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In this paper we study the geometry of fully augmented link complements in the thickened torus and describe their geometric properties, generalizing the study of fully augmented links in $S^3$. We classify which fully augmented links in the…
In this paper, we show that two flat fully augmented links with homeomorphic complements must be equivalent as links in $\mathbb{S}^{3}$. This requires a careful analysis of how totally geodesic surfaces and cusps intersect in these link…
Classical fully augmented links have explicit hyperbolic geometry, and have diagrams on the 2-sphere in the 3-sphere. We generalise to construct fully augmented links projected to the reflection surface of any 3-manifold obtained by…
Thurston's equations determine the hyperbolic structure of a 3-manifold with a triangulation. In work by Thistlethwaite and Tsvietkova, an alternative method was developed for link complements in $S^3$ depending on the link diagram, where a…
This paper examines number theoretic and topological properties of fully augmented pretzel link complements. In particular, we determine exactly when these link complements are arithmetic and exactly which are commensurable with one…
For hyperbolic 3-manifolds, the growth rate of their Turaev-Viro invariants, evaluated at a certain root of unity, is conjectured to give the hyperbolic volume of the manifold. This has been verified for a handful of examples and several…
We construct a family of hyperbolic link complements by gluing tangles along totally geodesic four-punctured spheres, then investigate the commensurability relation among its members. Those with different volume are incommensurable,…
We construct the augmentation representation. It is a representation of the fundamental group of the link complement associated to an augmentation of the framed cord algebra. This construction connects representations of two link invariants…
We investigate the geometry of hyperbolic knots and links whose diagrams have a high amount of twisting of multiple strands. We find information on volume and certain isotopy classes of geodesics for the complements of these links, based…
In this paper we continue the work of using the recent advances in algebraic $K$-theory to extend computations done in characteristic $p$ to the mixed characteristic setting using perfectoid rings. We extend the work of Hesselholt-Nikolaus…
We survey some tools and techniques for determining geometric properties of a link complement from a link diagram. In particular, we survey the tools used to estimate geometric invariants in terms of basic diagrammatic link invariants. We…
In this paper, we prove that the (orientation-preserving) symmetry groups of $b$-prime flat fully augmented links correspond exactly with the finite subgroups of $O(3)$. We accomplish this by first developing a dictionary between…
Classification of AS-regular algebras is one of the most important projects in noncommutative algebraic geometry. Recently, Itaba and the first author gave a complete list of defining relations of $3$-dimensional quadratic AS-regular…
(This is a report for the Proceedings of ``Journees Relativistes 1993'' written in September 1993. Containes a short description of the results published elsewhere in the joint paper with A. Ashtekar) Integral calculus on the space of gauge…
This report lists the link diagrams in S^3 for all principal congruence link complements for which such a link diagram is known. Several unpublished link diagrams are included. Related to this, we also include one link diagram for an…
The algebraic approach to the Constraint Satisfaction Problem (CSP) uses high order symmetries of relational structures -- polymorphisms -- to study the complexity of the CSP. In this paper we further develop one of the methods the…
The family of right-angled tiling links consists of links built from regular 4-valent tilings of constant-curvature surfaces that contain one or two types of tiles. The complements of these links admit complete hyperbolic structures and…
This article describes an entirely algebraic construction for developing conformal geometries, which provide models for, among others, the Euclidean, spherical and hyperbolic geometries. On one hand, their relationship is usually shown…
We determine the product structure on Hochschild cohomology of commutative algebras in low degrees, obtaining the answer in all degrees for complete intersection algebras. As applications, we consider cyclic extension algebras as well as…
We give a survey of analytic and geometric results on `fibred cusp spaces', a large class of non-compact Riemannian manifolds which include the regular parts of singular spaces with incomplete cusp singularities as well as complete spaces…