular, there is a weight…

We prove that the partial-dual genus polynomial considered as a function on chord diagrams satisfies the four-term relation. Thus it is a weight system from the theory of Vassiliev knot invariants.

To a finite type knot invariant, a weight system can be associated, which is a function on chord diagrams satisfying so-called $4$-term relations. In the opposite direction, each weight system determines a finite type knot invariant. In…

We introduce a universal weight system (a function on chord diagrams satisfying the $4$-term relation) taking values in the ring of polynomials in infinitely many variables whose particular specializations are weight systems associated with…

We show that the adjacency matrices of the intersection graphs of chord diagrams satisfy the 2-term relations of Bar-Natan and Garoufalides [bg], and hence give rise to weight systems. Among these weight systems are those associated with…

To a singular knot K with n double points, one can associate a chord diagram with n chords. A chord diagram can also be understood as a 4-regular graph endowed with an oriented Euler circuit. L. Traldi introduced a polynomial invariant for…

Color Lie algebras, which were introduced by Ree, are a graded extension of Lie (super)algebras by an abelian group. We show that the color Lie algebras can be used to construct universal weight systems for knot invariants of of Vassiliev…

We compute the universal weight system for Vassiliev invariants coming from the Lie superalgebra gl(1|1) applying the construction of \cite{YB}. This weight system is a function from the space of chord diagrams to the center $Z$ of the…

We discuss Vassiliev invariants for virtual knots, expanding upon the theory of quantum virtual knot invariants developed in arXiv:1509.00578. In particular, following the theory of quantum invariants we work with 'rotational' virtual…

The "fundamental theorem of Vassiliev invariants" says that every weight system can be integrated to a knot invariant. We discuss four different approaches to the proof of this theorem: a topological/combinatorial approach following M.…

This paper contains a proof that chromatic weight systems, introduced by Chmutov, Duzhin and Lando, can be expressed in terms of weight systems associated with direct sums of the Lie algebras gl_n and so_n. As a consequence the Vassiliev…

Ideas of Rozansky and Witten, as developed by Kapranov, show that a complex symplectic manifold X gives rise to Vassiliev weight systems. In this paper we study these weight systems by using D(X), the derived category of coherent sheaves on…

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. The converse statement is easy…

We use Polyak's skein relation to give a new proof that Milnor's string link homotopy invariants are finite type invariants, and to develop a recursive relation for their associated weight systems. We show that the obstruction to the…