arXiv research

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

Minimal crossing number found in arithmetic curve systems.

New formulas derived for lattice crossing coefficients, improving computation efficiency.

New method estimates spatial weights matrix for lattice data, improving prediction accuracy.

We introduce a new discrete system that arises from ellipsoidal billiards and is closely related to the double reflection nets. The system is defined on the lattice of a uniform honeycomb consisting of rectified hypercubes and cross polytopes. In the $2$-dimensional case, the lattice is regular and it incorporates dyna…

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…

Proof of wall-crossing formula using spectral networks.

Paper finds coefficients of Catalan states using Θ_A-state expansion.

For a Lattice crossing $L\left( m,n\right)$ we show which Catalan connection between $2\left( m+n\right)$ points on boundary of $m\times n$ rectangle $P$ can be realized as a Kauffman state and we give an explicit formula for the number of such Catalan connections. For the case of a Catalan connection with no arc sta…

The lattice stick number $s_L(K)$ of a knot $K$ is defined to be the minimal number of straight line segments required to construct a stick presentation of $K$ in the cubic lattice. In this paper, we find an upper bound on the lattice stick number of a nontrivial knot $K$, except trefoil knot, in terms of the minimal c…

For a Catalan state $C$ of a lattice crossing $L\left( m,n\right)$ with no returns on one side, we find its coefficient $C\left( A\right)$ in the Relative Kauffman Bracket Skein Module expansion of $L\left( m,n\right)$. We show, in particular, that $C\left( A\right)$ can be found using the plucking polynomial of a …

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…

We explore a simple lattice field model intended to describe statistical properties of high frequency financial markets. The model is relevant in the cross-disciplinary area of econophysics. Its signature feature is the emergence of a self-organized critical state. This implies scale invariance of the model, without tu…

We give an algorithmic computation for the height of Kauffman's clock lattice obtained from a knot diagram with two adjacent regions starred and without crossing information specified. We show that this lattice is more familiarly the graph of perfect matchings of a bipartite graph obtained from the knot diagram by over…

The study examines knot probabilities in confined lattice polygons.

Study on unimodality of plucking polynomial with delay function.

This thesis investigates cusp cross-sections of arithmetic real, complex, and quaternionic hyperbolic $n$--orbifolds. We give a smooth classification of these submanifolds and analyze their induced geometry. One of the primary tools is a new subgroup separability result for general arithmetic lattices.

Let $\mbox{Len}(K)$ be the minimum length of a knot on the cubic lattice (namely the minimum length necessary to construct the knot in the cubic lattice). This paper provides upper bounds for $\mbox{Len}(K)$ of a nontrivial knot $K$ in terms of its crossing number $c(K)$ as follows: $\mbox{Len}(K) \leq \min \left\{ \fr…

This thesis is concerned with the question of when the double branched cover of an alternating knot can arise by Dehn surgery on a knot in $S^3$. We approach this problem using a surgery obstruction, first developed by Greene, which combines Donaldson's Diagonalization Theorem with the $d$-invariants of Ozsv{á}th and S…

We extend our studies of a quantum field model defined on a lattice having the dilation group as a local gauge symmetry. The model is relevant in the cross-disciplinary area of econophysics. A corresponding proposal by Ilinski aimed at gauge modeling in non-equilibrium pricing is realized as a numerical simulation of t…

In this work, three lattice-free (LF) discriminative training criteria for purely sequence-trained neural network acoustic models are compared on LVCSR tasks, namely maximum mutual information (MMI), boosted maximum mutual information (bMMI) and state-level minimum Bayes risk (sMBR). We demonstrate that, analogous to L…

We study noncompact, complete, finite volume, negatively curved manifolds $M$. We construct $M$ with infinitely generated fundamental groups in all dimensions $n \geq 2$. We construct $M$ whose cusp cross sections are compact hyperbolic manifolds in all dimension $n\geq 3$. In contrast we show that if sectional curvatu…

Knots are commonly found in molecular chains such as DNA and proteins, and they have been considered to be useful models for structural analysis of these molecules. One interested quantity is the minimum number of monomers necessary to realize a molecular knot. The minimum lattice length $\mbox{Len}(K)$ of a knot $K$ i…

A family of TQFTs parametrised by G-crossed braided spherical fusion categories has been defined recently as a state sum model and as a Hamiltonian lattice model. Concrete calculations of the resulting manifold invariants are scarce because of the combinatorial complexity of triangulations, if nothing else. Handle deco…

We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improve…

Knots have been considered to be useful models for simulating molecular chains such as DNA and proteins. One quantity that we are interested on molecular knots is the minimum number of monomers necessary to realize a knot. In this paper we consider every knot in the cubic lattice. Especially the minimal length of a kno…

Investigates methods to regularize quantile regression for accurate predictions.

A discrete conformal map (DCM) maps the square lattice to the Riemann sphere such that the image of every irreducible square has the same cross-ratio. This paper shows that every periodic DCM can be determined from spectral data (a hyperelliptic compact Riemann surface, called the spectral curve, equipped with some mar…

New property identifies arithmetic lattices from nonuniform lattices.

Skew parallelogram nets factorize, encompassing discrete differential geometry.

We introduce a new fundamental domain for the cusp stabilizer of a Hilbert modular group over a real quadratic field K=Q(sqrt n). This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a foliation of the biplane. The region is the Cartesian product of the positive real…

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…

Quantum phase diagrams for Chern topological insulators show jumps at critical loci.

State-level minimum Bayes risk (sMBR) training has become the de facto standard for sequence-level training of speech recognition acoustic models. It has an elegant formulation using the expectation semiring, and gives large improvements in word error rate (WER) over models trained solely using cross-entropy (CE) or co…

This paper calculates stick numbers for rail arcs and knot classes.

Study automorphism groups of Artin groups, proving rigidity and classification results.

New rigidity theorem for product of lattices.

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…

The paper explores rigidity and proximality in dynamical systems, proving new results about $C^*$-algebras.

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

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

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 …