arXiv research

Paper introduces spherical knot mosaics for knot and link invariants.

Rectangular mosaics extend virtual knot studies to larger polygons.

New method uses mosaics to study wild knots.

In 2008, Kauffman and Lomonaco introduce the concepts of a knot mosaic and the mosaic number of a knot or link, the smallest integer $n$ such that a knot or link can be represented on an $n$-mosaic. In arXiv:1702.06462, the authors explore space-efficient knot mosaics and the tile number of a knot or link, the smallest…

Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on `Quantum knots and mosaics' to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an $m \! \times \! n$ matrix whose entries are eleven mosaic tiles, represent…

A new mosaic system for immersed surface-links is introduced.

Researchers found algorithms to construct toric mosaics and set upper bounds for their numbers.

Algorithm finds mosaic numbers for knots with 10 or fewer crossings.

Knot mosaics are used to model physical quantum states. The mosaic number of a knot is the smallest integer $m$ such that the knot can be represented as a knot $m$-mosaic. In this paper we establish an upper bound for the crossing number of a knot in terms of the mosaic number. Given an $m$-mosaic and any knot $K$ that…

Enhances knot counting using mosaic diagrams.

This paper studies virtual knots using mosaic diagrams.

Computes bounds on mosaic number of Legendrian knots.

Improved bounds for knot crossings in different mosaic patterns.

We investigate relationships between bounds on the crossing number and the mosaic number of mosaic knots.

KnotMosaics package simplifies knot theory computations in SageMath.

Lomonaco and Kauffman introduced knot mosaic system to give a definition of quantum knot system. This definition is intended to represent an actual physical quantum system. A knot $(m,n)$-mosaic is an $m \times n$ matrix of mosaic tiles which are $T_0$ through $T_{10}$ depicted as below, representing a knot or a link b…

Lomonaco and Kauffman introduced a knot mosaic system to give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This paper is inspired by an open question about the knot mosaic enumeration suggested by them. A knot $n$--mosaic is an $n \times n$ array of 11 mosaic…

Lomonaco and Kauffman developed knot mosaics to give a definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot $n$-mosaic is an $n \times n$ matrix of 11 kinds of specific mosaic tiles representing a knot or a link. The mosaic number $m(K)$ of a knot $K$ i…

The concepts of tile number and space-efficiency for knot mosaics were first explored by Heap and Knowles (arXiv:1702.06462), where they determined the possible tile numbers and space-efficient layouts for every prime knot with mosaic number 6 or less. In this paper, we extend those results to prime knots with mosaic n…

In 2008, Lomonaco and Kauffman introduced a knot mosaic system to define a quantum knot system. A quantum knot is used to describe a physical quantum system such as the topology or status of vortexing that occurs on a small scale can not see. Kuriya and Shehab proved that knot mosaic type is a complete invariant of tam…

Since the Jones polynomial was discovered, the connection between knot theory and quantum physics has been of great interest. Lomonaco and Kauffman introduced the knot mosaic system to give a definition of the quantum knot system that is intended to represent an actual physical quantum system. Recently the authors deve…

In this paper we introduce the concept of a space-efficient knot mosaic. That is, we seek to determine how to create knot mosaics using the least number of non-blank tiles necessary to depict the knot. This least number is called the tile number of the knot. We determine strict bounds for the tile number of a knot in t…

In this paper, we work to construct mosaic representations of knots on the torus, rather than in the plane. This consists of a particular choice of the ambient group, as well as different definitions of contiguous and suitably connected. We present conditions under which mosaic numbers might decrease by this projection…

New tiles allow efficient knot mosaics for small knots.

Paper proves corner connection tiles can represent knots with fewer tiles.

This paper presents a novel framework for generating texture mosaics with convolutional neural networks. Our method is called GANosaic and performs optimization in the latent noise space of a generative texture model, which allows the transformation of a content image into a mosaic exhibiting the visual properties of t…

We give an infinite family of knots such that for any given $r \geq 3$, the family contains a knot which can be embedded on a hexagonal $r$-mosaic, but cannot fit on a hexagonal $r$-mosaic in an embedding that achieves its crossing number. This extends the rectangular mosaic result of Ludwig, Evans, and Paat. We also i…

New tile types for knots and links reduce complexity.

Lomonaco and Kauffman introduced a knot mosaic system to give a definition of a quantum knot system which can be viewed as a blueprint for the construction of an actual physical quantum system. A knot $n$-mosaic is an $n \times n$ matrix of 11 kinds of specific mosaic tiles representing a knot or a link by adjoining pr…

Inspired by the paper on quantum knots and knot mosaics [23] and grid diagrams (or arc presentations), used extensively in the computations of Heegaard-Floer knot homology [2,3,7,24], we construct the more concise representation of knot mosaics and grid diagrams via mirror-curves. Tame knot theory is equivalent to knot…

MOSAIC detects change points in dynamic networks with low-rank and sparse changes.

Lomonaco and Kauffman developed a knot mosaic system to introduce a precise and workable definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an $m \times n$ matrix of mosaic tiles ($T_0$ through $T_{10}$ depicted in the introduction) re…

We address the problem of distinguishing cause from effect in bivariate setting. Based on recent developments in nonlinear independent component analysis (ICA), we train nonparametrically general nonlinear causal models that allow non-additive noise. Further, we build an ensemble framework, namely Causal Mosaic, which …

Representations of sets are challenging to learn because operations on sets should be permutation-invariant. To this end, we propose a Permutation-Optimisation module that learns how to permute a set end-to-end. The permuted set can be further processed to learn a permutation-invariant representation of that set, avoid…

MOSAIC selects few informative exemplars from high-dimensional data with non-linear structures.

Samuel J. Lomonaco Jr and Louis H. Kauffman conjectured that tame knot theory and knot mosaic theory are equivalent. We give a proof of the Lomonaco-Kauffman conjecture.

The AutoML task consists of selecting the proper algorithm in a machine learning portfolio, and its hyperparameter values, in order to deliver the best performance on the dataset at hand. Mosaic, a Monte-Carlo tree search (MCTS) based approach, is presented to handle the AutoML hybrid structural and parametric expensiv…

We introduce MosAIc, an interactive web app that allows users to find pairs of semantically related artworks that span different cultures, media, and millennia. To create this application, we introduce Conditional Image Retrieval (CIR) which combines visual similarity search with user supplied filters or "conditions". …

Theory explains creativity in diffusion models generating novel images.

We show that the orthogonal separation coordinates on the sphere $S^n$ are naturally parametrised by the real version of the Deligne-Mumford-Knudsen moduli space $\bar M_{0,n+2}(R)$ of stable curves of genus zero with $n+2$ marked points. We use the combinatorics of Stasheff polytopes tessellating $\bar M_{0,n+2}(R)$ t…

Neuromorphic-style inference only works well if limited hardware resources are maximized properly, e.g. accuracy continues to scale with parameters and complexity in the face of potential disturbance. In this work, we use realistic crossbar simulations to highlight that compact implementations of deep neural networks a…

In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to represent the "quantu…

Large datasets of sub-meter aerial imagery represented as orthophoto mosaics are widely available today, and these data sets may hold a great deal of untapped information. This imagery has a potential to locate several types of features; for example, forests, parking lots, airports, residential areas, or freeways in th…

Parametric generative deep models are state-of-the-art for photo and non-photo realistic image stylization. However, learning complicated image representations requires compute-intense models parametrized by a huge number of weights, which in turn requires large datasets to make learning successful. Non-parametric exem…

Extends Fisher's Discriminant Analysis for interval-valued data.

This paper explores how deep learning models can fit data exactly and why this is important.

Most digital cameras use sensors coated with a Color Filter Array (CFA) to capture channel components at every pixel location, resulting in a mosaic image that does not contain pixel values in all channels. Current research on reconstructing these missing channels, also known as demosaicing, introduces many artifacts, …

In this paper we show how to place Michael Berry's discovery of knotted zeros in the quantum states of hydrogen in the context of general knot theory and in the context of our formulations for quantum knots. Berry gave a time independent wave function for hydrogen, as a map from three space to the complex plane and suc…