Journal of Knot Theory and Its Ramifications

Random hypergraphs and random simplicial complexes on finite vertices were studied by [M. Farber, L. Mead and T. Nowik, Random simplicial complexes, duality and the critical dimension, J. Topol. Anal. 41(1) (2022) 1–32]. The map algebra on random sub-hypergraphs of a fixed simplicial complex, which detects relations between random sub-hypergraphs and random simplicial sub-complexes, was studied by the authors of this paper. In this paper, we study the map algebra on random sub-hypergraphs of a fixed hypergraph. We give some algorithms generating random hypergraphs and random simplicial complexes by considering the actions of the map algebra on the space of probability distributions. We prove some Künneth-type formulae for random hypergraphs and random simplicial complexes on finite vertices.

In this paper, we study conditions for $2$-component pretzel links to be concordant to the trivial link.

The $\mathrm{{\rm Y}}$ invariant is a concordance invariant using knot Floer homology. Földvári [The knot invariant $\mathrm{{\rm Y}}$ using grid homologies, J. Knot Theory Ramifications 30(7) (2021) 2150051] gives a combinatorial restructure of it using grid homology. We extend the combinatorial $\mathrm{{\rm Y}}$ invariant for balanced spatial graphs. Regarding links as spatial graphs, we give an upper and lower bound for the $\mathrm{{\rm Y}}$ invariant when two links are connected by a cobordism. Also, we show that the combinatorial $\mathrm{{\rm Y}}$ invariant is a concordance invariant for knots.

In this work, we find a closed form formula for the braid index of an $n$-bridge braid, a class of positive braid knots which simultaneously generalizes torus knots, 1-bridge braids, and twisted torus knots. Our proof is elementary, effective, and self-contained, and partially recovers work of Birman–Kofman. Along the way, we show that the disparate definitions of twisted torus knots in the literature agree.

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface (called a surface-link) embedded in 4-space. In this paper, we investigate surface-links by using charts. In [T. Nagase and A. Shima, The structure of a minimal $n$-chart with two crossings I: Complementary domains of ${\mathrm{\Gamma }}_1\cup {\mathrm{\Gamma }}_{n-1}$, J. Knot Theory Ramifactions 27(14) (2018) 1850078; T. Nagase and A. Shima, The structure of a minimal $n$-chart with two crossings II: Neighbourhoods of ${\mathrm{\Gamma }}_1\cup {\mathrm{\Gamma }}_{n-1}$, Revista de la Real Academia de Ciencias Exactas, Fiskcas y Natrales. Serie A. Math. 113 (2019) 1693–1738, arXiv:1709.08827v2] we gave an enumeration of the charts with two crossings. In particular, there are two classes for 4-charts with two crossings and eight black vertices. The first class represents surface-links each of which is connected. The second class represents surface-links each of which is exactly two connected components. In this paper, by using quandle colorings, we shall show that the charts in the second class represent different surface-links.

Several distinct Garside monoids having torus knot groups as groups of fractions are known. For $n,m\ge 2$ two coprime integers, we introduce a new Garside monoid ${ℳ}(n,m)$ having as Garside group the $(n,m)$-torus knot group, thereby generalizing to all torus knot groups a construction that we previously gave for the $(n,n+1)$-torus knot group. As a byproduct, we obtain new Garside structures for the braid groups of a few exceptional complex reflection groups of rank two. Analogous Garside structures are also constructed for a few additional braid groups of exceptional complex reflection groups of rank two which are not isomorphic to torus knot groups, namely, for $G_{13}$ and for dihedral Artin groups of even type.

In this paper, we introduce a polynomial invariant of virtual knots by using weights of crossings. It is motivated from the writhe polynomial and the affine index polynomial for virtual knots. The new polynomial invariant is distinct from these polynomials and is a Vassiliev invariant of degree 1. By using the polynomial invariant we give a lower bound for the Gordian distance of two homotopic virtual knots.

The finite topological quandles can be represented as $n\times n$ matrices, recently defined by Nelson and Wong. In this paper, we first study the finite topological quandles and we show how to use these matrices to distinguish all isomorphism classes of finite topological quandles for a given cardinality $n$. As an application, we classify finite topological quandles with up to four elements.