arXiv research

The paper finds incommensurable lattices in complex models of Baumslag-Solitar groups.

The paper classifies groups containing incommensurable lattices in Baumslag-Solitar complexes.

Minimal non-arithmetic hyperbolic 3-orbifold found with least volume.

New noncompact Coxeter polytopes found in various dimensions.

Suppose that two large, multi-dimensional data sets are each noisy measurements of the same underlying random process, and principle components analysis is performed separately on the data sets to reduce their dimensionality. In some circumstances it may happen that the two lower-dimensional data sets have an inordinat…

Infinite hyperbolic manifolds share same perimeter-to-volume ratio.

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, distinguished by their scissors congruence classes. Mutation produces arbitrarily large…

Reid has asked whether hyperbolic manifolds with the same geodesic length spectrum must be commensurable. Building toward a negative answer to this question, we construct examples of hyperbolic 3-manifolds that share an arbitrarily large portion of the length spectrum but are not commensurable. More precisely, for all …

In this paper we examine the geometry of minimal surfaces of arithmetic hyperbolic 3-manifolds. In particular, we give bounds on the totally geodesic 2-systole, construct infinitely many incommensurable manifolds with the same initial geometric genus spectrum in which volume and 1-systole are controlled, and analyze th…

This paper investigates the strength of the trace field as a commensurability invariant of hyperbolic 3-manifolds. We construct an infinite family of two-component hyperbolic link complements which are pairwise incommensurable and have the same trace field, and infinitely many 1-cusped finite volume hyperbolic 3-manifo…

Study general hyperbolic gluings, proving quasi-arithmeticity of building blocks.

Two number fields are said to be Brauer equivalent if there is an isomorphism between their Brauer groups that commutes with restriction. In this paper we prove a variety of number theoretic results about Brauer equivalent number fields (e.g., they must have the same signature). These results are then applied to the ge…

In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurabiltiy classes by analyzing their cusp sha…

New property identifies arithmetic lattices from nonuniform lattices.

Research finds bounds for knots in hexagonal lattice and classifies 11-stick knots.

We outline the theory of sets with distributive operations: multishelves and multispindles, with examples provided by semi-lattices, lattices and skew lattices. For every such a structure we define multi-term distributive homology and show some of its properties. The main result is a complete formula for the homology o…

Course on arithmetic lattices at EPFL.

We give a simple example showing that a knot or link diagram that lies in the ${\mathbb{Z}}^2$ lattice is not necessarily the projection of a lattice stick knot or link in the ${\mathbb{Z}}^3$ lattice, and we give a necessary and sufficient condition for when a knot or link diagram that lies in the ${\mathbb{Z}}^2$ lat…

We describe a collection of computer scripts written in PARI/GP to compute, for reflection groups determined by finite-volume polyhedra in $\mathbb{H}^3$, the commensurability invariants known as the invariant trace field and invariant quaternion algebra. Our scripts also allow one to determine arithmeticity of such gr…

New rigidity theorem for product of lattices.

A biperiodic alternating link has an alternating quotient link in the thickened torus. In this paper, we focus on semi-regular links, a class of biperiodic alternating links whose hyperbolic structure can be immediately determined from a corresponding Euclidean tiling. Consequently, we determine the exact volumes of se…

We explore hybrid subgroups of certain non-arithmetic lattices in $\mathrm{PU}(2,1)$. We show that all of Mostow's lattices are virtually hybrids; moreover, we show that some of these non-arithmetic lattices are hybrids of two non-commensurable arithmetic lattices in $\mathrm{PU}(1,1)$.

This paper studies the covolumes of nonuniform arithmetic lattices in PU(n, 1). We determine the smallest covolume nonuniform arithmetic lattices for each n, the number of minimal covolume lattices for each n, and study the growth of the minimal covolume as n varies. In particular, there is a unique lattice (up to conj…

This paper provides further investigation of the concept of shape m$_{\rm simpl}$-fibrators (previously introduced by the author). The main results identify shape m$_{\rm simpl}$-fibrators among direct products of Hopfian manifolds. First it is established that every closed orientable manifold homotopically determined …

Proves a lattice version of the Atiyah-Singer index theorem.

Sequence discriminative training criteria have long been a standard tool in automatic speech recognition for improving the performance of acoustic models over their maximum likelihood / cross entropy trained counterparts. While previously a lattice approximation of the search space has been necessary to reduce computat…

Vertex distortion measures how far lattice knots deviate from straight lines.

Regular integer lattices are characterized by k unit vectors that build up their generator matrices. These have rank k for D-lattices, and are rank-deficient for A-lattices, for E_6 and E_7. We count lattice points inside hypercubes centered at the origin for all three types, as if classified by maximum infinity norm i…

In this paper we use techniques from convex projective geometry to produce many new examples of thin subgroups of lattices in special linear groups that are isomorphic to the fundamental groups of finite volume hyperbolic manifolds. More specifically, we show that for a large class of arithmetic lattices in SO(n,1) it …

We show that the set of even positive definite lattices that arise from smooth, simply-connected 4-manifolds bounded by a fixed homology 3-sphere can depend on more than the ranks of the lattices. We provide two homology 3-spheres with distinct sets of such lattices, each containing a distinct nonempty subset of the ra…

The paper proves residual finiteness of certain lattices and constructs surfaces with specific fundamental groups.

A celebrated theorem of Hadwiger states that the Euler-Poincaré characteristic is the the unique invariant and continuous valuation on the distributive lattice of compact polyhedra in R^n that assigns value one to each convex non-empty such polyhedron. This paper provides an analogue of Hadwiger's result for finitely p…

The cubic lattice stick index of a knot type is the least number of sticks necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p,p+1)-torus knots, and show how composing and taking satellites can be used to obtain t…

New method finds lattice polygons that can be dissected into triangles with integer areas.

L-CNNs approximate gauge actions, revealing fixed points with no lattice artifacts.

The paper refines transformations of lattice diagrams and introduces dotted diagrams.

Classifies knots by lattice size, finding unknot ratios and crossing numbers.

Let $G$ be a simply connected, solvable Lie group and $Γ$ a lattice in $G$. The deformation space $\mathcal{D}(Γ,G)$ is the orbit space associated to the action of $\Aut(G)$ on the space $\mathcal{X}(Γ,G)$ of all lattice embeddings of $Γ$ into $G$. Our main result generalises the classical rigidity theorems of Mal'tsev…

Proves effective slope gaps for lattice surfaces.

We study the arithmeticity of the Couwenberg-Heckman-Looijenga lattices in PU(n,1), and show that they contain a non-arithmetic lattice in PU(3,1) which is not commensurable to the non-arithmetic Deligne-Mostow lattice in PU(3,1).

Study answers arithmeticity question for normal subgroup of lattices.

One type of switch simplifies operations on lattice knots.

New lattices in higher dimensions have dense surface subgroups.

Improved lattice field theory simulations with local-Autoregressive Conditional Normalizing Flow.

Study investigates lattices fibring over the circle, focusing on BNSR invariants.

The lattice stick number of knots is defined to be the minimal number of straight sticks in the cubic lattice required to construct a lattice stick presentation of the knot. We similarly define the lattice stick number $s_{L}(G)$ of spatial graphs $G$ with vertices of degree at most six (necessary for embedding into th…

A new surgery formula for knot lattice homology.

The work of Reid, Chinburg--Hamilton--Long--Reid, Prasad--Rapinchuk, and the author with Reid have demonstrated that geodesics or totally geodesic submanifolds can sometimes be used to determine the commensurability class of an arithmetic manifold. The main results of this article show that generalizations of these res…