Chord Diagrams and Coxeter Links

Link invariants from finite Coxeter racks

We study Coxeter racks over $\mathbb{Z}_n$ and the knot and link invariants they define. We exploit the module structure of these racks to enhance the rack counting invariants and give examples showing that these enhanced invariants are stronger than the unenhanced rack counting invariants.

Orientations of Chord Diagrams and Khovanov Homology

By adding or removing appropriate structures to Gauss diagram, one can create useful objects related to virtual links. In this paper few objects of this kind are studied: twisted virtual links generalizing virtual links; signed chord diagrams staying halfway between twisted virtual links and Kauffman bracket / Khovanov homology; alternatable virtual links intermediate between virtual and classical links. The most profound role here belongs to a structure that we dare to call orientation of chord diagram. Khovanov homology is generalized to oriented signed chord diagrams and links in oriented t…

从纽结到2-链环、弦图的映射以及增强瓦西里耶夫不变量的一种方法

Maps from knots to $2$-links, chord diagrams, and a way to enhance Vassiliev invariants

研究如何将瓦西里耶夫纽结不变量等推广到特定带框弦图，核心方法是探讨相关映射，主要贡献是提供推广方式。

通过弦图和涂鸦不变量表示Milnor三重环绕数

A representation of Milnor's triple linking number by chord diagrams and doodle invariants

本文借鉴二阶纽结不变量的分解思路，证明二阶的Milnor三重环绕数可表示为特定弦图计数与doodle不变量的组合，给出了该环绕数的新表示形式。

On a Basis for the Framed Link Vector Space Spanned by Chord Diagrams

In view of the result of Kontsevich, now often called ``the fundamental theorem of Vassiliev theory'', identifying the graded dual of the associated graded vector space to the space of Vassiliev invariants filtered by degree with the linear span of chord diagrams modulo the ``4T-relation'' (and in the unframed case, the ``1T-'' or ``isolated chord relation''), it is a problem of some interest to provide a basis for the space of chord diagrams modulo the 4T-relation. We construct the basis for the vector space spanned by chord diagrams with n chords and m distinguishable link components, modulo…

Poincare series of Klein groups, Coxeter polynomials, the Burau representation and Milnor invariants

For Poincare series of binary polyhedral groups and Coxeter polynomials there are obtained statements close to the Euclid algorithm and orthogonal polynomials theory: generalized Ebeling formula, decompositions into ramified continued fractions, Christoffel-Darboux identity, combinatorial formula. Known results about the factorization of the Alexander-Conway polynomial permit to connect Poincare series and Coxeter polynomials with the Burau representation and Milnor invariants of string links. One uses reconstractions of A'Campo links and Coxeter links. There is formulated a conjecture connect…

通过四维拓扑研究纽结与链环的交叉数

The crossing numbers of knots and links via 4D topology

该研究通过四维拓扑将链环交叉数与旋转带状环面纽结的弦指数关联，证明了链环连通和下交叉数的可加性。

Tree Diagrams for String Links

In previous work, the author defined the intersection graph of a chord diagram associated with string links (as in the theory of finite type invariants). In this paper, we classify the trees which can be obtained as intersection graphs of string link diagrams.

Filamentations for Virtual Links

In 2002, D. Hrencecin and L.H. Kauffman defined a filamentation invariant on oriented chord diagrams that may determine whether the corresponding flat virtual knot diagrams are non-trivial. A virtual knot diagram is non-classical if its related flat virtual knot diagram is non-trivial. Hence filamentations can be used to detect non-classical virtual knots. We extend these filamentation techniques to virtual links with more than one component. We also give examples of virtual links that they can detect as non-classical.

The Kontsevich Integral in Book Notation

We introduce a matrix representation of a chord on a tangle which leads us to representing tangle chord diagrams as stacks of matrices that we call books. We show that band sum moves, Reidemeister moves as well as orientation changes are implemented on \widetilde{Z}_f - a framed link invariant constructed from the Kontsevich integral that's well-behaved under band sum moves - by matrix congruences. We prove that being given the bare framed Kontsevich integral Z_f(L) in book notation for some unknown link L, we can determine what the link L is, as well as the projection of Z_f(L) in the origina…

A twisted link invariant derived from a virtual link invariant

Virtual knot theory is a generalization of knot theory which is based on Gauss chord diagrams and link diagrams on closed oriented surfaces. A twisted knot is a generalization of a virtual knot, which corresponds to a link diagram on a possibly non-orientable surface. In this paper, we discuss an invariant of twisted links which is obtained from the JKSS invariant of virtual links by use of double coverings. We also discuss some properties of double covering diagrams.

Weight systems and invariants of graphs and embedded graphs

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 constructions that are discussed in the paper are invariants of intersection graphs of chord diagrams that satisfy $4$-term relations for graphs and metrized Lie algebras. For the simplest nontrivial metrized Lie algebra $\mathfrak {sl}(2)$, we present recent results about the explicit form of generating functions for the values of the corresponding weight s…

The salient crossings of a crown diagram

A crown diagram of a smooth, closed oriented 4-manifold can be thought of as the projection of a link in the product of a closed surface and the circle, with chords in the circle direction connecting the strands of each crossing. This paper uses a straightforward assignment of integers to these chords to show that the smooth structures on the topological 4-manifold underlying a pair of Fintushel-Stern knot surgery 4-manifolds which are known to have the same Seiberg-Witten invariant are not isotopic. A natural question is to determine if the argument may be strengthened to show these manifolds…

Explicit special covers of alternating links

Given a prime, alternating link diagram, we build a special cover of the link complement whose degree is bounded by a factorial function of the crossing number. It follows that a subgroup of the link group of that index embeds into right-angled Artin and Coxeter groups. Corollaries of this result include a quantification of residual finiteness, control of the growth of Betti numbers in covers, and an explicit bound on the rank of a Z-module on which the link group acts faithfully.

Homotopy type of circle graphs complexes motivated by extreme Khovanov homology

It was proven by González-Meneses, Manchón and Silvero that the extreme Khovanov homology of a link diagram is isomorphic to the reduced (co)homology of the independence simplicial complex obtained from a bipartite circle graph constructed from the diagram. In this paper we conjecture that this simplicial complex is always homotopy equivalent to a wedge of spheres. In particular, its homotopy type, if not contractible, would be a link invariant and it would imply that the extreme Khovanov homology of any link diagram does not contain torsion. We prove the conjecture in many special cases and f…