Journal of Knot Theory and Its Ramifications

Biquandle brackets are a type of quantum enhancement of the biquandle counting invariant for oriented knots and links, defined by a set of skein relations with coefficients which are functions of biquandle colors at a crossing. In this paper, we use biquandle brackets to enhance the biquandle counting matrix invariant defined by the first two authors in (N. Gügümcü and S. Nelson, Biquandle coloring invariants of knotoids, J. Knot Theory Ramif. 28(4) (2019) 1950029). We provide examples to illustrate the method of calculation and to show that the new invariants are stronger than the previous ones. As an application we show that the trace of the biquandle bracket matrix is an invariant of the virtual closure of a knotoid.

Let ${{𝒰}}_n$ be the set of un-oriented and rational links with crossing number $n$, a precise formula for $\left|{{𝒰}}_n\right|$ was obtained by Ernst and Sumners in 1987. In this paper, we study the enumeration problem of oriented rational links. Let ${\mathrm{\Lambda }}_n$ be the set of oriented rational links with crossing number $n$ and let ${\mathrm{\Lambda }}_n(d)$ be the set of oriented rational links with crossing number $n$ ($n\ge 2$) and deficiency $d$. In this paper, we derive precise formulas for $\left|{\mathrm{\Lambda }}_n\right|$ and $|{\mathrm{\Lambda }}_n(d)|$ for any given $n$ and $d$ and show that

Solving tangle equations is deeply connected with studying enzyme action on DNA. The main goal of this paper is to solve the system of tangle equations $N(O+X_1)=b_1$ and $N(O+X_2)=b_2\#b_3$, where $X_1$ and $X_2$ are rational tangles, and $b_i$ is a 2-bridge link, for $i=1,2,3$, with $b_2$ and $b_3$ nontrivial. We solve this system of equations under the assumption $\tilde{O}$, the double branched cover of $O$, is not hyperbolic, i.e. $O$ is not $\pi$-hyperbolic. Besides, we also deal with tangle equations involving 2-bridge links only under the assumption $O$ is an algebraic tangle.

This paper is concerned with the problem of asphericity of 2-complexes associated to labeled oriented trees (LOTs), which are combinatorial models of ribbon disc complements. The concept of good enumeration for LOTs is introduced and applied to the study of asphericity and local indicability. We describe a large class of LOTs, which includes some families of non-injective LOTs of any diameter and complexity, that present locally indicable groups.

In this paper, we consider decompositions of 3-manifolds with three handlebodies. We classify such decompositions of the 3-sphere and lens spaces with small genera. These decompositions admit operations called stabilizations. We also determine whether these decompositions are obtained by stabilization from another decomposition.

Let $n\ge 3$. In this paper, we show that for any abelian subgroup $G$ of $s_n$ the crystallographic group $B_n/[P_n,P_n]$ has Bieberbach subgroups ${\mathrm{\Gamma }}_G$ with holonomy group $G$. Using this approach, we obtain an explicit description of the holonomy representation of the Bieberbach group ${\mathrm{\Gamma }}_G$. As an application, when the holonomy group is cyclic of odd order, we study the holonomy representation of ${\mathrm{\Gamma }}_G$ and determine the existence of Anosov diffeomorphisms and Kähler geometry of the flat manifold ${{𝒳}}_{{\mathrm{\Gamma }}_G}$ with fundamental group the Bieberbach group ${\mathrm{\Gamma }}_G\le B_n/[P_n,P_n]$.

We introduce two notions of quandle polynomials for $G$-families of quandles: the quandle polynomial of the associated quandle and a $G$-family polynomial with coefficients in the group ring of $G$. As an application we define image subquandle polynomial enhancements of the $G$-family counting invariant for trivalent spatial graphs and handlebody-links. We provide examples to show that the new enhancements are proper.

We give a presentation for a non-split compact surface embedded in the 3-sphere $S^3$ by using diagrams of spatial trivalent graphs equipped with signs and we define Reidemeister moves for such signed diagrams. We show that two diagrams of embedded surfaces are related by Reidemeister moves if and only if the surfaces represented by the diagrams are ambient isotopic in $S^3$.

Pseudo-alternating knots and links are defined constructively via their Seifert surfaces. By performing Murasugi sums of primitive flat surfaces, such a knot or link is obtained as the boundary of the resulting surface. Conversely, it is hard to determine whether a given knot or link is pseudo-alternating or not. A major difficulty is the lack of criteria to recognize whether a given Seifert surface is decomposable as a Murasugi sum.

In this paper, we propose a new idea to identify non-pseudo-alternating knots. Combining with the uniqueness of minimal genus Seifert surface obtained through sutured manifold theory, we demonstrate that two infinite classes of pretzel knots are not pseudo-alternating.

We have the generating function which determines the number of $2$-bridge knot groups admitting epimorphisms onto the knot group of a given $2$-bridge knot, in terms of crossing number. In this paper, we will refine this formula by taking into account genus as well as crossing number. Next, we determine the number of epimorphisms between fibered $2$-bridge knot groups. Moreover, we discuss degree one maps and $2$-bridge knots with unknotting number one.