In this paper we study principal congruence link complements in $S^3$. It is known that there are only finitely many such link complements, and we make a start on enumerating them using a combination of theoretical methods and computer calculations with MAGMA.

This paper investigates the productivity of the Zariski topology ${ℨ}_G$ of a group $G$. If $𝒢=\{G_i\mid i\in I\}$ is a family of groups, and $G={\prod }_{i\in I}{G}_i$ is their direct product, we prove that ${ℨ}_G\subseteq {\prod }_{i\in I}{ℨ}_{G_i}$. This inclusion can be proper in general, and we describe the doubletons $𝒢=\{G_1,G_2\}$ of abelian groups, for which the converse inclusion holds as well, i.e., ${ℨ}_G={ℨ}_{G_1}\times {ℨ}_{G_2}$. If $e_2\in G_2$ is the identity element of a group $G_2$, we also describe the class $\Delta$ of groups $G_2$ such that $G_1\times \left\{e_2\right\}$ is an elementary algebraic subset of $G_1\times G_2$ for every group $G_1$. We show among others, that $\Delta$ is stable...

It is proved that two planes that are properly homotopic in a noncompact, orientable, irreducible 3-manifold that is not homeomorphic to ${ℝ}^3$ are isotopic. The end-reduction techniques of E. M. Brown and C. D. Feustal and M. G. Brin and T. L. Thickstun are used.

Two virtual link diagrams are homotopic if one may be transformed into the other by a sequence of virtual Reidemeister moves, classical Reidemeister moves, and self crossing changes. We recall the pure virtual braid group. We then describe the set of pure virtual braids that are homotopic to the identity braid.

This paper is devoted to new algebraic structures, called qualgebras and squandles. Topologically, they emerge as an algebraic counterpart of knotted 3-valent graphs, just like quandles can be seen as an "algebraization" of knots. Algebraically, they are modeled after groups with conjugation and multiplication/squaring operations. We discuss basic properties of these structures, and introduce and study the notions of qualgebra/squandle 2-cocycles and 2-coboundaries. Knotted 3-valent graph invariants...

This article establishes the algebraic covering theory of quandles. For every connected quandle Q with base point q ∈ Q, we explicitly construct a universal covering p: (Q̃,q̃̃) → (Q,q). This in turn leads us to define the algebraic fundamental group $\pi ₁(Q,q):=Aut\left(p\right)=g\in Adj{\left(Q\right)}^{\text{'}}|q^g=q$, where Adj(Q) is the adjoint group of Q. We then establish the Galois correspondence between connected coverings of (Q,q) and subgroups of π₁(Q,q). Quandle coverings are thus formally analogous to coverings of topological spaces, and resemble Kervaire’s...

We give criteria for framed links and 3-manifolds to be periodic of prime order. As applications we show that the Poincaré sphere is of periodicity 2, 3, 5 only and the Brieskorn sphere Σ(2,3,7) is of periodicity 2, 3, 7 only.

We propose a direction of study of nonabelian theta functions by establishing an analogy between the Weyl quantization of a one-dimensional particle and the metaplectic representation on the one hand, and the quantization of the moduli space of flat connections on a surface and the representation of the mapping class group on the space of nonabelian theta functions on the other. We exemplify this with the cases of classical theta functions and of the nonabelian theta functions for the gauge group...