A few weight systems arising from intersection graphs

Mutant knots and intersection graphs

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 and well known. We discuss relationship between our results and certain Lie algebra weight systems.

Projections in the curve complex arising from covering maps

Let $P : Σ\rightarrow S$ be a finite degree covering map between surfaces. Rafi and Schleimer show that there is an induced quasi-isometric embedding $Π: \mathcal{C}(S) \rightarrow \mathcal{C}(Σ)$ between the associated curve complexes. We define an operation on curves in $\mathcal{C}(Σ)$ using minimal intersection number conditions and prove that it approximates a nearest point projection to $Π(\mathcal{C}(S))$. We also approximate hulls of finite sets of vertices in the curve complex, together with their corresponding nearest point projections, using intersection numbers.

Intersection Graphs for String Links

We extend the notion of intersection graphs for knots in the theory of finite type invariants to string links. We use our definition to develop weight systems for string links via the adjacency matrix of the intersection graph, and show that these weight systems are related to the weight systems induced by the Conway and Homfly polynomials.

关于交集图为完全二部图的弦图的 $\mathfrak{sl}_3$ 权系统取值

On values of $\mathfrak{sl}_3$ weight system on chord diagrams whose intersection graph is complete bipartite

本文针对交集图为完全二部图 $K_{2,n}$ 的弦图，计算了 $\mathfrak{sl}_3$ 权系统的取值，该权系统由李代数构造并满足 4 项关系。

On a weight system conjecturally related to $\mathfrak{sl}_2$

We introduce a new series $R_k$, $k=2,3,4,\dots$, of integer valued weight systems. The value of the weight system $R_k$ on a chord diagram is a signed number of cycles of even length $2k$ in the intersection graph of the diagram. We show that this value depends on the intersection graph only. We check that for small orders of the diagrams, the value of the weight system $R_k$ on a diagram of order exactly $2k$ coincides with the coefficient of $c^k$ in the value of the $\mathfrak{sl}_2$-weight system on the projection of the diagram to primitive elements.