61. 3–Manifolds And Knot Theory There will be a Conference on 3–Manifolds and knot theory at the University of Texas at Austin, Monday May 23 thru Wednesday May 25, 2005. Speakers are http://www.ma.utexas.edu/topcon4/

Pictures Speakers are: I. Agol (Illinois at Chicago) M. Khovanov (UC Davis) M. Bestvina (Utah) T. Li (Oklahoma State) M. Boileau (Toulouse) Y. Minsky (Yale) S. Boyer (Montreal) M. Scharlemann (UC Santa Barbara) M. Freedman (Microsoft) P. Shalen (Illinois at Chicago) T. Cochran (Rice) Z. Szabo (Princeton) Y. Eliashberg (Stanford) A. Thompson (UC Davis) D. Gabai (Princeton) There will be some funds available to support the attendance of graduate students and postdocs. To apply click on Registration. For details about Hotels please click on Local Information For insights into the work of Cameron Gordon see Other Links.

62. Louis H. Kauffman A topologist working in knot theory discusses the connection between knot theory and statistical mechanics. Sections on cybernetics and knots, Fourier knots and the author's research papers. http://bilbo.math.uic.edu/~kauffman/

Louis H. Kauffman Louis H. Kauffman Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago Chicago, IL 60607-7045 Phone: (312)-996-3066 E-Mail: kauffman AT uic.edu Research I am a topologist working in knot theory and its relationships with statistical mechanics, quantum theory, algebra, combinatorics and foundations. This material is based upon work supported by the National Science Foundation under Grant No. DMS - 0245588 and by a grant to study quantum computation and information theory under the auspices of the Defense Advanced Research Projects Agency (DARPA). I am teaching Calculus, Math 180 in Fall 2005. See Calculus. I taught Applied Linear Algebra, Math 310 in Spring 2005. See Vector Algebra. I taught Calculus, Math 181 in Spring 2005. See Calculus. In July 2004 I taught a workshop on knots and applications at the MSRI workshop at University of British Columbia in Vancouver, BC. See Knots in Vancouver for a course description and for downloads for the course. I visited from September 1, 2003 to August 31, 2004 at the University of Waterloo and the Perimeter Institute in Waterloo, Canada. In the winter term I taught a course on knots and physics at the university. See

63. Oscar Gonzalez: Global Curvature ... Global curvature, ideal knots and models of DNA packing Existence of ideal knots, Journal of knot theory and Its Ramifications 12 (2003) 123133. http://www.ma.utexas.edu/~og/curvature.html

Global curvature, ideal knots and models of DNA packing Global curvature characterizes knot tightness. Overview and acknowledgements During a lunch-time discussion on the ideal shapes of knots and such, the concept of the "global curvature function" of a space curve was born. This function is related to, but distinct from, standard local curvature, and is connected to various physically appealing properties of a curve. Global curvature provides a concise characterization of curve thickness, and of certain ideal shapes of knots as have been investigated within the context of DNA. Moreover, global curvature is connected to the writhing number of a space curve and has applications in the study of self-contact and packing problems for rods. Some of the figures here are based on numerical data from It is a pleasure to thank A. Stasiak V. Katritch and P. Pieranski for making their knot data available. Related articles Journal of Knot Theory and Its Ramifications Journal of Statistical Physics Contemporary Mathematics Calculus of Variations and Partial Differential Equations

64. Geometry And The Imagination Has a small section on knot theory at an introductory level. Also has sections on orbifolds, polyhedra and topology. http://math.dartmouth.edu/~doyle/docs/gi/gi/gi.html

Bicycle tracks C. Dennis Thron has called attention to the following passage from The Adventure of the Priory School , by Sir Arthur Conan Doyle: `This track, as you perceive, was made by a rider who was going from the direction of the school.' `Or towards it?' `No, no, my dear Watson. The more deeply sunk impression is, of course, the hind wheel, upon which the weight rests. You perceive several places where it has passed across and obliterated the more shallow mark of the front one. It was undoubtedly heading away from the school.' Problems Discuss this passage. Does Holmes know what he's talking about? Try to come up with a method for telling which way a bike has gone by looking at the track it has left. There are all kinds of possibilities here. Which methods do you honestly think will work, and under what conditions? For example, does your method only work if the bike has passed through a patch of wet cement? Would it work for tracks on the beach? Tracks on a patch of dry sidewalk between puddles? Tracks through short, dewy grass? Tracks along a thirty-foot length of brown package-wrapping paper, made by a bike whose tires have been carefully coated with mud, and which has been just ridden long enough before reaching the paper so that the tracks are not appreciably darker at one end of the paper than the other? Try to determine the direction of travel for the idealized bike tracks in Figure Figure 1: Which way did the bicycle go?

65. The Math Forum - Math Library - Knot Theory The Math Forum s Internet Math Library is a comprehensive catalog of Web sites and Web pages relating to the study of mathematics. This page contains sites http://mathforum.org/library/topics/knot_theory/

Browse and Search the Library Home Math Topics Topology : Knot Theory Library Home Search Full Table of Contents Suggest a Link ... Library Help Selected Sites (see also All Sites in this category The KnotPlot Site - Robert Scharein A collection of knots and links, viewed from a partly mathematical perspective. Images on this site were created with KnotPlot, a program designed to visualize and manipulate mathematical knots in three and four dimensions. A picture gallery, description of the features of the program, and links to other relevant sites on the Web are included. Also available at http://www.pims.math.ca/knotplot/ more>> Knots on the Web - Peter Suber, Earlham College A comprehensive list of knot resources on the Web, annotated and organized into three categories: knot tying, knot theory, and knot art. Also Knot books and a Knots Gallery displaying images from the newsgroup rec.crafts.knots. more>> Knot Theory (The Geometry Junkyard) - David Eppstein, Theory Group, ICS, UC Irvine An extensive annotated list of links to material on geometric questions arising from knot embeddings. more>> KT (Knot Theory) Online - Bryson R. Payne; North Georgia College and State University

66. ArXiv.org Search Search for papers held at LANL with the word 'knot' in them. http://arXiv.org/find/math/1/fr: knot/0/1/0/past,all/0/1

Search arXiv.org Search syntax error Usually this occurs because of: Misplaced parentheses, which are used for grouping only. Currently you can not search for items containing parentheses. Missing parentheses for grouping. 'a OR b OR c' must berepresented as either '(a OR b) OR c' or 'a OR (b OR c)'. Incorrect use of the operators like AND, OR, NOT, EXACT, or SOUNDEX. Note that you can't say 'NOT a'; you must use something like 'a NOT b'. These operators can currently not be searched for as words. Author/title/abstract search Select subject areas to search Computer Science Mathematics Nonlinear Sciences Physics [archive: All astro-ph cond-mat gr-qc hep-ex hep-lat hep-ph hep-th math-ph nucl-ex nucl-th physics quant-ph Quantitative Biology Select years to search (default is to search all years) Past year or the year or the years from to Author(s): Title: Abstract: Full Record: Comments: Journal-ref: Subject-class: Report-no: AND OR AND NOT Author(s): Title: Abstract: Full Record: Comments: Journal-ref: Subject-class: Report-no: AND OR AND NOT Author(s): Title: Abstract: Full Record: Comments: Journal-ref: Subject-class: Report-no: Show hits per page or selections to default values.

67. Math Forum - Ask Dr. Math Researching how knot theory is related to topology and modern algebra. Question What is knot theory? Purpose To prove that knot theory is related to http://mathforum.org/library/drmath/view/51653.html

Associated Topics Dr. Math Home Search Dr. Math What is Knot Theory? Date: 03/10/98 at 23:46:23 From: keatha Subject: the knot theory Dear Dr. Math, In order for me to complete my project on knot theory, I need a question, hypothesis, and a purpose. I have this information, I just want you to proofread it to see if I need to add anything or delete anything. Question: What is Knot Theory? Purpose: To prove that knot theory is related to topology. Hypothesis: If topology deals with the bending of objects, then it (knot theory) is related to topology. Thanks a million, Keatha Date: 03/11/98 at 10:32:24 From: Doctor Sonya Subject: Re: the knot theory Dear Keatha, I just wrote a big paper on knot theory. What a great topic to choose. Your question is good, and your hypothesis is correct that topology deals with the bending of objects, and that topology is related to knot theory. However, if you really want to answer the question, "What is knot theory?" I don't think, "Knot theory is related to topology." is enough of an answer. When you get to the real details of it, knot theory is also closely related to modern algebra (although you'd never know it from all the pretty pictures of knots!). The most exciting thing about knot theory that deals with the bending and twisting of knots is something called the "Jones Polynomial," discovered by a mathematician named Jones and written about by Louis Kauffman. If you still have to do more research, this might be interesting for you to include. Also, take a look at "The Knot Theory Home Page"

68. Liverpool Knot Theory Group: Publication List Articles and preprints from 1987 onward. Some are available for download in postscript format. http://www.liv.ac.uk/~su14/knotprints.html

Liverpool University Knot Theory Group Articles and Preprints (1987 onwards) Copies of the following papers can be requested by sending e-mail to morton@liv.ac.uk You can download PostScript versions of some of the recent preprints. This list can be found at URL http://www.liv.ac.uk/~su14/knotprints.html or by following links from http://www.liv.ac.uk/maths/ Programs for some knot theory calculations can be found in the list of programs of the Liverpool Knot Theory group You may also enjoy a collection of material on the Borromean Rings H.R.Morton and H.B.Short, `The 2-variable polynomial of cable knots'. Math. Proc. Camb. Philos. Soc. H.R.Morton, `Polynomials from braids'. In `Braids', ed. Joan S. Birman and Anatoly Libgober, Contemporary Mathematics 78, Amer. Math. Soc. (1988), 375-385. H.R.Morton and P.Traczyk, `The Jones polynomial of satellite links around mutants'. In `Braids', ed. Joan S. Birman and Anatoly Libgober, Contemporary Mathematics 78, Amer. Math. Soc. (1988), 587-592. P.R.Cromwell

69. Problems In Knot Theory There are already several collections of problems in knot theory available. So this is mainly a collection of problems of personal interest. http://math.ucr.edu/~xl/knotprob/knotprob.html

Problems in Knot Theory There are already several collections of problems in knot theory available. So this is mainly a collection of problems of personal interest. Feel free to send me any comments you might have: maybe the answer is obvious, maybe the answer exists already in literature, maybe the problem should be attributed more appropriately, etc. Also, it is welcome if anyone would like to suggest some problems to this collection. Nevertheless, since this is a personal collection, I reserve the right to decide whether to put the suggested problems into this collection according to my own taste. Surgery modification is a procedure of modifying a link in or by performing a Dehn surgery on an unknotted circle having linking number zero with all components of the link in question. Surgery equivalence is then the equivalence relation generated by surgery modification. The classification of links up to surgery equivalence is done by J. Levine (Topology,1987). We may refine the notion of surgery modification as follows. For simplicity, let us consider only links with vanishing linking numbers. If we assume further that the unknotted circle used to perform a surgery modification has vanishing Milnor's triple linking numbers with other componenets of the given link, we call such a surgery modification of "second order". It can be shown that the Sato-Levine invariant is invariant under surgery modification of second order. Classify links with zero linking numbers up to surgery equivalence of second order.

70. Jorge Pullin Quantizing general relativity brings knot theory into quantum gravity. The Jones polynomial is shown to give rise to physical states of quantum gravity. Links to research papers by the author. http://www.phys.lsu.edu/faculty/pullin

Jorge Pullin Horace Hearne Chair in theoretical Physics, Louisiana State University Adjunct Professor of Physics, PennState Head, "Coast to Cosmos" focus area, Center for Computation and Technology (CCT) Ph.D., Instituto Balseiro Honors and awards Phone/Fax: (225)578-0464 pullin@lsu.edu Horace Hearne Institute for Theoretical Physics Want to hear those pipes? Research. ... Background. Research My research interests cover many aspects of gravitational physics, both classical and quantum mechanical. I am currently focusing on two topics: quantum gravity and black hole collisions . You can also get my complete publication list , but if you want to get the latest, go to the Hearne Institute page and click on publications. The explanations that follow are a bit longish, feel free to skip to the next topic if you get bored! Quantum gravity I collaborate with Rodolfo Gambini, of the University of the Republic in Montevideo, Uruguay, our collaboration has been going on since 1990. We coauthored a book "Loops, knots, gauge theories and quantum gravity" in 1996 and have published many papers together. We study the quantization of general relativity using canonical methods. There is a small community pursuing this kind of research, which is complementary to the mainstream approach to quantum gravity: string theory. String theorists believe that one cannot quantize general relativity because it is not a fundamental theory and one has to replace it with string theory in order to quantize it. General relativity will be an "effective" "low energy" theory.

71. Knot Theory knot theory studies the placement of onedimensional objects called strings The importance of knot theory in the study of three-dimensional flows comes http://www.drchaos.net/drchaos/Book/node141.html

Next: Crossing Convention Up: Knots and Templates Previous: Example: Duffing Equation Knot Theory Knot theory studies the placement of one-dimensional objects called strings [23,24,25] in a three-dimensional space. Figure 5.6: Planar diagrams of knots: (a) the trivial or unknot ; (b) figure-eight knot ; (c) left-handed trefoil ; (d) right-handed trefoil; (e) square knot ; (f) granny knot. A simple and accurate picture of a knot is formed by taking a rope and splicing the ends together to form a closed curve. A mathematician's knot is a non-self-intersecting smooth closed curve (a string ) embedded in three-space. A two-dimensional planar diagram of a knot is easy to draw. As illustrated in Figure , we can project a knot onto a plane using a solid (broken) line to indicate an overcross (undercross). A collection of knots is called a link (Fig. Figure 5.7: Link diagrams: (a) Hopf link ; (b) Borromean rings ; (c) Whitehead link The same knot can be placed in space and drawn in planar diagram in an infinite number of different ways. The equivalence of two different presentations of the same knot is usually very difficult to see. Classification of knots and links is a fundamental problem in topology. Given two separate knots or links we would like to determine when two knots are the same or different. Two knots (or links) are said to be

72. Knots And Links In Braid Notation Listing of every knot of up to 11 crossings in braid notation. Useful for computer calculations in knot theory. http://www.scoriton.demon.co.uk/knots.html

a Knots and Links in Braid notation This document (this page with its associated tables) is intended as a resource for anyone interested in knots and links. Every knot up to 11 crossings and every link up to 10 crossings is listed here. The main feature of the listings is that every knot and link is shown with both its numeric notation and a braid notation. The numeric notation allows cross-reference with other works, while the braid notation is by far the easiest form to manipulate by computer. An n -crossing knot or link has a braid notation on at most 5/9.( n +2) strings [A] History of this document This section records the significant changes to this document, so that you can quickly tell if you need to download any information since your last visit. The most recent entries are at the top. 2002-01-06: Links added. This has expanded the tables to include all links up to 10 crossings, and expanded the list of errors in [C] to include those relating to links. 2000-10-19: My thesis [A] uploaded, and pointers to it added to this document. 2000-07-24: Braid index of 11(C123) is at least 4. No ? left anywhere in the tables. Removed 10(6VI) because it is the same as 10(5II), the Perko pair. Thanks to Richard Hadji and Hugh Morton.

73. PlanetMath: Knot Theory Much of knot theory is devoted to telling when two knot diagrams represent the This definition is used by Charles Livingston in his book knot theory. http://planetmath.org/encyclopedia/KnotTheory.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About knot theory (Topic) Knot theory is the study of knots and links Roughly a knot is a simple closed curve in , and two knots are considered equivalent if and only if one can be smoothly deformed into another. This will often be used as a working definition as it is simple and appeals to intuition. Unfortunately this definition can not be taken too seriously because it includes many pathological cases, or wild knots, such as the connected sum of an infinite number of trefoils. Furthermore one must be careful about defining a ``smooth deformation'', or all knots might turn out to be equivalent! (We shouldn't be allowed to shrink part of a knot down to nothing.) Links are defined in terms of knots, so once we have a definition for knots we have no trouble defining them.

74. Sam Ruby: Knot Theory knot Robert G. Scharein knot theory is a branch of algebraic topology where Sam Ruby’s entry knot theory really gets at the essence of what he finds http://www.intertwingly.net/blog/1811.html

intertwingly Knot Theory Robert G. Scharein 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. In response to a question from Christian Romney: the above web page was the inspiration for my current favicon.ico. When asked to describe my passions, I typically reply " Sam Ruby takes a perverse pleasure in integrating disparate things ". PHP and Java. Bean Scripting Framework. Common Language Runtime. SOAP. Protocols and Formats. Placement problems. Embedding of one topological space into another. Intertwingled knotty problems. Tue, 29 Jun 2004 Posted by Christian Romney at I'd say that depends very much on the subject matter and the IQ of the people generating inbound links. Posted by Mark at I honestly hope your readers are smart enough not to care about IQs or percentiles. Smart enough to see beyond that. Posted by Marco at Wed, 30 Jun 2004 Perhaps it would have been better stated that Sam's readers (and of course Sam) are all probably well Cheers

75. Knot Theory RAP homepage for knot theory RAP. First, we will discuss some of the basics of knot theory such as knot projections, types of knots, knot groups, etc., http://www.math.uiuc.edu/~brinkman/teaching/rap/

Knot Theory RAP When Th 2:00-3:50 (with a break) Where Altgeld Organizers Peter Brinkmann and Nadya Shirokova Email brinkman@math.uiuc.edu or nadya@math.uiuc.edu This RAP serves two purposes. First, we will discuss some of the basics of knot theory such as knot projections, types of knots, knot groups, etc., loosely based on some introductory texts (such as Adams, 'The Knot Book'; Crowell-Fox, 'Introduction to Knot Theory'; Burde-Zieschang, 'Knots'; Rolfsen, 'Knots and Links'; Kauffman, 'On Knots'). Second, the RAP will serve as a platform for talks about current research (such as Vassiliev invariants, geometric knot theory, and slalom knots). Archive: Fall 01 Spring 02 Peter Brinkmann $Date: 2002-06-03 01:33:18-05 $

76. DNA AND KNOT THEORY Question How can knot theory help us understand DNA packing? By thinking of DNA as a knot, we can use knot theory to estimate how hard DNA is to unknot http://www.tiem.utk.edu/~gross/bioed/webmodules/DNAknot.html

DNA AND KNOT THEORY Introduction: DNA is the genetic material of all cells, containing coded information about cellular molecules and processes. DNA consists of two polynucleotide strands twisted around each other in a double helix. The first step in cellular division is to replicate DNA so that copies can be distributed to daughter cells. Additionally, DNA is involved in transcribing proteins that direct cell growth and activities. However, DNA is tightly packed into genes and chromosomes. In order for replication or transcription to take place, DNA must first unpack itself so that it can interact with enzymes. DNA packing can be visualized as two very long strands that have been intertwined millions of times, tied into knots, and subjected to successive coiling. However, replication and transcription are much easier to accomplish if the DNA is neatly arranged rather than tangled up in knots. Enzymes are essential to unpacking DNA. Enzymes act to slice through individual knots and reconnect strands in a more orderly way. Importance: We can gain insight into the unknotting of DNA by using principles of topology. Topologists study the invariant properties of geometric objects, such as knots. Tightly packed DNA in the genes must quickly unknot itself in order for replication or transcription to occur. This is a topological problem.

77. Classical Knot Theory Classical knot theory. Epiphany Term 2003. Lecturer Dr C. Kearton. I shall distribute copies of the lecture notes and problem sheet at the first lecture. http://maths.dur.ac.uk/~dma0ck/gradtop.html

Classical Knot Theory Epiphany Term 2003 Lecturer: Dr C. Kearton I shall distribute copies of the lecture notes and problem sheet at the first lecture. If you want one before that, come and see me. Course material Format Lecture Notes Postscript PDF Problem Sheet Postscript PDF Problem Sheet with (some) solutions Postscript PDF Course Description Postscript PDF Software: GSview (viewer for Postcript files) Adobe Acrobat Reader (viewer for PDF files) Return to page top Recommended Books: See the lecture notes. Return to page top Links: e-mail to me. Durham Pure Mathematics Home page (formal page). Durham Pure Mathematicians' page (intended for members of the department but open to all). Department of Mathematical Sciences home page The University of Durham home page application/x-dvi; xdvi %s application/postscript; ghostview %s (or similar) to your .mailcap file. Info on HTML etc. NCSAA Beginner's Guide to HTML Computing Information for the Dept. of Mathematical Sciences - homepages University of Durham - ITS - About the Internet A Quick Review of HTML 3.0 ... Hypertext Markup Language - 2.0 - The HTML Coded Character Set

78. Dror Bar-Natan: Classes: 2001-02: Knot Theory Seminar WB Raymond Lickorish, An Introduction to knot theory, GTM 175, SpringerVerlag, See almost any book on knot theory. None. Seifert surfaces and knot http://www.math.toronto.edu/~drorbn/classes/0102/KnotTheory/

Dror Bar-Natan Classes Seminar on Knot Theory Instructor: Dror Bar-Natan drorbn@math.huji.ac.il Meetings: Mondays 12:00-14:00 at Upper Papik. Office hours: Tuesdays 14:00-15:00 in my office, Mathematics 309. Agenda: Have every student give at least one fun lecture on elementary knot theory. Prerequisites: Meant for advanced undergraduate students. Reading material: W.B. Raymond Lickorish An Introduction to Knot Theory , GTM 175, Springer-Verlag, New York 1997. The books by Kauffman and Rolfsen V. V. Prasolov and A. B. Sossinsky's Knots, links, braids and 3-manifolds: an introduction to the new invariants in low-dimensional topology, Translations of Mathematical Monographs 154, American Mathematical Society 1997. Thanks, Raz , for the tushim! Some suggested topics for student lectures: (more may be added later, and you may choose subjects not on this list!) Topic Speaker and Date Details Dependencies Reidemeister's theorem Lior Zaibel, March 18th Prove that any two diagrams for the same knot are connected by a sequence of Reidemeister moves. See almost any book on knot theory. None.

79. Dror Bar-Natan Classes 2003-04 Math 1350F - Knot Theory Agenda Use knot theory as an excuse to learning deep and beautiful mathematics. Instructor Dror BarNatan, drorbn@math.toronto.edu, Sidney Smith 5016G, http://www.math.toronto.edu/~drorbn/classes/0304/KnotTheory/

Dror Bar-Natan Classes FEEDBACK Math 1350F - Knot Theory Fall Semester 2003 Agenda: Use knot theory as an excuse to learning deep and beautiful mathematics. Instructor: Dror Bar-Natan drorbn@math.toronto.edu , Sidney Smith 5016G, 416-946-5438. Office hours: Thursdays 12:30-1:30. Classes: Tuesdays 1-3 at Sidney Smith 5017A and Thursdays 2-3 at Sidney Smith 2128 Announcements September 8: Welcome back to UofT! Course Calendar Week of ... September 8 Handout: About This Class Handout: Some Non Obvious Examples Handout: Pathologies Class notes for Tuesday September 9, 2003 (non obvious examples, pathologies, Reidemeister's theorem, 3-colorings, the Kauffman bracket) Homework Assignment 1 Class notes for Thursday September 11, 2003 (R-II and R-III for the Kauffman bracket, R-I, the writhe, the Jones polynomial, a little programming) September 15 Handouts: The Rolfsen Knot Table Torus Knots The Knot 9-32 Class notes for Tuesday September 16, 2003 (The reverse and the mirror of a knot, sketch of the proof of Redemeister's theorem, linking numbers, connected sum, classification of surfaces and the genus of a knot) Homework Assignment 2 Class notes for Thursday September 18, 2003

80. KNOT THEORY LINKS Galois Relations on Knot Invariants, by T. Gannon and MA Walton, 95/09 Noncommutative Geometry of Finite Groups, by K. Bresser et al., 95/09 http://web.mit.edu/afs/athena.mit.edu/user/r/e/redingtn/www/netadv/knots.html

KNOTS AND BRAIDS (AND) LINKS Galois Relations on Knot Invariants, by T. Gannon and M. A. Walton, 95/09 Noncommutative Geometry of Finite Groups, by K. Bresser et al., 95/09 The Many-Anyon Problem, by Claus Montonen, 95/02 Braided Geometry and q-Minkowski Space, by Shahn Majid, 94/10 ... Spin networks, by John C. Baez To contribute to this page, write Norman Redington, redingtn@mit.edu RETURN TO THEORETICAL PHYSICS LINKS RETURN TO NET ADVANCE OF PHYSICS HOMEPAGE

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