Journal of Knot Theory and Its Ramifications

Joyce showed that for a classical knot $K$, the order of the involutory medial quandle is $\left|\mathrm{\text{det}}K\right|$. Generalizing Joyce’s result, we show that for a classical link $L$ of $\mu \ge 1$ components, the order of the involutory medial quandle is $\mu \left|\mathrm{\text{det}}L\right|/2^{\mu -1}$. In particular, $\mathrm{\text{IMQ}}(L)$ is infinite if and only if $detL=0$. We also relate $\mathrm{\text{IMQ}}(L)$ to several other link invariants.

We investigate embeddings from the set of long flat virtual knot diagrams to the set of long virtual knot diagrams so that we can construct invariants for long flat virtual knots. Also, we give properties and examples of several invariants for long flat virtual knots via these embeddings and invariants for long virtual knots.

We characterize the para-associative ternary quasi-groups (flocks) applicable to knot theory, and show which of these structures are isomorphic. We enumerate them up to order 64. We note that the operation used in knot-theoretic flocks has its non-associative version in extra loops. We use a group action on the set of flock colorings to improve the cocycle invariant associated with the knot-theoretic flock (co)homology.

We introduce some polynomial invariants for flat virtual links which are similar to the Jones–Kauffman polynomial, the Miyazawa polynomial and the arrow polynomial for virtual link diagrams, and we give several properties and examples.

A region crossing change at a region of a spatial-graph diagram is a transformation changing every crossing on the boundary of the region. In this paper, it is shown that every spatial graph consisting of theta-curves can be unknotted by region crossing changes.

We describe an effective method for simultaneously computing $d$-invariants of infinite families of Brieskorn spheres $\mathrm{\Sigma }(p,q,r)$ with $pq+pr-qr=1$.

We explore the relationship between two knot invariants, unknotting number and crosscap number. Specifically, we present two infinite families of knots with a specific crossing change that lowers the unknotting number while raising the crosscap number. One of these families is hyperbolic. This is a non-orientable parallel of Scharlemann and Thompson’s 1988 result for orientable surfaces.

The problem concerning which Gauss diagrams can be realized by knots is an old one and has been solved in several ways. In this paper, we present a direct approach to this problem. We show that the needed conditions for realizability of a Gauss diagram can be interpreted as follows “the number of exits = the number of entrances” and the sufficient condition is based on the Jordan curve theorem. Further, using matrices, we redefine conditions for realizability of Gauss diagrams and then we give an algorithm to construct meanders.

A braided fusion category is said to have Property F if the associated braid group representations factor through a finite group. We verify integral metaplectic modular categories have property F by showing these categories are group-theoretical. For the special case of integral categories ${𝒞}$ with the fusion rules of $\mathrm{\text{SO}}{(8)}_2$ we determine the finite group $G$ for which $\mathrm{\text{Rep}}(D^{\omega }G)$ is braided equivalent to ${𝒵}({𝒞})$. In addition, we determine the associated classical link invariant, an evaluation of the 2-variable Kauffman polynomial at a point.

We investigate representations of mapping class groups of surfaces that arise from the untwisted Drinfeld double of a finite group $G$, focusing on surfaces without marked points or with one marked point. We obtain concrete descriptions of such representations in terms of finite group data. This allows us to establish various properties of these representations. In particular, we show that they have finite images, and that for surfaces of genus at least $3$ their restriction to the Torelli group is non-trivial if and only if $G$ is non-abelian.