Visual Algebraic proofs for Unknot Detection

A knot diagram looks like a two-dimensional drawing of aknotted rubberband. Proving that a given knot diagram can be untangled(that is, is a trivial knot, called an unknot) is one of the most famousproblems of knot theory. For a small knot diagram, one can try to finda sequence of untangling moves explicitly, but for a larger knot diagramproducing such a proof is difficult, and the produced proofs are hardto inspect and understand. Advanced approaches use algebra, with anadvantage that since the proofs are algebraic, a computer can be usedto produce the proofs, and, therefore, a proof can be produced evenfor large knot diagrams. However, such produced proofs are not easy toread and, for larger diagrams, not likely to be human readable at all.We propose a new approach combining advantages of these: the proofsare algebraic and can be produced by a computer, whilst each part ofthe proof can be represented as a reasonably small knot-like diagram(a new representation as a labeled tangle diagram), which can be easilyinspected by a human for the purposes of checking the proof and findingout interesting facts about the knot diagram

Towards human readability of automated unknottedness proofs

© 2018 CEUR-WS. All rights reserved. When is a knot actually unknotted? How does one convince a human reader of the correctness of an answer to this question for a given knot diagram? For knots with a small number of crossings, humans can be efficient in spotting a sequence of untangling moves. However, for knot diagrams with hundreds of crossings, computer assistance is necessary. There have been recent developments in algorithms for both (indirectly) (i) detecting unknotedness and (directly) (ii) producing such sequences of untangling moves. Automated reasoning can be applied to (i) and, to some extent, (ii), but the computer output is not necessarily human-readable. We report on work in progress towards bridging the gap between the computer output and human readability, via generating human-readable visual proofs of unknottedness

Where the Wild Knots Are

The new work in this document can be broken down into two main parts. In the first, we introduce a formalism for viewing the signed Gauss code for virtual knots in terms of an action of the symmetric group on a countable set. This is achieved by creating a standard unknot whose diagram contains countably-many crossings, and then representing tame knots in terms of the action of permutations with finite support. We present some preliminary computational results regarding the group operation given by this encoding, but do not explore it in detail. To make the encoding above formal, we require the aforementioned unknot with a countable sequence of crossings; building up the machinery to work with these kinds of objects is the focus of the second part of the project. Initially, the presence of infinitely-many crossing might appear to be a contradiction to the finiteness constraint in Reidemeister\u27s theorem; we show that this is not the case, and introduce the notion of feral points to represent areas of our diagrams in which it is not immediately obvious whether the knot is wild or tame. We employ uniform convergence to create sufficient conditions for guaranteeing ambient isotopy under limits and resolve a seeming contradiction given by the wild arc of Fox-Artin. Finally, we show that any knot whose crossings are topologically discrete is ambient isotopic to a countable union of polygonal segments, and discuss implications for extending Reidemeister\u27s theorem in this context

Flexible Object Manipulation

Flexible objects are a challenge to manipulate. Their motions are hard to predict, and the high number of degrees of freedom makes sensing, control, and planning difficult. Additionally, they have more complex friction and contact issues than rigid bodies, and they may stretch and compress. In this thesis, I explore two major types of flexible materials: cloth and string. For rigid bodies, one of the most basic problems in manipulation is the development of immobilizing grasps. The same problem exists for flexible objects. I have shown that a simple polygonal piece of cloth can be fully immobilized by grasping all convex vertices and no more than one third of the concave vertices. I also explored simple manipulation methods that make use of gravity to reduce the number of fingers necessary for grasping. I have built a system for folding a T-shirt using a 4 DOF arm and a fixed-length iron bar which simulates two fingers. The main goal with string manipulation has been to tie knots without the use of any sensing. I have developed single-piece fixtures capable of tying knots in fishing line, solder, and wire, along with a more complex track-based system for autonomously tying a knot in steel wire. I have also developed a series of different fixtures that use compressed air to tie knots in string. Additionally, I have designed four-piece fixtures, which demonstrate a way to fully enclose a knot during the insertion process, while guaranteeing that extraction will always succeed

New Directions in Geometric and Applied Knot Theory

The aim of this book is to present recent results in both theoretical and applied knot theory—which are at the same time stimulating for leading researchers in the field as well as accessible to non-experts. The book comprises recent research results while covering a wide range of different sub-disciplines, such as the young field of geometric knot theory, combinatorial knot theory, as well as applications in microbiology and theoretical physics

Cyclic Cellular Automata on Networks and Cohomological Waves

A dynamic coverage problem for sensor networks that are sufficiently dense but not localized is considered. By maintaining only a small fraction of sensors on at any time, we are aimed to find a decentralized protocol for establishing dynamic, sweeping barriers of awake-state sensors. Network cyclic cellular automata is used to generate waves. By rigorously analyzing network-based cyclic cellular automata in the context of a system of narrow hallways, it shows that waves of awake-state nodes turn corners and automatically solve pusuit/evasion-type problems without centralized coordination. As a corollary of this work, we unearth some interesting topological interpretations of features previously observed in cyclic cellular automata (CCA). By considering CCA over networks and completing to simplicial complexes, we induce dynamics on the higher-dimensional complex. In this setting, waves are seen to be generated by topological defects with a nontrivial degree (or winding number). The simplicial complex has the topological type of the underlying map of the workspace (a subset of the plane), and the resulting waves can be classified cohomologically. This allows one to program pulses in the sensor network according to cohomology class. We give a realization theorem for such pulse waves

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library

Contact fibrations over the 2-disk

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 17-04-201