Journal of Knot Theory and Its Ramifications

Although it is known that the dimension of the Vassiliev invariants of degree three of long virtual knots is seven, the complete list of seven distinct Gauss diagram formulas has been unknown explicitly, where only one known formula was revised without proof. In this paper, we give seven Gauss diagram formulas to present the seven invariants of the degree three (Proposition 4). We further give $23$ Gauss diagram formulas of classical knots (Proposition 5). In particular, the Polyak–Viro Gauss diagram formula [M. Polyak and O. Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Not. 1994 (1994) 445–453] is not a long virtual knot invariant; however, it is included in the list of $23$ formulas. It has been unknown whether this formula would be available by arrow diagram calculus automatically. In consequence, as it relates to the conjecture of Goussarov-Polyak-Viro [Finite-type invariants of classical and virtual knots, Topology 39 (2000) 1045–1068, Conjecture 3.C], for all the degree three finite type long virtual knot invariants, each Gauss diagram formula is represented as those of Vassiliev invariants of classical knots (Theorem 1).

It is known that the category ${\mathcal{𝒞}}^H$ of nilpotent weight modules over the quantum group associated with the super Lie algebra $𝔰𝔩(2|1)$ is a relative pre-modular $G$-category. Its modified trace enables one to define an invariant of $3$-manifolds. In this paper, we show that the category ${\mathcal{𝒞}}^H$ is a relative modular $G$-category which allows one to constructing a family of non-semi-simple extended topological quantum field theories, and the family of TQFTs is anomaly free. The quantum group associated with $𝔰𝔩(2|1)$ is considered at odd roots of unity.

This paper resolves the problem of comparing the skein modules defined using the skein relations discovered by Melvin and Kirby that underlie the quantum group-based Reshetikhin–Turaev model for $\mathrm{\text{SU}}(2)$ Chern–Simons theory to the Kauffman bracket skein modules. Several applications and examples are presented.

We show that a decorated knot concordance ${𝒞}$ from $K_0$ to $K_1$ induces an $𝔽[U]$-module homomorphism

The singular braids with $n$ strands, $n\ge 3$, were introduced independently by Baez and Birman. It is known that the monoid formed by the singular braids is embedded in a group that is known as singular braid group, denoted by $S{G}_n$. There has been another generalization of braid groups, denoted by $GV{B}_n$, $n\ge 3$, which was introduced by Fang as a group of symmetries behind quantum quasi-shuffle structures. The group $GV{B}_n$ simultaneously generalizes the classical braid group, as well as the virtual braid group on $n$ strands.

We investigate the commutator subgroups $S{G}_n^{\prime }$ and $GV{B}_n^{\prime }$ of these generalized braid groups. We prove that $S{G}_n^{\prime }$ is finitely generated if and only if $n\ge 5$, and $GV{B}_n^{\prime }$ is finitely generated if and only if $n\ge 4$. Further, we show that both $S{G}_n^{\prime }$ and $GV{B}_n^{\prime }$ are perfect if and only if $n\ge 5$.

Given a knot in the 3-sphere, the non-orientable 4-genus or 4-dimensional crosscap number of a knot is the minimal first Betti number of non-orientable surfaces, smoothly and properly embedded in the 4-ball, with boundary the knot. In this paper, we calculate the non-orientable 4-genus of knots with crossing number 10.

The height of a knotoid is a measure of how far it is from being a knot. Here, we define the positive and negative parts of the height, and we prove that they determine the unsigned height. Some polynomial invariants provide lower bounds for the signed heights. We also study a set of sequences associated to a knotoid.