Narrow Results

In this paper, we shall give an explicit Gauss diagram formula for the Kontsevich integral of links up to degree four. This practical formula enables us to actually compute the Kontsevich integral in a combinatorial way.

In this paper a classification of Reidemeister moves, which is the most refined, is introduced. In particular, this classification distinguishes some Ω₃-moves that only differ in how the three strands that are involved in the move are ordered on the knot.

To transform knot diagrams of isotopic knots into each other one must in general use Ω₃-moves of at least two different classes. To show this, knot diagram invariants that jump only under Ω₃-moves are introduced.

Knot diagrams of isotopic knots can be connected by a sequence of Reidemeister moves of only six, out of the total of 24, classes. This result can be applied in knot theory to simplify proofs of invariance of diagrammatical knot invariants. In particular, a criterion for a function on Gauss diagrams to define a knot invariant is presented.

The purpose of this note is to give a formula for the Casson knot invariant (the second coefficient of the Conway polynomial) of a 2-bridge knot, calculating without using a continued fraction expansion.

The purpose of this note is to give a formula for the Vassiliev invariant of type three for 2-bridge knots, calculating without using a continued fraction expansion.

Gauss diagram formulas are extensively used to study Vassiliev link invariants. Now we apply this approach to invariants of 3-manifolds, considering manifolds given by surgery on framed links in the 3-sphere. We study the lowest degree case — the celebrated Casson–Walker invariant of rational homology spheres. This paper is dedicated to a detailed treatment of 2-component links; a general case will be considered in a forthcoming paper. We present simple Gauss diagram formulas for the Casson–Walker invariant. This enables us to understand/separate its dependence on the unframed link and on the framings. We also obtain skein relations for the Casson–Walker invariant under crossing changes, and study its asymptotic behavior when framings tend to infinity. Finally, we present results of extensive computer calculations.

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.

Every virtual knot has a pair of groups called the upper and lower groups of the knot. In this paper we treat two topics on those groups: We first give a sufficient condition for a pair of groups which are realized by a certain virtual knot as the upper and lower groups. Secondly we give a necessary condition for a virtual knot to be almost classical in terms of the first elementary ideals of the groups.

Parity mappings (weights) from the chords of a Gauss diagram to the integers are defined. The parity of the chords is used to construct families of invariants of Gauss diagrams and consequently, virtual knots. Each family forms a set of degree n Vassiliev invariants for n ≥ 1.

A state of a virtual knot diagram D is a disjoint union of circles obtained from D by splicing all real crossings. For each integer n > 0, we denote by s_n(D) the number of states of D consisting of n circles. The state number of a virtual knot K is the minimal number of s_n(D) for all diagrams D of K and denoted by s_n(K). The aim of this paper is to study s_n(D) and s_n(K).

In this paper, we study the Gauss diagrams for periodic virtual knots (Theorem 3.1) and show that the virtual knot corresponding to a periodic Gauss diagram is equivalent to the periodic virtual knot whose factor is the virtual knot corresponding to the factor Gauss diagram (Theorem 3.2). We give formulae for the writhe polynomial and the affine index polynomial of periodic virtual knots by using those of factor knots (Corollary 4.2, Corollary 4.6).

When the signed weighted resolution set (or were-set) was defined as an invariant of pseudoknots, it was unknown whether this invariant was complete. Using the Gauss-diagrammatic invariants of pseudoknots introduced by Dorais et al., we show that the signed were-set cannot distinguish all non-equivalent pseudoknots. This goal is achieved through studying the effects of a flype-like local move on a pseudodiagram.

We introduce a sequence of polynomial invariants for Gauss diagrams which are one-to-one correspondence with virtual knot diagrams. Also, we give some properties of these polynomials and examples.

We introduce a family of polynomial invariants for knotoids. Via the corresponding Gauss diagrams for knotoids, we investigate the method to calculate these polynomial invariants. Also, we give some properties and examples.

A state of a virtual knot diagram $D$ is a collection of circles obtained from $D$ by splicing all the real crossings. For each integer $n\ge 1$, we denote by $s_n(D)$ the number of states of $D$ with $n$ circles. The $n$-state number ${\mathrm{\text{s}}}_n(K)$ of a virtual knot $K$ is the minimum number of $s_n(D)$ for $D$ of $K$. Let ${{𝒦}}_n(i)$ be the set of virtual knots $K$ with ${\mathrm{\text{s}}}_n(K)=i$ for an integer $i\ge 0$. In this paper, we study the finiteness of ${{𝒦}}_n(i)$. We determine the finiteness of ${{𝒦}}_n(1)$ for any $n\ge 1$ and ${{𝒦}}_1(i)$ for any $i\ge 0$.

We consider a problem of realizability of Gauss diagrams by closed plane curves where the plane curves have only double points of transversal self-intersection. We formulate the necessary and sufficient conditions for realizability. These conditions are based only on the parity of double and triple intersections of the chords in the Gauss diagram.

Satoh and Taniguchi introduced the $n$-writhe $J_n$ for each non-zero integer $n$, which is an invariant for virtual knots. The $n$-writhes give the coefficients of some polynomial invariants for virtual knots including the index polynomial, the odd writhe polynomial and the affine index polynomial. It is obvious that the virtualization of a real crossing is an unknotting operation for virtual knots. The values of $n$-writhes changed by some local moves are calculated. However for the virtualization, it is unknown. In this paper, we show that for any given non-zero integer $n$ and any given integer $N$, there exists a virtual knot whose unknotting number by the virtualization is one and the value of the $n$-writhe equals $N$. Namely, the virtualization of a real crossing changes the value of $n$-writhe by any given integer $N$.

For a virtual knot $K$ and an integer $r\ge 0$, the $r$-covering $K^{(r)}$ is defined by using the indices of chords on a Gauss diagram of $K$. In this paper, we prove that for any finite set of virtual knots $J_0,J_2,J_3,\dots ,J_m$, there is a virtual knot $K$ such that $K^{(r)}=J_r(r=0\mathrm{\text{ and }}2\le r\le m)$, $K^{(1)}=K$, and otherwise $K^{(r)}=J_0$.

We prove that the Gaussian parity on free two-dimensional knots is universal.

Twisted knot theory, introduced by Bourgoin, is a generalization of virtual knot theory. It is well known that any virtual knot can be deformed into a trivial knot by a finite sequence of generalized Reidemeister moves and two forbidden moves $F1$ and $F2$. Similarly, we show that any twisted knot also can be deformed into a trivial knot or a trivial knot with a bar by a finite sequence of extended Reidemeister moves and three forbidden moves $T4$, $F1$ (or $F2$) and $F3$ (or $F4$).

A state model for Kauffman polynomial of Dubrovnik-version is described using Gauss diagrams. Based on the state model, the Gauss diagram formulae for Vassiliev invariants are given from the coefficients of Kauffman polynomial following the method of Chmutov and Polyak. Some arrow diagram identities are given to simplify the Gauss diagram formulae of order 3, which give Polyak–Viro and Chmutov–Polyak formulae for the Vassiliev invariant of order 3. The models of Kauffman polynomial and HOMFLY-PT polynomial give different Gauss diagram expressions when specializing in Jones polynomial.