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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. [ABSTRACT FROM AUTHOR]

We create an invariant of virtual Y -oriented trivalent spatial graphs using colorings by virtual Niebrzydowski algebras. This paper generalizes the color invariants using virtual tribrackets and Niebrzydowski algebras by Nelson–Pico, and Graves-Nelson-T. We computed all tribrackets, Niebrzydowski algebras and virtual Niebrzydowski algebras of orders 3 and 4, and provide generative code for all data sets. [ABSTRACT FROM AUTHOR]

In this paper, the discovery of the Jones polynomial will be discussed, emphasizing the way in which it illustrated the remarkable unity between distinct parts of mathematics, each with its own language, but initially without a dictionary. [ABSTRACT FROM AUTHOR]

With the untimely passing of Patrick Dehornoy in September 2019, the world of mathematics lost a brilliant scholar who made profound contributions to set theory, algebra, topology, and even computer science and cryptography. And I lost a dear friend and a strong influence in the direction of my own research in mathematics. In this paper, I will concentrate on his remarkable discovery that the braid groups are left-orderable, and its consequences, and its strong influence on my own research. I'll begin by describing how I learned of his work. [ABSTRACT FROM AUTHOR]

Based on the presentation of the Kauffman bracket skein algebra of the thickened torus given by the third author in previous work [4], Frohman and Gelca established a complete description of the multiplicative operation leading to a famous product-to-sum formula. In this paper, we study the multiplicative structure of the Kauffman bracket skein algebra of the thickened four-holed sphere. We present an algorithm to compute the product of any two elements of the algebra, and give an explicit formula for some families of curves. We surmise that the algorithm has quasi-polynomial growth with respect to the number of crossings of a pair of curves. Further, we conjecture the existence of a positive basis for the algebra. [ABSTRACT FROM AUTHOR]