Related papers: Partially abelian representations of knot groups

Cannon, Swenson, and others have proved numerous theorems about subdivision rules associated to hyperbolic groups with a 2-sphere at infinity. However, few explicit examples are known. We construct an explicit subdivision rule for many…

It is known that any tame hyperbolic 3-manifold with infinite volume and a single end is the geometric limit of a sequence of finite volume hyperbolic knot complements. Purcell and Souto showed that if the original manifold embeds in the…

In this article we investigate the existence, uniqueness and exponential decay of asymptotically almost periodic solutions of the parabolic-elliptic Keller-Segel system on a real hyperbolic manifold. We prove the existence and uniqueness of…

Given a homomorphism from a knot group to a fixed group, we introduce an element of a $K_1$-group, which is a generalization of (twisted) Alexander polynomials. We compare this $K_1$-class with other Alexander polynomials. In terms of…

In this paper, we consider the knot complement problem for not null-homologous knots in homology lens spaces. Let $M$ be a homology lens space with $H_1(M; \mathbb{Z}) \cong \mathbb{Z}_p$ and $K$ a not null-homologous knot in $M$. We show…

We study the recently discovered isomorphisms between hyperbolic Weyl groups and unfamiliar modular groups. These modular groups are defined over integer domains in normed division algebras, and we focus on the cases involving quaternions…

A geometric interpretation is given for certain elliptic-hyperbolic systems in the plane. Among several examples, one which reduces in the elliptic region to the equations for harmonic 1-forms on the projective disc is studied in detail. A…

The Pontryagin dual of the twisted Alexander module for a d-component link and GL(N,Z) representation is an algebraic dynamical system with an elementary description in terms of colorings of a diagram. In the case of a knot, its associated…

This article pursues the study of the knot state asymptotics in the large level limit initiated in "Knot sate Asymptotics I". As a main result, we prove the Witten asymptotic expansion conjecture for the Dehn fillings of the figure eight…

We construct examples of hyperbolic rational homology spheres and hyperbolic knot complements in rational homology spheres containing closed embedded totally geodesic surfaces.

We present a statistical approach for the discovery of relationships between mathematical entities that is based on linear regression and deep learning with fully connected artificial neural networks. The strategy is applied to…

Suppose that a hyperbolic knot in $S^3$ admits a finite surgery, Boyer and Zhang proved that the surgery slope must be either integral or half-integral, and they conjectured that the latter case does not happen. Using the correction terms…

We study holographic renormalization and the variational problem in four-dimensional Einstein gravity coupled to a self-interacting scalar field in asymptotically AdS spacetimes with mixed, designer-gravity boundary conditions. For static…

Kapustin-Witten (KW) equations are encountered in the localization of the topological N=4 SYM theory. Mikhaylov has constructed model solutions of KW equations for the boundary 't~Hooft operators on a half space. Direct proof of the…

We prove that, for a hyperbolic two bridge knot, infinitely many Dehn fillings are rigid in $SO_0(4,1)$. Here rigidity means that any discrete and faithful representation in $SO_0(4,1)$ is conjugate to the holonomy representation in…

We explore the combination theorem for a group G splitting as a graph of relatively hyperbolic groups. Using the fine graph approach to relative hyperbolicity, we find short proofs of the relative hyperbolicity of G under certain…

We develop skein theory for 3-manifolds in the presence of codimension-one defects, focusing especially on defects arising from parabolic induction/restriction for quantum groups. We use these defects as a model for the quantum decorated…

We provide a method for gluing (small) resolutions of singularities of Schubert varieties \(X_w\). An explicit isomorphism of \(X_w\) with an (iterated) bundle is constructed when \(w\) has an (iterated) BP decomposition. Combined with the…

Associated with every quaternionic representation of a compact, connected Lie group there is a Seiberg-Witten equation in dimension three. The moduli spaces of solutions to these equations are typically non-compact. We construct Kuranishi…

General Relativity in 4 dimensions can be equivalently described as a dynamical theory of SO(3)-connections rather than metrics. We introduce the notion of asymptotically hyperbolic connections, and work out an analog of the…