Big Data Approaches to Knot Theory: Understanding the Structure of the Jones Polynomial
We examine the structure and dimensionality of the Jones polynomial using manifold learning techniques. Our data set consists of more than 10 million knots up to 17 crossings and two other special families up to 2001 crossings. We introduce and describe a method for using filtrations to analyze infinite data sets where representative sampling is impossible or impractical, an essential requirement for working with knots and the data from knot invariants. In particular, this method provides a new approach for analyzing knot invariants using Principal Component Analysis. Using this approach on the Jones polynomial data we find that it can be viewed as an approximately 3 dimensional manifold, that this description is surprisingly stable with respect to the filtration by the crossing number, and that the results suggest further structures to be examined and understood.
Code (1)
Similar Papers 제목 키워드 기반
Applications of Knot Theory for the Improvement of the AlphaFold Protein Database
AlphaFold, a groundbreaking protein prediction model, has revolutionized protein structure prediction, populating the AlphaFold Protein Database (AFDB) with millions of predicted structures. However, AlphaFold's accuracy…
Protein Structure PredictionModeling knotted proteins with tangles
Although rare, an increasing number of proteins have been observed to contain entanglements in their native structures. To gain more insight into the significance of protein knotting, researchers have been investigating …
Mean-field Analysis of Piecewise Linear Solutions for Wide ReLU Networks
Understanding the properties of neural networks trained via stochastic gradient descent (SGD) is at the heart of the theory of deep learning. In this work, we take a mean-field view, and consider a two-layer ReLU network…
Geometric deep learning approach to knot theory
In this paper, we introduce a novel way to use geometric deep learning for knot data by constructing a functor that takes knots to graphs and using graph neural networks. We will attempt to predict several knot invariant…
Deep LearningThe possibility of making \$138,000 from shredded banknote pieces using computer vision
Every country must dispose of old banknotes. At the Hong Kong Monetary Authority visitor center, visitors can buy a paperweight souvenir full of shredded banknotes. Even though the shredded banknotes are small, by using …