arXiv research

Upper bound for lattice stick number of spatial graphs.

This paper finds upper bounds for lattice stick numbers of rational links with specific stick configurations.

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

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…

Knots and links have been considered to be useful models for structural analysis of molecular chains such as DNA and proteins. One quantity that we are interested on molecular links is the minimum number of monomers necessary to realize them. In this paper we consider every link in the cubic lattice. Lattice stick numb…

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…

The lattice stick number of a knot type is defined to be the minimal number of straight line segments required to construct a polygon presentation of the knot type in the cubic lattice. In this paper, we mathematically prove that the trefoil knot $3_1$ and the figure-8 knot $4_1$ are the only knot types of lattice stic…

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

New lattice stick knot condition identified.

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

Improved bounds on stick numbers of knots up to 13 crossings.

Exact stick number of two knots with 10 crossings found.

New bounds on stick number of knots found using random polygon generation.

Upper bounds on stick and equilateral stick numbers of spatial graphs derived.

An equilateral stick number $s_{=}(K)$ of a knot $K$ is defined to be the minimal number of sticks required to construct a polygonal knot of $K$ which consists of equal length sticks. Rawdon and Scharein [12] found upper bounds for the equilateral stick numbers of all prime knots through 10 crossings by using algorithm…

The stick index of a knot is the least number of line segments required to build the knot in space. We define two analogous 2-dimensional invariants, the planar stick index, which is the least number of line segments in the plane to build a projection, and the spherical stick index, which is the least number of great c…

Utilizing both twisting and writhing, we construct integral tangles with few sticks, leading to an efficient method for constructing polygonal 2-bridge links. Let L be a two bridge link with crossing number c, stick number s, and n tangles. It is shown that s is less than or equal to 2/3 c + 2n+3 . We also show that if…

Negami found an upper bound on the stick number $s(K)$ of a nontrivial knot $K$ in terms of the minimal crossing number $c(K)$ of the knot which is $s(K) \leq 2 c(K)$. Furthermore McCabe proved $s(K) \leq c(K) + 3$ for a $2$-bridge knot or link, except in the case of the unlink and the Hopf link. In this paper we const…

Study shows stick numbers for specific graphs and explains a protein structure.

The study proves all prime knots up to 10 crossings have superbridge index ≤ 5.

In 1991, Negami found an upper bound on the stick number $s(K)$ of a nontrivial knot $K$ in terms of the minimal crossing number $c(K)$ of the knot which is $s(K) \leq 2 c(K)$. In this paper we improve this upper bound to $s(K) \leq \frac{3}{2} (c(K)+1)$. Moreover if $K$ is a non-alternating prime knot, then $s(K) \leq…

New method finds exponential growth in knot types from sticks.

We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The ribbonlength is the length to width ratio of such a ribbon, and it turns out that the way the ribbon is folded influences the ribbonlength. We give an upper bound of $n\cot(π/n)$ for the ribbonlength of $…

Study on knots formed by Coxeter galleries, finding bounds and symmetric trefoils.

Directly proves CRP from stick-breaking process without measure theory.

New upper bounds on superbridge index for 49 knots, increasing known results to 49.

Study on folded ribbon knots and their minimum length.

To model categorical response variables given their covariates, we propose a permuted and augmented stick-breaking (paSB) construction that one-to-one maps the observed categories to randomly permuted latent sticks. This new construction transforms multinomial regression into regression analysis of stick-specific binar…

The beta-Bernoulli process provides a Bayesian nonparametric prior for models involving collections of binary-valued features. A draw from the beta process yields an infinite collection of probabilities in the unit interval, and a draw from the Bernoulli process turns these into binary-valued features. Recent work has …

New superbridge index calculations for knots with odd edges.

We extend Stochastic Gradient Variational Bayes to perform posterior inference for the weights of Stick-Breaking processes. This development allows us to define a Stick-Breaking Variational Autoencoder (SB-VAE), a Bayesian nonparametric version of the variational autoencoder that has a latent representation with stocha…

Improved Gaussian process experts model for complex data.

We show that the number of conjugacy classes of maximal finite subgroups of a lattice in a semisimple Lie group is linearly bounded by the covolume of the lattice. Moreover, for higher rank groups, we show that this number grows sublinearly with covolume. We obtain similar results for isotropy subgroups in lattices. Ge…

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

Growth rate of Dehn twist lattice points in Teichmüller space is slower than mapping class group lattice points.

Smooth knots with odd Conway polynomial terms have inscribed trefoils.

Many data are naturally modeled by an unobserved hierarchical structure. In this paper we propose a flexible nonparametric prior over unknown data hierarchies. The approach uses nested stick-breaking processes to allow for trees of unbounded width and depth, where data can live at any node and are infinitely exchangeab…

The study assesses sensitivity to prior choices in Bayesian nonparametric models.

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…

Minimal crossing number found in arithmetic curve systems.

Study of subgroups in complex hyperbolic lattice triangle groups.

Expectation maximization (EM) has recently been shown to be an efficient algorithm for learning finite-state controllers (FSCs) in large decentralized POMDPs (Dec-POMDPs). However, current methods use fixed-size FSCs and often converge to maxima that are far from optimal. This paper considers a variable-size FSC to rep…

Research examines lattices in Lie groups with specific geometric properties.

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…

We show that the number of noncommensurable lattices, hence also that of maximal lattices in SO(1,n) is at least exponential. To do so we construct large families of noncommensurable hybrid hyperbolic (Gromov/Piatetski-Shapiro) manifolds.

Study on geodesics and dihedral groups in lattices.

The study examines knot probabilities in confined lattice polygons.

Study links weaving knots with polynomial coefficients and lattice numbers.