Journal of Knot Theory and Its Ramifications

A parity is a labeling of the crossings of knot diagrams which is compatible with Reidemeister moves. We define the notion of parity for based matrices — algebraic objects are introduced by Turaev in his research on virtual strings. We present the reduced stable parity on based matrix which gives a new example of a parity of virtual knots.

An $n$-crossing projection of a link $L$ is a projection of $L$ onto a plane such that $n$ points on $L$ are superimposed on top of each other at every crossing. We prove that for all $k\in \mathbb{N}$ and all links $L$, the inequality

To better understand the fundamental quandle of a knot or link, it can be useful to look at finite quotients of the quandle. One such quotient is the $n$-quandle (or, when $n=2$, the involutory quandle). Hoste and Shanahan [J. Hoste and P. D. Shanahan, Links with finite n-quandles, Algebraic Geom. Topol. 17 (2017) 2807–2823.] gave a complete list of the links which have finite $n$-quandles; it remained to give explicit descriptions of these quandles. This has been done for several cases in [A. Crans, J. Hoste, B. Mellor and P. D. Shanahan, Finite n-qundles of torus and two-bridge links, J. Knot Theory Ramifications 28 (2019) 1950028; J. Hoste and P. D. Shanahan, Involutory quandles of (2, 2, r)-Montesinos links, J. Knot Theory Ramifications 26 (2017)]; in this work, we continue this project and explicitly describe the Cayley graphs for the finite involutory quandles of two-bridge links with an axis.

In this paper, we generalize unoriented handlebody-links to the twisted virtual case, obtaining Reidemeister moves for handlebody-links in ambient spaces of the form $\mathrm{\Sigma }\times [0,1]$ for $\mathrm{\Sigma }$ a compact closed 2-manifold up to stable equivalence. We introduce a related algebraic structure known as twisted virtual bikeigebras whose axioms are motivated by the twisted virtual handlebody-link Reidemeister moves. We use twisted virtual bikeigebras to define $X$-colorability for twisted virtual handlebody-links and define an integer-valued invariant ${\mathrm{\Phi }}_X^{\mathbb{Z}}$ of twisted virtual handlebody-links. We provide example computations of the new invariants and use them to distinguish some twisted virtual handlebody-links.

A finite subset $M\subset {ℝ}^d$ is basic, if for any function $f:M\to ℝ$ there exists a collection of functions $f_1,\dots ,f_d:ℝ\to ℝ$ such that for each element $(x_1,\dots ,x_d)\in M$, we have $f(x_1,\dots ,x_d)=f_1(x_1)+\dots +f_d(x_d)$. For certain finite sets, we prove a criterion for a set to be basic, and we show that it cannot be extended to the general case. In addition, we interpret the above criterion in terms of doubly-weighted graphs and give an estimation for the number of elements in certain basic and non-basic subsets.

For a connected cobordism $S$ between two knots $K_0,K_1$ in $S^3$, we establish an inequality involving the number of local maxima, the genus of $S$, and the torsion orders of ${\mathrm{\text{Kh}}}_t(K_0),{\mathrm{\text{Kh}}}_t(K_1)$, where ${\mathrm{\text{Kh}}}_t$ denotes Lee’s perturbation of Khovanov homology. This shows that the torsion order gives a lower bound for the band-unlinking number.

This paper mainly studies the minimum number of colorings for all non-trivially 19-colored diagrams of any 19-colorable knot K. By using some special Reidemeister move, we successfully eliminated 13 colors from 19 colors. It can be seen that for any 19-colorable knot K, at least six colors are enough to color K, that is, the minimum number of 19-colorable knot is six.

In the study of finite (Vassiliev’s) knot invariants, Vogel introduced the so-called universal parameters, belonging to the projective plane, in particular parametrizing the simple Lie algebras by Vogel’s table. Subsequently, a number of quantities, such as some universal knot invariants and (quantum) dimensions of simple Lie algebras, have been represented in terms of these parameters, i.e., in the universal form. We prove that at the points from Vogel’s table all known universal quantum dimension formulae are linearly resolvable, i.e., yield finite answers even if these points are singular, provided one restricts them to the appropriate lines. We show that the same phenomenon takes place for another three distinguished points in Vogel’s plane — $E_{7\frac{1}{2}}$, $X_1,$ and $X_2$. We also examine the same formulae on linear resolvability at the remaining 48 distinguished points in Vogel’s plane, which correspond to the so-called $Y$-objects. Among them, three points were found to be regular for all known quantum dimension formulae. Two of them happen to be sharing a remarkable similarity with the simple Lie algebras, namely, the universal formulae yield integer-valued outputs (dimensions) at the indicated points in the classical limit.