Journal of Knot Theory and Its Ramifications

We give a procedure that provides, given any unoriented pretzel link $L$, a braid whose closure is $L$. Moreover, we give a specific braid word in terms of the entries of the pretzel link.

We define a type of Niebrzydowski tribracket we call $\mathrm{\Delta }$-tribrackets and show that their counting invariants are invariants of link-homotopy. We further identify several classes of tribrackets whose counting invariants for oriented classical knots and links are trivial, including vertical tribrackets satisfying the center-involutory condition and horizontal tribrackets satisfying the late-commutativity condition. We provide examples and end with questions for future research.

The notion of a virtual link is a generalization of a classical link. Alexander numbering is a numbering of $\mathbb{Z}$ to arcs of a classical link diagram which is due to a numbering to disks of a complement of a link diagram in $S^2$. Every classical link diagram admits Alexander numbering. A virtual link diagram corresponds to a link diagram in a closed oriented surface. Some virtual link diagrams do not admit any Alexander numbering. If a virtual link diagram admits Alexander numbering, we call it an almost classical virtual link diagram. In this paper, we construct a map from the set of virtual link diagrams to that of almost classical virtual link diagrams. It induces a map from the set of virtual links to that of almost classical virtual links. Using this map, we define a kind of Goeritz matrix of virtual link diagrams and introduce invariants of virtual links.

In this paper, we provide two new congruences of the generalized Alexander polynomial $Z_L$ for periodic virtual links $L$. We use the Yang–Baxter state model of $Z_L$ introduced by Kauffman and Radford.

In this paper, we study the structures of homomorphisms induced from “Arc selection game” and “Arc freeze selection game” based on the knot theory. As a result, we give necessary and sufficient conditions for the games to be cleared, and show how many solutions exist. Also, we show that local moves on knot diagrams related with the games are unknotting operations.

A spiral knot or link diagram (introduced in [C. Adams, R. Hudson, R. Morrison, W. George, L. Starkston, S. Taylor and O. Turanova, The spiral index of knots, Math. Proc. Cambridge Philos. Soc. 149(2) (2010) 297–315]) is an oriented knot or link diagram where, when traversing through the planar diagram, the curvature does not change sign. An oriented knot or link type is called curly if it admits a spiral diagram with fewer maxima than the braid index of that knot or link type. In this paper, we exhibit families of curly knots and links.

In this paper, we attempt to find counterexamples to the conjecture that the ideal form of a knot, that which minimizes its contour length while respecting a no-overlap constraint, also minimizes the volume of the knot, as determined by its convex hull. We measure the convex hull volume of knots during the length annealing process, identifying local minima in the hull volume that arise due to buckling and symmetry breaking. We use $T(p,2)$ torus knots as an illustrative example of a family of knots whose locally minimal-length embeddings are not necessarily ordered by volume. We identify several knots whose central curve has a convex hull volume that is not minimized in the ideal configuration, and find that $8_{19}$ has a non-ideal global minimum in its convex hull volume even when the thickness of its tube is taken into account.

In this paper, we generalize the embedded homology groups of hypergraphs initially given in [S. Bressan, J. Li, S. Ren and J. Wu, The embedded homology of hypergraphs and applications, Asian J. Math. 23(3) (2019) 479–500] and study the relative embedded homology groups of hypergraph pairs. We prove some long exact sequences as well as a Mayer–Vietoris sequence for the relative embedded homology groups of hypergraph pairs. Moreover, we briefly discuss the two-dimensional persistence for the relative embedded homology groups of hypergraph pairs.

This paper presents a formula for the determinant of the twisted generalized hybrid weaving knot ${\widehat{Q}}_3(m_1,-m_2,n,l)$ which is a closed 3-braid. As a corollary, we prove Conjecture 2 given in Singh and Chbili [Nuclear Phys. B 980 (2022) 115800].