Narrow Results

In this paper, we give a definition of $\mathbb{Z}$-valued functions from the ambient isotopy classes of spherical/plane curves derived from chord diagrams, denoted by ${\sum }_i{\alpha }_i{x}_i$. Then, we introduce certain elements of the free $\mathbb{Z}$-module generated by the chord diagrams with at most $l$ chords, called relators of Type (I) ((SII), (WII), (SIII), or (WIII), respectively), and introduce another function ${\sum }_i{\alpha }_i{\tilde{x}}_i$ derived from ${\sum }_i{\alpha }_i{x}_i$. The main result (Theorem 1) shows that if ${\sum }_i{\alpha }_i{\tilde{x}}_i$ vanishes for the relators of Type (I) ((SII), (WII), (SIII), or (WIII), respectively), then ${\sum }_i{\alpha }_i{x}_i$ is invariant under the Reidemeister move of type RI (strong RII, weak RII, strong RIII, or weak RIII, respectively) that is defined in [N. Ito and Y. Takimura, $(1,2)$ and weak $(1,3)$ homotopies on knot projections, J. Knot Theory Ramifications22 (2013) 1350085 14 pp].

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, originally considered by Vassiliev, 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 link components, modulo 4T relations for n ≤ 5.

We characterize planar diagrams which may be divided into n arc embeddings in terms of their chord diagrams, generalizing a result of Taniyama for the case n = 2. Two algorithms are provided, one which finds a minimal arc embedding (in quadradic time in the number of crossings), and one which constructs a minimal subdiagram having the same arc number as D.

For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane-closed curve whose chord diagram contains the given chord diagram as a sub-chord diagram. For any generic immersion of the complete graph on six vertices to the plane, the sum of averaged invariants of all Hamiltonian plane curves in it is congruent to one quarter modulo one-half.

A ribbon chord diagram, or simply a chord diagram, of a ribbon surface-link in the 4-space is introduced. Links, virtual links and welded virtual links can be described naturally by chord diagrams with the corresponding moves, respectively. Some moves on chord diagrams are introduced by overseeing these special moves. Then the faithful equivalence on ribbon surface-links is stated in terms of the moves on chord diagrams. This answers questions by Nakanishi and Marumoto affirmatively. The faithful TOP-equivalence on ribbon surface-links derives the same result. By combining a previous result on TOP-triviality of a surface-knot, a ribbon surface-knot is DIFF-trivial if and only if the fundamental group is an infinite cyclic group. This corrects an erroneous proof in Yanagawa's old paper.

We consider framed chord diagrams, i.e. chord diagrams with chords of two types. It is well known that chord diagrams modulo $4$T-relations admit a Hopf algebra structure, where the multiplication is given by any connected sum with respect to the orientation. But in the case of framed chord diagrams a natural way to define a multiplication is not known yet. In this paper, we first define a new module ${{ℳ}}_2$ which is generated by chord diagrams on two circles and factored by $4$T-relations. Then we construct a “parity” map from the module of framed chord diagrams into ${{ℳ}}_2$ and a weight system on ${{ℳ}}_2$. Using the map and weight system we show that a connected sum for framed chord diagrams is not a well-defined operation. In the end of the paper we touch linear diagrams, the circle replaced by a directed line.

To each ribbon graph we assign a so-called $L$-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassiliev moves on ribbon graphs are reinterpreted nicely in the language of $L$-spaces, becoming changes of bases in this vector space. Finally, we define a bialgebra structure on the span of $L$-spaces, which is analogous to the 4-bialgebra structure on chord diagrams.

A revised proof is given to an assertion on chord diagrams of a ribbon surface-link.

A chord graph in $3$-space is constructed from a ribbon surface-link in $4$-space. In earlier papers, the three moves on the diagrams of chord graphs (namely, the chord diagrams) were introduced to describe the faithful equivalence of a ribbon surface-link. In this paper, it is shown that any two equivalent ribbon surface-links are faithfully equivalent, so that any chord diagrams of any two equivalent ribbon surface-links are connected by a finite number of these three moves. By combining it with an earlier result, it is shown that any two TOP-equivalent ribbon surface-links are equivalent. In other words, there is no exotic ribbon surface-link, generalizing an earlier result on the trivial ribbon surface-knot. In another earlier result, the three moves on the chord diagrams were modified into the 16 moves on the chord diagrams without base crossing. In this paper, further modified moves of the 16 moves on the chord diagrams without base crossing are also introduced to describe how the set of ribbon torus-links is produced from the set of welded virtual links.

In this paper, we introduce a distance ${\tilde{d}}_{w3}$ on the equivalence classes of spherical curves under deformations of type RI and ambient isotopies. We obtain an inequality that estimate its lower bound (Theorem 1). In Theorem 2, we show that if for a pair of spherical curves $P$ and $P^{\prime }$, ${\tilde{d}}_{w3}([P],[P^{\prime }])=1$ and $P$ and $P^{\prime }$ satisfy a certain technical condition, then $P^{\prime }$ is obtained from $P$ by a single weak RIII only. In Theorem 3, we show that if $P$ and $P^{\prime }$ satisfy other conditions, then $P^{\prime }$ is ambient isotopic to a spherical curve that is obtained from $P$ by a sequence of a particular local deformations, which realizes ${\tilde{d}}_{w3}([P],[P^{\prime }])$.

The knotting probability of an arc diagram is defined as the quadruplet of four kinds of finner knotting probabilities which are invariant under a reasonable deformation containing an isomorphism on an arc diagram. In a separated paper, it is shown that every oriented spatial arc admits four kinds of unique arc diagrams up to isomorphisms determined from the spatial arc and the projection, so that the knotting probability of a spatial arc is defined. The definition of the knotting probability of an arc diagram uses the fact that every arc diagram induces a unique chord diagram representing a ribbon 2-knot. Then the knotting probability of an arc diagram is set to measure how many nontrivial ribbon genus 2 surface-knots occur from the chord diagram induced from the arc diagram. The conditions for an arc diagram with the knotting probability 0 and for an arc diagram with the knotting probability 1 are given together with some other properties and some examples.

We give a definition of an integer-valued function ${\sum }_i{\alpha }_i{x}_i^{\ast }$ derived from arrow diagrams for the ambient isotopy classes of oriented spherical curves. Then, we introduce certain elements of the free $\mathbb{Z}$-module generated by the arrow diagrams with at most $l$ arrows, called relators of Type ($\check{\mathrm{\text{I}}}$) (($\check{\mathrm{\text{SII}}}$), ($\check{\mathrm{\text{WII}}}$), ($\check{\mathrm{\text{SIII}}}$) or ($\check{\mathrm{\text{WIII}}}$), respectively), and introduce another function ${\sum }_i{\alpha }_i{\tilde{x}}_i^{\ast }$ to obtain ${\sum }_i{\alpha }_i{x}_i^{\ast }$. One of the main results shows that if ${\sum }_i{\alpha }_i{\tilde{x}}_i^{\ast }$ vanishes on finitely many relators of Type ($\check{\mathrm{\text{I}}}$) (($\check{\mathrm{\text{SII}}}$), ($\check{\mathrm{\text{WII}}}$), ($\check{\mathrm{\text{SIII}}}$) or ($\check{\mathrm{\text{WIII}}}$), respectively), then ${\sum }_i{\alpha }_i{x}_i^{\ast }$ is invariant under the deformation of type RI (strong RII, weak RII, strong RIII or weak RIII, respectively). The other main result is that we obtain new functions of arrow diagrams with up to six arrows explicitly. This computation is done with the aid of computers.