Journal of Knot Theory and Its Ramifications

The classical knot recognition problem is the problem of determining whether the virtual knot represented by a given diagram is classical. We prove that this problem is in NP, and we give an exponential time algorithm for the problem.

In [B. Kwon and S. Kang, Rectangle conditions and families of 3-bridge prime knots, Topol. Appl. 291 (2021) 107453], using the set $EA{T}^k$ of all essential alternating rational 3-tangles for positive integer $k$, the authors showed that all knot diagrams in the numerator closure set $C_N(EA{T}^{2l+1})$ and the denominator closure set $C_D(EA{T}^{2l+2})$ with $l>0$ are 3-bridge prime knot diagrams. In this paper, for $n>4$ we construct a set $AA{T}_4^n$ of additions of alternating rational tangles in $EA{T}^4$. The set $AA{T}_4^n$ generalizes $EA{T}^k$ and contains it as a subset for some $k$. We show that any closure set $C(AA{T}_4^n)$ on $AA{T}_4^n$ so that the resulting diagrams are reduced and alternating knot diagrams represent alternating 3-bridge prime knot diagrams. Since a tangle diagram in $AA{T}_4^{n+1}$ is constructed inductively from a tangle diagram in $AA{T}_4^n$ by adding only one crossing positively, the result of this paper supports the conjecture that 3-bridge property is preserved under one-crossing alternating addition positively to alternating 3-bridge knots in 3-bridge representations.

For immersed surfaces in the four-space, we have a generating set of the Swenton–Hughes–Kim–Miller spatial moves that relate singular banded diagrams of ambient isotopic immersions of those surfaces. We also have Yoshikawa–Kamada–Kawauchi–Kim–Lee planar moves that relate marked graph diagrams of ambient isotopic immersions of those surfaces. One can ask if the former moves form a minimal set and if the latter moves form a generating set. In this paper, we derive a minimal generating set of spatial moves for diagrams of surfaces immersed in the four-space, which translates into a generating set of planar moves. We also show that the complements of two equivalent immersed surfaces can be transformed one another by a Kirby calculus not requiring the 1-1-handle or 2-1-handle slides. We also discuss the fundamental group of the immersed surface-link complement in the four-space and a quandle coloring invariant of an oriented immersed surface-link.

Torsion in the concordance group $\mathcal{𝒞}$ of knots in $S^3$ can be studied with the algebraic concordance group ${\mathcal{𝒢}}^{𝔽}$. Here, $𝔽$ is a field of characteristic $\chi (𝔽)\ne 2$. The group ${\mathcal{𝒢}}^{𝔽}$ was defined by Levine, who also obtained an algebraic classification when $𝔽=\mathbb{Q}$. While the concordance group $\mathcal{𝒞}$ is abelian, it embeds into the non-abelian virtual knot concordance group $\mathcal{𝒱}\mathcal{𝒞}$. It is unknown if $\mathcal{𝒱}\mathcal{𝒞}$ admits non-classical finite torsion. Here, we define the virtual algebraic concordance group ${\mathcal{𝒱}\mathcal{𝒢}}^{𝔽}$ for Seifert surfaces of almost classical knots. This is an analogue of ${\mathcal{𝒢}}^{𝔽}$ for homologically trivial knots in thickened surfaces $\mathrm{\Sigma }\times [0,1]$, where $\mathrm{\Sigma }$ is closed and oriented. The main result is an algebraic classification of ${\mathcal{𝒱}\mathcal{𝒢}}^{𝔽}$. A consequence of the classification is that ${\mathcal{𝒢}}^{\mathbb{Q}}$ embeds into ${\mathcal{𝒱}\mathcal{𝒢}}^{\mathbb{Q}}$ and ${\mathcal{𝒱}\mathcal{𝒢}}^{\mathbb{Q}}$ contains many nontrivial finite-order elements that are not algebraically concordant to any classical Seifert matrix. For $𝔽=\mathbb{Z}/2\mathbb{Z}$, we give a generalization of the Arf invariant.

We enhance the psyquandle counting invariant for singular knots and pseudoknots using quivers analogously to quandle coloring quivers. This enables us to extend the in-degree polynomial invariants from quandle coloring quiver theory to the case of singular knots and pseudoknots. As a side effect, we obtain biquandle coloring quivers and in-degree polynomial invariants for classical and virtual knots and links.

In this paper, we consider two applications of the pentagon equation. The first deals with actions of flips on edges of triangulations labeled by rational functions in some variables. The second can be formulated as a system of linear equations with variables corresponding to triangles of a triangulation. The general method says that if there is some general data (say, edge lengths or areas) associated with states (say, triangulations) and a general data transformation rule (say, how lengths or areas are changed under flips) then after returning to the initial state we recover the initial data.