Estimation of knotting properties of a protein We need to construct a knot diagram from a protein conformation in order to obtain the knot invariant. Show activity on this post. Haken originally used this algorithm to show that unknotting is decidable, but did not analyze its complexity in more detail. The basic problem of knot theory, the recognition problem, is determining the equivalence of two knots. Algorithms exist to solve this problem, with the first given by Wolfgang Haken in the late 1960s ( Hass 1998 ). The Knot Book - An Elementary Introduction to the Mathematical Theory of Knots. ⁡. Click the arrow to see the un- The theorems below gives a homeomorphism classification for the surfaces. In the rst chapter, we will see how the research on this subject changed during the time. In science, knot theory and its applications are applied to use knots to inspect the capacity of topoisomerase proteins to add or eliminate tangles from DNA. How to represent a knot/link: Link diagram Sage 9.5 Reference Manual: Knot Theory Release 9.5 The Sage Development Team Jan 31, 2022 Abstract: Journal of Knot Theory and Its Ramifications, Volume 30, Issue 14, December 2021. In 1984, after nearly half a century in which the main focus in knot theory was the knot invariants derived from the geometry of knot, that is, the knot invariants which had been well-studied were based on the shape of the knot, V. Jones announced the discovery of a new invariant. Theory TutICTS program Knot Theory and Its Applications (KTH-2013), held from December 10–20, 2013, at IISER Mohali, India. … This type of topology investigates the fundamental traits and limitations of … The second part explores braid theory, including braids in different spaces and simple word recognition algorithms. Theorem 1.8 (Classification I) Every … In this setting, experimentation knot theory, representation theory, and combinatorics. AUTHORS: Miguel Angel Marco Buzunariz. Knot theory and other areas of topology are no more considered as abstract areas of classical mathematics remote from anything of practical interest. Theory and algorithms for the optimization of unconstrained problems Topology is the study of abstract ... known as knot theory, but even there Conway who’s programming computers and building a tough knot to crack Our 7th Annual Neuroscience Virtual Event is now available On-Demand! Another title, perhaps “Surveys of Recent Advances in Knot Theory might better suit this book. January 16, 2022. , 3:35 pm. There is an interesting puzzle related to the Braids Theory. On the higher-end of the knots textbook world, Burde and Zieschang's "Knots" covers quite a lot of ground in much detail. The real-time simulation of rope, and knot tying in particular, raises difficult issues in contact detection and management. Knot theory: Algorithms, complexity and computation Organizers: Ryan Budney (University of Victoria, Canada) Benjamin Burton (The University of Queensland, Australia) Kazuhiro Ichihara (Nihon University, Japan) April 28{May 1, 2014 Knot theory is, at its most basic level, concerned with the topology of closed loops in 3-dimensional space. Haken originally used this algorithm to show that unknotting is decidable, but did not analyze its complexity in more detail. Knot theory applications in chemistry allow us to depict topological stereoisomers or atoms with identical particles but various designs. This Journal is intended as a forum for new developments in knot theory, particularly developments that create connections between knot theory and other aspects of mathematics and natural science. I have no idea what this is supposed to mean, but it does seem interesting. CR Categories: I.3.7 [Computer Graphics]: Three-Dimensional c, where c is the crossing number). We initiate the study of classes of knots where the genus problem and even the equivalence problem have very low computational … Knot (data, check = True) ¶. Amit Jamadagni. The thesis of this paper is: Protein similarity detection leads naturally to an algorithm operating at the metric, relational, and homotopic scales. Knots 2 3. I've seen many links to knot theory in general, spanning from punctual references to general information.. Before trying to devise something from scratch by myself, I'd like to know about the … A special case of this problem is one of the fundamental questions of Knot Theory: Given a knot, is it the unknot? 1 Introduction In mathematics, a knot refers to an entangled loop. Knot Theory, Second Edition is notable not only for its expert presentation of knot theory’s state of the art but also for its accessibility. The Trieste look at knot theory, by Jozef Przytycki. An advanced gauge theory approach. Agol, Haas and Thurston showed that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP-complete. The Knot Book Reviewed by Alexey Sossinsky The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots Colin C. Adams Reprinted with corrections, 2004, American Mathematical Society 307 pages, Paperback US$29.00 ISBN 0-8218-3678-1 Knot theory has been very fortunate with books—from the first one, Kurt Reidemeister’s We show that the genus problem for alternating knots with n crossings has linear time complexity and is in Logspace(n). This is not true generally for links with more than one component. Theorem 1.7 Every knot is the boundary of an orientable surface. A Classical Result two knots with homeomorphic oriented complements are equivalent. 2.2.2. The Jones polynomial. Request PDF | Recognition algorithms in knot theory | In this paper the problem of constructing algorithms for comparing knots and links is discussed. Since then other algorithms have been devised, notably by Joan Birman and Michael Hirsch, and at least partially imple-mented. This presentation is intended for a general mathematical audience. Pub Date: March 2018 arXiv: arXiv:1803.04908 Bibcode: 2018arXiv180304908K Keywords: Mathematics - Geometric Topology; Mathematics - Combinatorics; even the study of the structure of DNA. Theory and Algorithms Theory and algorithms form the mathematical core of computer science. Leonidas Guibas & Jorge Stolfi. Knot theory is essential to finding the grand equation of knitting. If you run Haken's algorithm on the Rolfsen/Conway table of knots, the first thing you have to do is triangulate the knot complement. Ever since Sir William Thomson's vortex theory, mathematicians have tried to classify and tabulate all possible knots.As of May 2008, all prime knots up to 16 crossings have been tabulated. Jeff Weeks, Computation of hyperbolic structures in knot theory Two Lectures On The Jones Polynomial And Khovanov Homology, by Edward Witten. The farther apart two areas are, the more dramatic the implications when relationships are discovered. Almost all alternating knots of given genus possess additional combinatorial structure, we call them standard. Vogel’s Algorithm 7 Acknowledgments 13 References 13 1. low complexity algorithms in knot theor y 15 [13] L. H. Kauffman, State models and the Jones polynomial , Topolo gy 26 (1987), 395–407. Section2. Furthermore, there is an algorithm for its explicit construction, and the resulting surface is called a Seifert surface for the knot. This affinity I think exist between knot theory and string theory. Those are usually acquired through basic courses of linear algebra and real analysis. additional module uses recent results from knot theory to identify which topological knots have been tied, also in real-time. For your purposes, where you have free ends, a few more moves are available that allow you to untangle actual knots as well. Braids are collections of lines whose ends are attached to two parallel straight lines. This field of topology focuses on issues such as 1. algorithm for deciding if a knot is the unknot by applying Haken’s theory to the complement of the knot. Cardinal. It also illustrates how the topics and techniques come together to deliver efficient solutions to important practical problems. 30, No. Because this enumeration is subject to extra combinatorial constraints, the only practical algorithms to date have been variants of the classical double description … Along with a new elementary exposition of the standard ideas from Thurston's work, the article includes never-before-published explanations of SnapPea's … Orientation and writhe For any knot, we can define an orientation or direction of … Kawauchi's "A survey of knot theory" covers much more ground but in less detail. In low-dimensional topology, many important decision algorithms are based on normal surface enumeration, which is a form of vertex enumeration over a high-dimensional and highly degenerate polytope. Several algorithms solving the unknotting problem are based on Haken's theory of normal surfaces: Haken's algorithm uses the theory of normal surfaces to find a disk whose boundary is the knot. Several algorithms solving the unknotting problem are based on Haken's theory of normal surfaces: Haken's algorithm uses the theory of normal surfaces to find a disk whose boundary is the knot. Gamifying Knot Theory. Given a tangled loop of string, is it really knotted or can it, with Recent papers in Knot Theory. Artificial Intelligence in Knot Theory. The main issue is that Haken's algorithm is extremely memory-intensive. The second part explores braid theory, including braids in different spaces and simple word recognition algorithms. 550 Followers. ... We conclude that the interplay between Knot Theory and Topological Field Theories is manifested not only at the quantum level, but in a classical context as well. introduction to high dimensional knot theory and briefly indicates how the methods of classical knot theory can be applied. Dijkstra's Algorithm: Dijkstra's algorithm is arguably one of the most common algorithm used to find the shortest path between the source vertex to every other vertex in the graph. The major challenge of the process is that many apparently different knots may actually be different geometrical presentations of the same topological entity, and that proving or disproving knot … The loops need to be closed to ensure that their properties stay the same when we deform them. The present volume, written by a well-known specialist, gives a complete survey of this theory from its very beginnings to today's most recent research results. "In topology, knot theory is the study of mathematical knots." An indispensable book for everyone concerned with knot theory. In particular, diverse combinatorial methods for representing links are discussed, the Haken algorithm for recognizing a trivial knot (the unknot) and a scheme for constructing a general algorithm (using Haken's ideas) for comparing links are presented, an approach based on representing links by closed braids is described, the known algorithms for solving the word … algorithm is used to nd a closed braid representation from a knot projection. Answer (1 of 7): The most important prerequisite for any topic of higher math is understanding the concepts of proofs and abstract definitions. A section devoted to the Vassiliev knot invariants follows, wherein the author proves that Vassiliev invariants are stronger than all polynomial invariants and introduces Bar-Natan's theory on Lie algebra respresentations and knots. ABSTRACT: Knot theory is an exciting area of study, with many applications in the sciences. W. H. Freeman and Company, New York, 1994. of the knot theory is also the history of the development of this new geometry theorized by Leibniz. Geometrical structures—typically hyperbolic—provide deep insight into the topology of knot and link complements. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot. The writhe is the number of times the molecule crosses over, or under, itself due to the e ects of supercoiling. This chapter from the upcoming Handbook of Knot Theory (eds. ... quantum groups, combinatorics of Gauss codes, combinatorics, algorithms and computational complexity, category theory and categorification … Twist the bottom ends of the first pair ptimes (counterclockwise if p>0and clockwise if p<0). Braids Theory. Proof: If the knot is unknotted, the firs t process will end after a finite amount Knot Theory and Its Applications. Contents 1. Equations in Groups and the Group of a Knot. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2022-04-13T07:00:00Z DOI: 10.1142/S0218216521990017 Issue No: Vol. A Distributed Graph Algorithm" Knot Detection J. MISRA and K. M. CHANDY University of Texas at Austin A knot in a directed graph is a useful concept in deadlock detection. The trunk of the knot is the maximum number of intersections across all of the regular levels tr(K) is the minimum tr(k) across all k in K. Image Illustrating Bridge, Trunk and Width Number ... Our mentor, Zhenkun Li, for teaching us about knot theory This work was motivated by surgical suturing, but simulation in other domains, such as sailing and rock climbing, could benefit from it. Formally "Knot theory is the mathematical branch of topology that studies mathematical knots, which are defined as embeddings of a circle in 3-dimensional Euclidean space, R3." , new. Knot theory is a broad field involving dimensional tangles and the work of untangling them. A mathematical knot is not the same as a knot in the real world. This forms a continuous loop which cannot be undone by manipulation. AUTHOR : C. Adams : EDITOR/ART : R. Ghrist : To move mathematics forward, researchers are constantly seeking new relationships between various areas of mathematics. In this dissertation, we are going to give a brief introduction of knot theory, looking at di erent aspects. Introductory Lectures On Knot Theory Author: autoadvisor.stevens.edu-2022-04-19T00:00:00+00:01 Subject: Introductory Lectures On Knot Theory Keywords: introductory, lectures, on, knot, theory Created Date: 4/19/2022 7:11:54 AM Such surfaces can be used to study the properties of the associated knot or link. Also designed art based on sandpile groups later turned into a rug. Knot theory has many relations to topology, physics, and (more recently!) 3:30 p.m. - 3:50 p.m. Jennifer Townsend, Bellevue College Twist the bottom ends of the second pair q times and the bottom end of the third pair rtimes. We claim that we can compute the Alexander polynomial of a knot with the following algorithm, which we call “Fox’s Algorithm.” Step 1. Computational algorithms and the complexity of link polynomials. Python code that explores two ways to derive the Alexander polynomial for a knot: the first using the knot’s braid and the second using the knot's quandles. from knot theory include unknot recognition (given a polygonal representation of a knot, de-cide whether it is equivalent to a trivial unknotted loop), and the more general problem ... algorithm (with respect to the combined input and output size) exists.2 For topological The book is divided into six thematic sections. Braids 6 5. The simplest form of knot theory involves the embedding of the unit circle into three-dimensional space. knot theory. algorithm. After discussing the history of the subject and covering basic definitions, we’ll discuss the property of tricolorability and prove its use in answering a fundamental question in knot theory: whether a given knot is equivalent to the unknot. The first such term is orientation. Enumerating fundamental normal surfaces: Algorithms, experiments and invariants Benjamin A. Burtony Abstract Computational knot theory and 3-manifold topology have seen signi cant breakthroughs in recent years, despite the fact that many key algorithms have complexity bounds that are exponential or greater. Does knot theory ever intersect with machine learning? 6.4. Knot Theory . Unknotting algorithms . : Over the last fifteen years, the face of knot theory has changed due to various new theories and invariants coming from physics, topology, combinatorics and alge-bra. Efficient ideal triangulations are readily generated by the algorithm used in SnapPea (originally due to Thurston, as far as I know). We also show, that the equivalence problem for such knots with n … A few terms need to be defined in order for the polynomials to be possible to calculate. In this presentation, we will discuss my work with undergraduate students to use computer algorithms to count the number of distinct knot mosaics that can occur on square mosaic board. Knot Theory: Why Everything Tangles : The Bryant Park Project Knot theory: Science for people putting away the Christmas lights. ... for building and testing optimal ecosystems that ensure worry-free data processing and the running of different algorithms. An introduction to knot Floer homology, by Ciprian Manolescu. The central problem of Knot Theory is determining whether two knots can be rearranged (without cutting) to be exactly alike. Knot theory is a rapidly developing field of research with many applications, not only for mathematics. Knot Groups and the Alexander Polynomial 5 Note 6.4.A. The program was Page 5/155 The practice of mathematics involves discovering patterns and using these to formulate and prove conjectures, resulting in theorems. A distributed algorithm for identifying a knot in a graph by using a network of processes is … Keywords: real algebraic geometry, knot theory, algorithms, symbolic compu-tation, SMT solving. Introduction 1 2. One such algorithm is Dijkstra's Algorithm. Seifert surfaces are also interesting in their own right, and the … A knot is defined as embedding of the circle \(\mathbb{S}^1\) in the 3-dimensional sphere \(\mathbb{S}^3\), considered up to ambient isotopy.They represent the physical idea of a knotted rope, but with … These modules provide you with two methods of input/output (user and file) and the ability to turn on “tracing” to see how the algorithms are working. We show that the genus problem for these knots belongs to TC0 circuit complexity class. Knot theory is a young branch of mathematics that started about a century ago. Chapter 10 describes new combinatorial meth-ods. Knot and Braid Notation Some input/output Python modules are common to all three programs. I'm looking for an algorithm that provides a way to generate graphical representations of knots (either 2D or 3D, but the former, with vector graphics is preferable).. MATH OUTREACH. They focused on two fields: knot theory, which is the mathematical study of knots; and representation theory, which is a field that focuses on … Papers; People; Real-time knot-tying simulation. Knots¶. Knot Theory: Second Edition. 3 Knot Theory applications to medicine 3.1 Knot Theory and DNA Biologists and mathematicians can picture knotted strands of DNA as links and use knot theory to gain insight into how topoisomerases interact with the double helix. Capable math communicator, having taught math at an individual and class level, spoken at professional ... algorithms, and the game of Nim. These methods greatly extend those of Chapter 3; the study of these combinatorial invariants is one of the most active and fascinating areas of knot theory today. While most areas in computer science study specific concrete systems, the goal of theory and algorithms is to abstract away these details in order to study the question: What sort of computations are possible to do, and possible to do efficiently, by any computer? 1. Namely, by estimating the knot invariant, we can determine whether a conformation has knots or not. The paper introduces a definition of similarity based on atomic motions that preserve local backbone topology without incurring significant distance errors. ... With integers, even though complicated, there are still several algorithms that ensures the eventual factorization. Louis H Kauffman was born February 3, 1945 in knot theory. In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable surface whose boundary is a given knot or link.. Braids Theory was invented by Emil Artin and is a part of the Knot Theory. Plenty of nice exercises. Knot Theory. 14 (2022) Geography of … In knot theory, the Reidemeister moves are the only three moves sufficient to turn any representation of a knot into another representation (such as a simpler one with fewer crossings). Introduction to knots and a survey of knot colorings. The most common data structure used to implement this algorithm is a min-priority queue. Instead, mathematical knots are closed loops in three-dimensional space. I recently found myself needing to know some things about knots – calculating fundamental groups and polynomial invariants, specifically. A complete algorithmic solution to this problem exists, which has unknown complexity . Primitives for the Manipulation of General Subdivisions and the Computation of Voronoi Diagrams. ACM Transactions on Graphics, vol 4, no 2, April 1985, pp 74-123. 2) The Central Problem of Knot Theory. (See also What Is Algorithm?) The presentation will Seifert Circles 3 4. For example, many knot invariants are most easily calculated using a Seifert surface. They have made deep impact on quantum field theory, quantum computation and complexity of computation. p;q;ris constructed as follows: take three pairs of string segments and arrange them vertically. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, (in topology, a circle isn't bound to the classical geometric concept, but to all of its homeomorphisms ). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of Title: Low complexity algorithms in knot theory Authors: Olga Kharlampovich , Alina Vdovina (Submitted on 13 Mar 2018 ( v1 ), last revised 27 Mar 2018 (this version, v2)) Graduate students beginning a program of advanced study will find a worthwhile overview, and the reader will need no training … 1 (F = 1), if it is other knots, the knot invariant is a polynomial except 1 (F 6= 1). Knot theory forms a fundamental source of examples in 3-manifold theory. Knot and link complements enjoy a geometry of crystalline beauty, rigid enough that simple cut-and-paste techniques meet geometrical as well as topological needs, yet surprisingly complex in their inexhaustible variety. He conjectures (but indicates at the time that it is work-in-progress) that his algorithm runs in quasi-polynomial time ( c log. Adams, Colin C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. W. H. Freeman and Company, New York, New York, 2001. Crowell, R. H. and Fox, R. H. Introduction to Knot Theory. Graduate Texts in Mathematics, Volume Fifty Seven. Springer-Verlag, New York-Heidelberg-Berlin, 1977. Livingston, C. In mathematical terminology we say that a knot1 is an embedding of S1 (a circle) in R3 that does not intersect itself. ... together with Python-oriented explanations of selected topics in optimization theory and optimization algorithms. The meeting focused on the broad area of knot theory and its interaction with other disciplines of theoretical science. a mathematical knot di ers in that the two loose ends of a strand are joined to-gether. Answer (1 of 3): Knot theory could be used in string theory to explain the beginning of the universe. 11. Louis H Kauffman, University of Illinois at Chicago, Mathematics Statistic and Computer Science Department, Faculty Member. This is how Wikipedia defines knot theory. Knot theory may seem to stand alone as a eld of study, but it has strong Some of these connections were explored in the second part of the class, and in some of the class projects. A section devoted to the Vassiliev knot invariants follows, wherein the author proves that Vassiliev invariants are stronger than all polynomial invariants and introduces Bar-Natan's theory on Lie algebra respresentations and knots. When we concentrate on knots themselves, start by mathematicizing them. Since solving the problem in 2018, Lisa Piccirillo has … Now, for a simple loop, that’s an easy question. Studies Knots, Topological Quantum Computing, and Foundations of Physics. Bases: sage.knots.link.Link, sage.structure.element.Element A knot. The first part discusses "pre-Vassiliev" knot theory, from knot arithmetics through the Jones polynomial and the famous Kauffman-Murasugi theorem. Recently, Marc Lackenby discussed a new algorithm for unknot recognition in a talk at the Newton Institute (see time around 1:03). Knot Theory Manuela Almeida Applied Mathematics and Computation, IST February 8, 2012 1 Introduction In mathematics, knot theory is the study of knots. Menasco and Thistlethwaite) shows how to construct hyperbolic structures on link complements and perform hyperbolic Dehn filling. any knot there is some surface having that knot as its boundary. Theorem 10 The algebraic algorithm describe d above gives an algorithm to decide whether a knot is the unknot. Figure 1: Escher’s Knots, 1965 1 1 Knot Theory Knot theory is an appealing subject because the objects studied are familiar in everyday physical space. Algorithms and Theory of Computation Handbook, Second Edition: General Concepts and Techniques provides an up-to-date compendium of fundamental computer science topics and techniques. class sage.knots.knot. The event will remain open 6 months 1 Talk delivered at the … The how do you get universes space and mass in a singular point could unite both … Haken’s paper was 130 pages long and his algorithm has never been implemented. Each student in the class will participate in a class-related project (see information below). Knot theory is a branch of algebraic topology where one studies what is known as the placement problem, or the embedding of one topological space into another. This shows that (unless P = NP) the genus problem has high computational complexity even for knots in a 3-manifold. Knot Theory, a lively exposition of the mathematics of knotting, will appeal to a diverse audience from the undergraduate seeking experience outside the traditional range of studies to mathematicians wanting a leisurely introduction to the subject. Take any presentation of the group of the knot, as described in Section 5.4. Knot Theory. The second part explores braid theory, including braids in different spaces and simple word recognition algorithms. Rama Mishra Polynomials in knot theory. Going to the sources (Fox’s 1962 classic “A Quick Trip Through Knot Theory” is really great mathematical reading) reveals there is a pretty straightforward algorithm for doing these kinds of things, but it seems like I’m going to have to … Quantum field theory, looking at di erent aspects algorithms for untangling knots in... Of nice exercises presentation of the class projects clockwise if p < 0 ) complexity! We deform them fclid=837bb15f-c045-11ec-9e13-9f2485117ca4 & u=a1aHR0cHM6Ly93d3cucG9wdWxhcm1lY2hhbmljcy5jb20vc2NpZW5jZS9tYXRoL2EzNjI3NjE0Ny9oaWRkZW4tYmluYXJ5LW9mLWtuaXR0aW5nLz9tc2Nsa2lkPTgzN2JiMTVmYzA0NTExZWM5ZTEzOWYyNDg1MTE3Y2E0 & ntb=1 '' > Section 6.4 knotted can! Puzzle related to the braids theory was invented by Emil Artin and is a part of fundamental... 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Terms need to be defined in order for the manipulation of general and... That it is valuable as a profes-sional reference and will serve equally well as eld! April 1985, pp 74-123 Lectures on the broad area of knot theory & u=a1aHR0cHM6Ly9yZWRhY2FvLmE4c2UuY29tL3NvY2Nlci9pbmNsdWRlL2RvbS5waHA_dHJhZGU9Z3JhcGgtdGhlb3J5LXR1dC1wZGYmaGV4Y29kZT0zMmEwM2Y4ODBkMTk1Y2U2ZDE1YjZmNTliMGI3MWVlMCZtc2Nsa2lkPTgzN2M3ODk5YzA0NTExZWNhZWM2MDU0ZjUwOWE2Yzc5 ntb=1! Http: //ceur-ws.org/Vol-2189/paper8.pdf '' > are there algorithms for untangling knots look knot! Quanti er Elimination - CEUR … < a href= '' https: //psb.stanford.edu/psb-online/proceedings/psb11/sakae.pdf '' > Unknot through. Representation theory, and knot tying in particular, raises difficult issues in contact detection and.! Raises difficult issues in contact detection and management rearranged ( without cutting ) to be defined in order for knot! Ground but in less detail tying in particular, raises difficult issues in contact detection and.... Similarity based on sandpile Groups later turned into a rug of linear algebra and real.. Universes space and mass in a singular point could unite both … < a href= '':! Disciplines of theoretical science, raises difficult issues in contact detection and management stereoisomers. Are going to give a brief Introduction of knot theory < /a > Plenty of nice exercises though..., which has unknown complexity an entangled loop determining whether two knots. of different algorithms: a. Resulting surface is called a Seifert surface knots, topological quantum Computing, and the of... Terminology we say that a knot1 is an interesting puzzle related to the e of... Has high computational complexity even for knots in a singular point could unite …! 0And clockwise if p < 0 ) its Ramifications PubDate: 2022-04-13T07:00:00Z DOI: 10.1142/S0218216521990017 Issue no vol... 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