Journal of Knot Theory and Its Ramifications

A multi-variable polynomial invariant for knotoids and linkoids, which is an enhancement of the bracket polynomial for knotoids introduced by Turaev, is given by using the concept of a pole diagram which originates in constructing a virtual link invariant. Several features of the polynomial are revealed.

For a polygon in the $d$-dimensional Euclidean space, we give two kinds of normalizations of its $m$th midpoint polygon by a homothetic transformation and an affine transformation, respectively. As $m$ goes to infinity, the normalizations will approach “regular” polygons inscribed in an ellipse and a generalized Lissajous curve, respectively, where the curves may be degenerate.

The most interesting case is when $d=3$, where polygons with all its $m$th midpoint polygons knotted are discovered and discussed. Such polygonal knots can be seen as a discrete version of the Lissajous knots.

We construct a group ${\mathrm{\Gamma }}_n^4$ corresponding to the motion of points in ${ℝ}^3$ from the point of view of Delaunay triangulations. We study homomorphisms from pure braids on $n$ strands to the product of copies of ${\mathrm{\Gamma }}_n^4$. We will also study the group of pure braids in ${ℝ}^3$, which is described by a fundamental group of the restricted configuration space of ${ℝ}^3$, and define the group homomorphism from the group of pure braids in ${ℝ}^3$ to ${\mathrm{\Gamma }}_n^4$. At the end of this paper, we give some comments about relations between the restricted configuration space of ${ℝ}^3$ and triangulations of the 3-dimensional ball and Pachner moves.

We generalize three invariants, first discovered by Henrich, to the long and/or framed virtual knot case. These invariants are all finite-type invariants of order one, and include a universal invariant. The generalization will require us to extend the notion of a based matrix of a virtual string, first introduced by Turaev and later generalized by Henrich, to the long and framed cases.

A symmetric union in the 3-space ${ℝ}^3$ is obtained from a knot in ${ℝ}^3$ and its mirror image, which is symmetric with respect to a 2-plane in ${ℝ}^3$, by connecting them with some vertical 2-string tangles with twists and a horizontal 2-string tangle with no twists along the plane. We introduce the minimal twisting number of a symmetric union and show that if a knot $K$ is a composite symmetric union with minimal twisting number one, then $K$ has a non-trivial knot and its mirror image as connected summands. As an application, we study the minimal twisting number of a symmetric union and show that there exist infinitely many symmetric unions with minimal twisting number two.

We study 2-adjacency between a classical (3-strand) pretzel knot and the trefoil knot or the figure-eight knot by using the early results about classical pretzel knots and their polynomials and elementary number theory. We show that except for the trefoil knot or the figure-eight knot, a nontrivial classical pretzel knot is not 2-adjacent to either of them, and vice versa.

We consider classification of the oriented ribbon 2-knots presented by virtual arcs with up to four crossings. We show the difference by the 2-fold branched covering space, the Alexander polynomial, the number of representations of the knot group to SL$(2,𝔽)$, $𝔽$ a finite field, and the twisted Alexander polynomial.

We first prove the following: Let $p\ge 2$ and $p\in \mathbb{N}$. Let $K$ and $J$ be closed, oriented, $(2p+1)$-dimensional $(p-1)$-connected, simple submanifolds of $S^{2p+3}$. Then $K$ and $J$ are isotopic if and only if a Seifert matrix associated with a simple Seifert hypersurface for $K$ is ${(-1)}^p$-$S$-equivalent to that for $J$. We also discuss the $p=1$ case. This result implies one of our main results: Let $\mu \in \mathbb{N}$. A 1-link $A$ is pass-equivalent to a 1-link $B$ if and only if $A{\otimes }^{\mu }\mathrm{\text{Hopf}}$ is $(2\mu +1,2\mu +1)$-pass-equivalent to $B{\otimes }^{\mu }\mathrm{\text{Hopf}}$. Here, $J\otimes K$ means the knot product of $J$ and $K$, and $J{\otimes }^{\mu }K$ means $J\underset{\mu }{\underbrace{\otimes K\cdots \otimes K}}$. See the body of the paper for the definition of knot products. It also implies the other main results: We strengthen the authors’ old result that two-fold cyclic suspension commutes with the performance of the twist move for spherical $(2k+1)$-knots. See the body for the precise statement. Furthermore, it implies the following: Let $p\ge 2$ and $p\in \mathbb{N}$. Let $K$ be a closed oriented $(2p+1)$-submanifold of $S^{2p+3}$. Then $K$ is a Brieskorn submanifold if and only if $K$ is $(p-1)$-connected, simple and has a $(p+1)$-Seifert matrix associated with a simple Seifert hypersurface that is ${(-1)}^p$-$S$-equivalent to a $KN$-type (see the body of the paper for a definition). We also discuss the $p=1$ case.

In this paper, we discuss algebraic, combinatorial and topological properties of singular virtual braids. On the algebraic side, we state the relations between classical and singular virtual objects, in addition we discuss a Birman-like conjecture for the virtual case. On the topological and combinatorial side, we prove that there is a bijection between singular abstract braids, singular horizontal Gauss diagrams up to a certain equivalence relation, and singular virtual braids, in particular using singular horizontal Gauss diagrams we obtain a presentation of the singular pure virtual braid monoid.

We describe a presentation for the augmented fundamental rack of a link in the lens space $L(p,1)$. Using this presentation, the (enhanced) counting rack invariants that have been defined for the classical links are applied to the links in $L(p,1)$. In this case, the counting rack invariants also include the information about the action of ${\pi }_1(L(p,1))$ on the augmented fundamental rack of a link.