Related papers: Quandle rings
In this paper, we introduce and initiate the study of quandle products of groups, a family of groups that includes graph products of groups, cactus groups, wreath products, and the recently introduced trickle groups. Our approach is…
This article is devoted to the investigation of wrap groups of connected fiber bundles. CW-groups associated with wrap groups are studied.
E. Bunch, P. Lofgren, A. Rapp and D. N. Yetter [J. Knot theory Ramifications (2010)] pointed out that by considering inner automorphism groups of quandles, one have a functor from the category of quandles with surjective homomorphisms to…
Tangles were originally introduced as a concept to formalize regions of high connectivity in graphs. In recent years, they have also been discovered as a link between structural graph theory and data science: when interpreting similarity in…
This paper studies an algebraic invariant of virtual knots called the biquandle. The biquandle generalizes the fundamental group and the quandle of virtual knots. The approach taken in this paper to the biquandle emphasizes understanding…
We define biquandle structures on a given quandle, and show that any biquandle is given by some biquandle structure on its underlying quandle. By determining when two biquandle structures yield isomorphic biquandles, we obtain a…
We show how the theory of tangles is equivalent to that of well-connected tangles. These are drawn on a surface with boundary, and equivalent via Reidemeister moves of a restricted kind. This reworking of the graphical foundations for link…
Consideration of certain properties of group rings and their ideals search
In this paper, we attempt to develop the Quillen Suslin theory for the algebraic fundamental group of a ring. We give a surjective group homomorphism from the algebraic fundamental group of the field of the real numbers to the group of…
We construct a family of rings. To a plane diagram of a tangle we associate a complex of bimodules over these rings. Chain homotopy equivalence class of this complex is an invariant of the tangle. On the level of Grothendieck groups this…
This article will explore the K- and L-theory of group rings and their applications to algebra, geometry and topology. The Farrell-Jones Conjecture characterizes K- and L-theory groups. It has many implications, including the Borel and…
The theory of quadratic forms and class numbers has connections to many classical problems in number theory. Recently, class numbers have appeared in the study of black holes in string theory. We describe this connection and raise questions…
In this paper, we give a characterization of homogeneous quandles with abelian inner automorphism groups. In particular, we show that such a quandle is expressed as an abelian extension of a trivial quandle. Our construction is a…
A quandle is an algebraic system originated in knot theory, and can be regarded as a generalization of symmetric spaces. The inner automorphism group of a quandle is defined as the group generated by the point symmetries (right…
We review recent developments in the theory of Thompson group representations related to knot theory.
In a previous paper, we introduce and study formal manifolds, which generalize smooth manifolds. In this paper, we establish the basic theory of formal Lie groups, which are group objects in the category of formal manifolds. In particular,…
Quandle Coloring Quivers are directed graph-valued invariants of classical and virtual knots and links associated to finite quandles. Quandle action quivers are subquivers of the full quandle coloring quiver associated to quandle actions by…
In this note we introduce the notion of bundle gerbe K-theory and investigate the relation to twisted K-theory. We provide some examples. Possible applications of bundle gerbe K-theory to the classification of D-brane charges in non-trivial…
Traditional clustering identifies groups of objects that share certain qualities. Tangles do the converse: they identify groups of qualities that often occur together. They can thereby discover, relate, and structure types: of behaviour,…
The principal aim of this article is to introduce and study n-valued quandles and n-corack bialgebras. We elaborate the basic methods of this theory, reproduce the coset construction known in the theory of n-valued groups. We also consider…