81. PHYSICAL AND NUMERICAL MODELS IN KNOT THEORY PHYSICAL AND NUMERICAL MODELS IN knot theory. Together, the chapters explore four major themes physical knot theory, knot theory in the life sciences, http://www.worldscibooks.com/mathematics/5766.html

Home Browse by Subject Bestsellers New Titles ... Browse all Subjects Search Bookshop New Titles Editor's Choice Bestsellers Book Series ... Series on Knots and Everything - Vol. 36 PHYSICAL AND NUMERICAL MODELS IN KNOT THEORY Including Applications to the Life Sciences edited by Jorge A Calvo (Ave Maria University, USA) , Kenneth C Millett (University of California, Santa Barbara, USA) , Eric J Rawdon (Duquesne University, USA) (Université de Lausanne, Switzerland) The physical properties of knotted and linked configurations in space have long been of interest to mathematicians. More recently, these properties have become significant to biologists, physicists, and engineers among others. Their depth of importance and breadth of application are now widely appreciated and valuable progress continues to be made each year. This volume presents several contributions from researchers using computers to study problems that would otherwise be intractable. While computations have long been used to analyze problems, formulate conjectures, and search for special structures in knot theory, increased computational power has made them a staple in many facets of the field. The volume also includes contributions concentrating on models researchers use to understand knotting, linking, and entanglement in physical and biological systems. Topics include properties of knot invariants, knot tabulation, studies of hyperbolic structures, knot energies, the exploration of spaces of knots, knotted umbilical cords, studies of knots in DNA and proteins, and the structure of tight knots. Together, the chapters explore four major themes: physical knot theory, knot theory in the life sciences, computational knot theory, and geometric knot theory.

82. FUNCTORIAL KNOT THEORY FUNCTORIAL knot theory. All the definitions from both knot theory and category theory are included, as well as proofs of many basic results which are http://www.worldscibooks.com/mathematics/4542.html

Home Browse by Subject Bestsellers New Titles ... Browse all Subjects Search Bookshop New Titles Editor's Choice Bestsellers Book Series ... Series on Knots and Everything - Vol. 26 FUNCTORIAL KNOT THEORY Categories of Tangles, Coherence, Categorical Deformations, and Topological Invariants by David N Yetter (Kansas State University) Almost since the advent of skein-theoretic invariants of knots and links (the Jones, HOMFLY, and Kauffman polynomials), the important role of categories of tangles in the connection between low-dimensional topology and quantum-group theory has been recognized. The rich categorical structures naturally arising from the considerations of cobordisms have suggested functorial views of topological field theory. This book begins with a detailed exposition of the key ideas in the discovery of monoidal categories of tangles as central objects of study in low-dimensional topology. The focus then turns to the deformation theory of monoidal categories and the related deformation theory of monoidal functors, which is a proper generalization of Gerstenhaber's deformation theory of associative algebras. These serve as the building blocks for a deformation theory of braided monoidal categories which gives rise to sequences of Vassiliev invariants of framed links, and clarify their interrelations. Contents: Knots and Categories: Monoidal Categories, Functors and Natural Transformations

83. Knot Theory Maths knot theory. knot theory is a branch of mathematics dealing with tangled loops. When there s just one loop, it s called a knot. http://f2.org/maths/kt/

Up to Home Maths Site Map Text version Knot Theory Fred Curtis - Mar 2001] This page is a tiny introduction to Knot Theory. It describes some basic concepts and provides links to my work and other Knot Theory resources. What is Knot Theory? My Interests Old papers I'm typing up References What is Knot Theory? Knot theory is a branch of mathematics dealing with tangled loops. When there's just one loop, it's called a knot . When there's more than one loop, it's called a link and the individual loops are called components of the link. A picture of a knot is called a knot diagram or knot projection . A place where parts of the loop cross over is called a crossing . The simplest knot is the unknot or trivial knot , which can be represented by a loop with no crossings. The big problem in knot theory is finding out whether two knots are the same or different. Two knots are regarded as being the same if they can be moved about in space, without cutting, to look exactly like each other. Such a movement is called an ambient isotopy - the ambient refers to moving the knot through 3-dimensional space, and

84. Knot Theory History History. This web site is for the basic understanding of where knot theory came from. The history of knot theory began in the very early 1800 s. http://www.kpbsd.k12.ak.us/kchs/JimDavis/CalculusWeb/Knot Theory History.htm

Knot Theory History This web site is for the basic understanding of where Knot Theory came from. Johann Frederich Carl Gauss (1775-1855) The history of Knot Theory began in the very early 1800's. Johann Frederich Carl Gauss was a German mathematician who was interested in the idea of knots. His contribution to the Knot Theory was "analysis situs." Analysis situs deals with the mathematical differences between the simple and complex knots. The mathematical differences are very different to the human eye, but until then, they were difficult to distinguish them mathematically. Lord Kelvin, William Thomson (1824-1907) Until the late 1800's no one was really interested in the idea of knots. Lord Kelvin, also known as William Thomson, was an English scientist who taught at the University of Glasgow. One of his greatest achievements was the idea of "ether." His belief was that the universe was filled with an invisible and frictionless fluid called "ether," and the atoms were the vortices in this fluid in the shape of knots. Thus a table of knots would be a table of elements. This of course was later on disproved when atomic ideas began to emerge. The Knot Theory soon faded away until the 1950's. James Watson (left) and Francis Crick (right) The race to discover the shape of deoxyribonucleic acid (DNA) was all over the world. When it was finally finish, the idea of knots came back to life. In 1953, James Watson and Francis Crick discovered that DNA was in the shape of a double helix. This concept led to discovering that DNA had parts of it knotted in many locations. With the help of both of these scientists, Knot Theory came back with scientist thinking up new ideas fighting diseases and understanding the make-up of all living things.

85. Louis H. Kauffman I am a topologist working in knot theory and its relationships with statistical KNOTS A short course in knot theory from Reidemeister moves to state http://www.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

86. Math 486-S Knot Theory Math 486S knot theory At Left Photo of Gordian Knot by sculptor John Robinson, the creator of many mathematically-themed artworks. http://web.library.emory.edu/subjects/science/math/486-S.html

EUCLID Catalog Library Info Research Assistance Subject and Course Guides Research Tutorial EndNote Use ... Library Publications Math 486-S: Knot Theory EUCLID Databases e-Journals Reserves Direct Table Of Contents Getting Started The Knot Book , by Colin C. Adams Books ... Contact Information At Left: Photo of " Gordian Knot " by sculptor John Robinson, the creator of many mathematically-themed artworks . His " Immortality " is a Möbius band in a trefoil knot. Getting Started top Knot theory , the mathematical study of knots, is part of a branch of mathematics called topology (the definitions linked here are from MathWorld ). It has applications in chemistry, biology, and physics. It is almost paradoxical that while many questions in knot theory are simple to state and easy to understand, the solutions to those questions can require a deep and intricate understanding of higher mathematics.

87. Syllabus For Topics In Geometry & Topology: Knot Theory, Fall 2003 Supplementary Texts The Knot Book by Adams and knot theory Its Applications by Murasugi. Requirements We will assume familiarity with linear algebra and http://www.math.columbia.edu/~ikofman/knotsyllabus.html

Prof. Ilya Kofman Office: 607 Mathematics, phone: 854-3210 Email: ikofman@math.columbia.edu Web site: http://www.math.columbia.edu/~ikofman/ Course Time and Place: 2:40pm - 3:55pm Tuesday and Thursday, 520 Mathematics Building Text: Knots, Links, Braids and 3-Manifolds: An Introduction to the New Invariants in Low-Dimensional Topology by Prasolov and Sossinsky Supplementary Texts: The Knot Book by Adams and by Murasugi Requirements: We will assume familiarity with linear algebra and some group theory. Basic Topology is not required, though it provides a useful perspective for knot theory. You will be asked to think about mathematical ideas and to prove what you think is true, so a certain amount of mathematical maturity is expected. Material Covered: Basics of knots, links, and their diagrams, elementary knot invariants, Jones-type polynomials, Vassiliev invariants, braids, Seifert matrices and Alexander polynomial, 3-manifold topology, including Heegaard diagrams, surgery, and branched covers of knots. Homework: Assignments will be announced in class and then posted on this website. Any changes will be announced in class. Incomplete work with good progress will be rewarded.

88. Knot Theory: Information From Answers.com knot theory Trefoil knot, the simplest nontrivial knot. knot theory is a branch of topology that was inspired by observations, as the name suggests, http://www.answers.com/topic/knot-theory

showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Wikipedia Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping knot theory Wikipedia knot theory Trefoil knot, the simplest non-trivial knot. Knot theory is a branch of topology that was inspired by observations, as the name suggests, of knots . But progress in the field no longer depends on experiments with twine. Knot theory concerns itself with abstract properties of theoretical knots In mathematical jargon , knots are embeddings of the closed circle in three-dimensional space. An ordinary knot is converted to a mathematical knot by splicing its ends together. The topological theory of knots asks whether two such knots can be rearranged to match, without opening the splice. The question of untying an ordinary knot has to do with unwedging tangles of rope pulled tight. A knot can be untied in the topological theory of knots if and only if it is equivalent to the unknot , a circle in 3-space. History Knot theory originated in an idea of Lord Kelvin 's (1867), that atoms were knots of swirling vortices in the

89. QA Quantum Algebra Key College Publishing Why Knot? An Introduction to the knot theory! “Why Knot? is a witty, lighthearted exploration of a topic of knot theory is appealing and accessible it naturally lends itself to lots http://eprints.math.duke.edu/archive/q-alg

Fri 16 Sep 2005 Search Submit Retrieve Subscribe ... iFAQ QA Quantum Algebra Calendar Search Authors: All AB CDE FGH ... U-Z New articles (last 12) 15 Sep math.QA/0509302 Planar algebras and Kuperberg's 3-manifold invariant. Vijay Kodiyalam , V. S. Sunder . 19 pages. QA OA 14 Sep math.QA/0509284 Duality and equivalence of module categories in noncommutative geometry I. Jonathan Block QA OA 13 Sep math.QA/0509264 The Gromov-Witten potential associated to a TCFT. Kevin J. Costello . 38 pages. QA AG 13 Sep math.QA/0509260 Factorizations of Polynomials over Noncommutative Algebras and Sufficient Sets of Edges in Directed Graphs. Israel Gelfand , Sergei Gelfand , Vladimir Retakh , Robert Lee Wilson . 15 pages. QA CO 13 Sep math.QA/0509254 On the Hochschild homology of quantum SL(N). Tom Hadfield , Ulrich Kraehmer . 4 pages. QA RA 13 Sep math.QA/0509251 On R-matrix representations of Birman-Murakami-Wenzl algebras. A. P. Isaev , O. V. Ogievetsky , P. N. Pyatov . 9 pages. Proc. of Steklov Math. Institute vol. 246, 2004, pp.134-141. QA Cross-listings 14 Sep math.RT/0509276 Weyl modules, affine Demazure modules, fusion products and limit constructions. Ghislain Fourier , Peter Littelmann RT QA 13 Sep math.RT/0509252

90. A Family Of Impossible Figures Studied By Knot Theory studied by knot theory. Corinne Cerf. Mathematics Dept., CP 216. Université Libre de Bruxelles. B1050 Bruxelles, Belgium. ccerf@ulb.ac.be. 1. Introduction http://www.mi.sanu.ac.yu/vismath/cerf/

A family of impossible figures studied by knot theory Corinne Cerf Mathematics Dept., CP 216 B-1050 Bruxelles, Belgium ccerf@ulb.ac.be 1. Introduction Impossible figures are fascinating objects, related to art, psychology, and mathematics [ ]. Lionel and Roger Penrose (father and son) introduced the impossible tribar in 1958 [ Fig. 1 ). A figure is called impossible when "a contradiction in our interpretation is noticed but does not result in our rejecting it in favour of a consistent one" [ ]. The object represented in Fig. 1 is an impossible figure because our mind tries to interpret it as a three-dimensional (3D) object in the Euclidean space, with straight edges and planar faces, instead of interpreting it, for example, as a two-dimensional object drawn on the paper plane (which is perfectly possible). Fig. 1 Impossible figures have inspired researchers with more than one hundred papers (see Kulpa [ ] for an extensive bibliography), and the Dutch artist Escher [ ] with some famous drawings (see e.g. Fig. 2

91. Differential Geometry And Knot Theory Differential Geometry and knot theory. The knot theory part of the course is concerned with various ways to embed a circle in Euclidean space and how http://www.ma.umist.ac.uk/kd/ma351/ma351.html

Next: Introduction Differential Geometry and Knot Theory General Information General description: The differential geometry part of this course involves the study of curves and surfaces in three-dimensional Euclidean space. Using vector calculus and moving frames of reference on curves embedded in surfaces we can define quantities such as Gaussian curvature that allow us to distinguish among surfaces. The knot theory part of the course is concerned with various ways to embed a circle in Euclidean space and how two knots can be distinguished from one another. Course description: Tangent vectors, vector fields, differentiable maps, curves, Frenet frames, surfaces, shape operator, Gaussian and mean curvature, knots, knot groups, Alexander polynomials. See the online documents relating to this course: Curves Surfaces Knots Author: Professor C.T.J. Dodson Homepage You can email me for any further information at: ctdodson@manchester.ac.uk Other On-Line Mathematical Materials: Mathematical Formulae and Tables (PDF format). Revision Notes on Series (PDF format).

92. NSF Knot Theory Grant knot theory for Preservice and Practicing Secondary Mathematics Teachers NSF DUE Award 0126685. Principal Investigators. Neil Portnoy Thomas W. Mattman http://www.csuchico.edu/math/mattman/NSF.html

Knot Theory for Preservice and Practicing Secondary Mathematics Teachers NSF DUE Award #0126685 Principal Investigators Neil Portnoy Thomas W. Mattman Department of Mathematics Department of Mathematics and Statistics Stony Brook University California State University, Chico Project Summary Anticipated Learning Outcomes Lecture Notes Mathlet ... Links

93. Knot Theory Links knot theory for Teachers. http//www.csuchico.edu/math/mattman/NSF.html A knot theory Primer. http//www.inst.bnl.gov/~wei/contents.html http://www.csuchico.edu/math/mattman/NSF/Links.html

Knot Theory Links Knot Theory for Teachers http://www.csuchico.edu/math/mattman/NSF.html Knots on the Web http://www.earlham.edu/~peters/knotlink.htm Knot Plot http://www.pims.math.ca/knotplot/ A Knot Zoo http://www.pims.math.ca/knotplot/zoo/zoo1.html Knot Theory Online http://www.freelearning.com/knots/ Tie Me Up http://www.math.cuhk.edu.hk/publect/lecture4/ A Knot Theory Primer http://www.inst.bnl.gov/~wei/contents.html Mathematics and Knots http://www.cpm.informatics.bangor.ac.uk/exhib/knotexhib.html MathWorld's Knot Theory http://mathworld.wolfram.com/KnotTheory.html Planet Math's Knot Theory http://planetmath.org/encyclopedia/KnotTheory.html The Mathematical Atlas http://www.math.niu.edu/~rusin/known-math/index/57MXX.html Knots to a Mathematician http://129.49.101.235/math/index.html#lectures Hyperbolic Knots http://www.pims.math.ca/knotplot/hyper/ Last Updated: June 6, 2005 Please let us know about any links that are not working, or send additional link suggestions to TMattman@csuchico.edu

94. Knot Theory (190) Course, Fall 2003 With a little care we can do quite a lot of knot theory without needing to talk about the If you get really interested in knot theory and want a less http://www.math.ucsd.edu/~justin/190.html

Knot theory (190) course, Fall 2003 Monday-Wednesday-Friday from 2.00-2.50 in room HSS 2321. Lecturer: Justin Roberts The traditional first course in topology deals with ``point-set topology'': the study of metric and topological spaces, continuity, compactness, connectedness, and other properties beginning with ``c''. This branch of the subject is really just a part of analysis, and while it is important for the foundations of the subject and can help you learn to write proofs properly, it can all seem very abstract and dry. Where are the doughnuts, coffee cups, pretzels, rubber sheets, knots and so on of popular topology? Traditionally, we teach the more geometric, visual side of the subject after teaching all the basic tools. This is not unreasonable, but it does take a long time to do properly, and is quite hard to motivate because it turns history on its head. After all, people have been using and thinking about knots for thousands of years, but the definition of a topological space is only a hundred years old. Fortunately it isn't necessary to work this way round. With a little care we can do quite a lot of knot theory without needing to talk about the foundational aspects of topology. I intend to start this way, and then try to develop a little bit of the abstract theory only if we really start to need it later on.

95. Open Problems In Knot Theory A List of Approachable Open Problems in knot theory Suggested by Colin Adams during the knot theory Workshop at Wake Forest University during June 2428 http://www.williams.edu/Mathematics/cadams/knotproblems.html

A List of Approachable Open Problems in Knot Theory .ps and .pdf files also available) Suggested by Colin Adams during the Knot Theory Workshop at Wake Forest University during June 24-28, 2002. Problems. What knots with high symmetries have projections that demonstrate this symmetry? (eg. the Figure-8 knot) Find specific families of knots satisfying the property c(K_1#K_2) = c(K_1)+c(K_2), where c=c(K) is the crossing number and # means knot composition. (eg. This is known for alternating knots.) What about torus knots? [In a 2003 preprint, Yuanan Diao demonstrated that this does hold for compositions of torus knots, as well. This was also independently proved by Herman Gruber. His paper is available at arxiv.org under math.GT/0303273.] When is a knot equivalent to its inverse? (The inverse has the same projection but with an opposite orientation). (eg. the trefoil and its inverse) Hass and Lagrias proved that if you have an n-crossing projection of the trivial knot, you can turn it into a trivial projection by using no more than 2^(1,000,000,000n) Reidemeister moves. Find a better upper bound. Find a pair of non-tricolorable knots whose composition istricolorable or show that this is not possible. (To show it's false, it's enough to show that an open knot is tricolorable if and only if its closure is tricolorable.)

96. Knot Theory Find It Science Math Topology knot theory Sections on knot tying, mathematical knot theory, knot art, and knot books. http://www.ebroadcast.com.au/dir/Science/Math/Topology/Knot_Theory/

SEARCH GUIDE NEWS AUSTRALIAN TV GUIDE DVD RENTALS ... Topology : Knot Theory Science The A to Z of science is right here. Research Oriented Knots on the Web (Peter Suber) The most comprehensive collection of knotting resources on the web. Sections on knot tying, mathematical knot theory, knot art, and knot books. BraidLink Braidlink is software for knot and braid theory computations. It performs both analytic and numerical manipulations of knots and braids. A Circular History of Knot Theory Starting with the flawed theory of Kelvin's knotted vortex to the work of Thurston, Jones and Witten, knot theory has circled back to its ancestral origins of theoretical physics. Cook's Borromean Ring Links Links to pages and two outlines of proofs that show the Borromean rings can't be made from circular rings. Geometry and the Imagination Has a small section on knot theory at an introductory level. Also has sections on orbifolds, polyhedra and topology. The Geometry Junkyard: Knot Theory A page of links on geometric questions arising from knot embeddings. Harmonic Knots An introduction to harmonic knots. Gives (parametric) formulas for knots of up to 7 crossings.

97. Knot Theory With KnotPlot knot theory with KnotPlot. Equilateral Stick Numbers. This research is in collaboration with Eric Rawdon of the Department of Mathematics at Duquesne http://www.colab.sfu.ca/KnotPlot/ktheory.html

Knot Theory with KnotPlot Equilateral Stick Numbers This research is in collaboration with Eric Rawdon of the Department of Mathematics at Duquesne University in Pittsburgh. We've used KnotPlot to find equilateral polygonal representatives of all knots to 10 crossings with the fewest number of sides. This number of sides is known as the equilateral stick number. It can be compared to the stick number, which is the same quantity when the constraint of being equilateral is dropped. Surprisingly, for 242 of the 249 prime knots examined, all have an equilateral stick number equal to their stick numbers. Title: Upper Bounds for Equilateral Stick Numbers Authors: Eric J. Rawdon and Robert G. Scharein Abstract: We use algorithms in the software KnotPlot to compute upper bounds for the equilateral stick numbers of all prime knots through 10 crossings, i.e. the least number of equal length line segments it takes to construct a conformation of each knot type. We find seven knots for which we cannot construct an equilateral conformation with the same number of edges as a minimal non-equilateral conformation, notably the 8 knot.

98. Alexa - Browse: Knot Theory Alexa web search a new kind of search engine. With traffic rankings, user reviews and other information about sites, Alexa is a web site discovery tool. http://www.alexa.com/browse?&CategoryID=26948

99. Knot Theory -- Facts, Info, And Encyclopedia Article knot theory is a branch of (The configuration of a communication network) knot theory concerns itself with abstract properties of (Click link for more http://www.absoluteastronomy.com/encyclopedia/k/kn/knot_theory.htm

Knot theory [Categories: Knot theory, Geometric topology, Algebraic topology] Knot theory is a branch of (The configuration of a communication network) topology that was inspired by observations, as the name suggests, of (Any of various fastenings formed by looping and tying a rope (or cord) upon itself or to another rope or to another object) knot s. But progress in the field no longer depends on experiments with twine. Knot theory concerns itself with abstract properties of (Click link for more info and facts about theoretical knots) theoretical knots In (Click link for more info and facts about mathematical) mathematical (Specialized technical terminology characteristic of a particular subject) jargon , knots are (Click link for more info and facts about embedding) embedding s of the closed circle in three-dimensional space. An ordinary knot is converted to a mathematical knot by splicing its ends together. The topological theory of knots asks whether two such knots can be rearranged to match, without opening the splice. The question of untying an ordinary knot has to do with unwedging tangles of rope pulled tight. A knot can be untied in the topological theory of knots if and only if it is equivalent to the unknot , a circle in 3-space.

100. [math/0411115] Knot Theory Of Complex Plane Curves knot theory of complex plane curves. Authors Lee Rudolph Comments 26 figures; to appear in Handbook of knot theory (W. Menasco and M. Thistlethwaite, eds. http://arxiv.org/abs/math.GT/0411115

Mathematics, abstract math.GT/0411115 From: Lee Rudolph [ view email ] Date ( ): Fri, 5 Nov 2004 20:51:01 GMT (257kb) Date (revised v2): Sun, 7 Nov 2004 18:27:50 GMT (260kb) Knot theory of complex plane curves Authors: Lee Rudolph Comments: 26 figures; to appear in Handbook of Knot Theory (W. Menasco and M. Thistlethwaite, eds.) Subj-class: Geometric Topology; Algebraic Geometry MSC-class: 57M25 (Primary) 14B05, 20F36, 32S22, 32S55, 51M99 (Secondary) The primary objects of study in the ``knot theory of complex plane curves'' are C-links: links (or knots) cut out of a 3-sphere in the complex plane by complex plane transverse and totally tangential. Transverse C-links are naturally oriented. There are many natural classes of examples: links of singularities; links at infinity; links of divides, free divides, tree divides, and graph divides; andmost generallyquasipositive links. Totally tangential C-links are unoriented but naturally framed; they turn out to be precisely the real-analytic Legendrian links, and can profitably be investigated in terms of certain closely associated transverse C-links. The knot theory of complex plane curves is attractive not only for its own internal results, but also for its intriguing relationships and interesting contributions elsewhere in mathematics. Within low-dimensional topology, related subjects include braids, concordance, polynomial invariants, contact geometry, fibered links and open books, and Lefschetz pencils. Within low-dimensional algebraic and analytic geometry, related subjects include embeddings and injections of the complex line in the complex plane, line arrangements, Stein surfaces, and Hilbert's 16th problem.

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