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Topic: Figure-eight knot (mathematics)

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Figure-eight knot (mathematics) - Wikipedia, the free encyclopedia In knot theory, a figure-eight knot is the unique knot with a crossing number of four, the smallest possible except for the unknot and trefoil knot. The name is given because joining the ends of a string with a normal figure-eight knot tied in it, in the most natural way, gives a model of the mathematical knot. (Robert Riley and Troels Jorgensen, working independently of each other, had earlier shown that the figure-eight knot was hyperbolic by other means.) This construction, new at the time, led him to many powerful results and methods. en.wikipedia.org /wiki/Figure-eight_knot_(mathematics)   (309 words)

Figure-of-eight knot - Wikipedia, the free encyclopedia The figure-of-eight knot is a type of knot. Variant Name(s): Figure Eight Knot, Savoy Knot, Flemish Knot, Double Stopper. Grog's Animated Knots: How to tie the figure of eight knot en.wikipedia.org /wiki/Figure-of-eight_knot   (125 words)

KNOTS Figure 19 - Alexander Polynomial of the Trefoil Knot Figure 17 - Equations for the Trefoil Knot The Alexander polynomial is an algebraic modulus for the knot. www2.math.uic.edu /~kauffman/Tots/Knots.htm   (16146 words)

Articles - Chirality (mathematics) For example the unknot and the figure-eight knot are achiral, whereas the trefoil knot is chiral. In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone. In three dimensions, every figure which possesses a plane of symmetry or a center of symmetry is achiral. lastring.com /articles/Chirality_(mathematics)?...   (529 words)

String Figures and Knot Theory: Part III The look-alikes of such a 'symmetric' string figure may be grouped by rotational equivalence, one look-alike of a group being transformable into another by rotating the figure in space. This figure, Jayne’s (Second) Worm, is illustrated again to demonstrate the application of the analysis to a three-dimensional figure. The sequences of 'unravellings by motifs' that relate the eight look-alikes are given on the right. website.lineone.net /~m.p/sf/exhibits.html   (870 words)

Poster Project, Knots Poster Knot Theory has a very pretty presentation and quite a bit of mathematics. Knot theory is the rigorous mathematics which can distinguish between different knots. Mathematical knots are an imagined model of the physical knot-making process. www.math.sunysb.edu /posterproject/www/materials/knots/knots.html   (440 words)

Figure 8/Horoball It also produces diagrams that describe pictorially the behavior of the space as one approaches "infinity." The diagram at right--in technical language, the horoball diagram--is associated with the complement of the figure eight knot. Part of the research of Allan Edmonds, Professor of Mathematics, Indiana University Bloomington, has included the study of symmetries of knots. The program then investigates the existence of a geometric structure on the complement of the knot, and computes a number of different numerical invariants of the space, such as its volume. www.indiana.edu /~rcapub/v17n2/10c.html   (181 words)

The Chirality of a Knot The animation above shows that the figure eight knot seen at the beginning can be transformed into its mirror image, which appears at the end. A wonderful and accessible book on Knot Theory is Colin C. Adams' The Knot Book, published by Freeman and Co. It requires no advanced mathematics but looks at some deep problems involving knots. The branch of mathematics called knot theory provides us will tools to check whether one knot can be continuously transformed into another. www.stats.uwaterloo.ca /~cgsmall/knot.html   (228 words)

Math Trek: The Tangled Task of Distinguishing Knots, Science News Online, Feb. 22, 2003 Jones’s discovery prompted a great deal of excitement in the mathematics community because his polynomial detects the difference between a knot and its mirror image, something that the Alexander polynomial had failed to do. One approach to labeling knots is to use the arrangement of the crossings in a knot diagram to produce an algebraic expression for that knot. To solve the problem of distinguishing among knots, mathematicians have tried to find schemes for labeling them in such a way that two knots having the same label are really equivalent—even when their diagrams may appear different—and that two knots with different labels are truly different. www.sciencenews.org /20030222/mathtrek.asp   (1143 words)

Why knot: knots, molecules and stick numbers Fig 3: The trivial knot, the trefoil knot, and the figure of eight knot The trefoil knot and the figure-eight knot are the two simplest nontrivial knots, the first having a picture with three crossings and the second with four. Since the trefoil knot is a (q,q-1)-torus knot for q =3, we obtain 6 for the number of sticks to construct it, as we expected. pass.maths.org.uk /issue15/features/knots   (2287 words)

Physical Knots Session (#152) at IMACS (Aug 21-25, 2000) Among the simple knots we tested the overhand knot proved to be the weakest while the figure eight knot the strongest. Spaces of polygonal knots, subject to specified constraints such as the number of nondegenerate edges or the requirement of having fixed edge lengths, provide the context within which it is natural to study configurations which are ideal with respect to a variety of natural physically motivated constraints. For a model of random polygons with N nodes, we define the knotting probability of a knot K by the probability that a given random polygon is equivalent to the knot K. www.haverford.edu /math/rmanning/imacs2000/152abstracts.html   (1772 words)

Carving for Mathematical Understanding of Surfaces The eight knot basis for this non-orientable sculpture was formed using piecewise special Bezier quartic curves [4] defined as an interpolating curve for the eight points shown here, which stem from lines drawn from the center to the vertices of a regular tetrahedron. Friedman is a professor of mathematics at University of Albany, SUNY. You may even be lucky enough to attend a Mathematics and Art Conference at SUNY Albany and participate in a stone carving session with mathematicians and master carvers Helaman Ferguson[7] and Nate Friedman[8]. www.wcnet.org /~clong/carving/carving.html   (841 words)

Figure eight halter hitch. (from knot, hitch, and splice) --  Britannica Student Encyclopedia To tie a figure eight halter hitch, the end is passed through a hitching ring or a similar secure object, then turned against itself to form a closed loop; the remaining part of the… The term knot derives from its former use as a length measure on ships' log lines, which were used to measure the speed of a ship through the water. Knots have existed from the time humans first used vines and cordlike fibres to bind stone heads to wood in primitive axes. www.britannica.com /ebi/article-203466?tocId=203466   (956 words)

The Most Useful Rope Knots for the Average Person to Know The most-mentioned loop knots which the average person might find useful in a variety of situations tend to be the Alpine Butterfly, the Bowline, the Bowline on the Bight, the Figure-Eight Loop, and the Double Figure-Eight Loop. Considering the number of people who are trusting their lives to the knots that they tie in ropes (rock climbers, cavers, search-and-rescue workers, etc.), it is surprisingly difficult to find solid research on the strengths of the main knots that are being used. Therefore, it is probably best to interpret the common strength ratings of knots as general guidelines, and also to be guided by the combined experiences of people such as search-and-rescue workers who are trusting their lives and other people's lives to the knots that they use. www.layhands.com /knots   (3165 words)

lego.html I decided to parameterize the knot as a curve on the surface of a torus, thickening it to a diameter of about three LEGO studs. Certain areas of mathematics have a distinctly tactile pleasure in addition to their abstractly mathematical aesthetic qualities. Eric Harshbarger (http:/www.ericharshbarger.com) is the undisputed champion of this subgenre and has made life-size figures, a grandfather clock, and even a desk out of LEGO. www.maa.org /features/lego.html   (1510 words)

Berkeley Lab Mathematician Coauthors Two New Books on Experimental Mathematics Bailey is well known for his work in computational mathematics and has written numerous papers on using modern computer technology in mathematical research. Published by A. Peters, Ltd., the experimental mathematics books generated an enthusiastic review in the May 2003 issue of Scientific American, well in advance of publication, and are selections of Bookspan, a scientific book club. A section of Mathematics by Experiment on "proof versus truth" is an example of the gems even a mathematical tyro can find among these equations. www.lbl.gov /Science-Articles/Archive/CRD-expmath-Bailey.html   (672 words)

The Geometry Junkyard: Knot Theory There is of course an enormous body of work on knot invariants, the 3-manifold topology of knot complements, connections between knot theory and statistical mechanics, etc. I am instead interested here primarily in geometric questions arising from knot embeddings. Atlas of oriented knots and links, Corinne Cerf extends previous lists of all small knots and links, to allow each component of the link to be marked by an orientation. With a proof of the origami-folklore that this folded-flat overhand knot forms a regular pentagon. www.ics.uci.edu /~eppstein/junkyard/knot.html   (673 words)

Princeton Mathematician Speaks at CMI Inauguration His sculpture is called “Figure Eight Knot Complement,” and is made of black granite. The president of the International Mathematical Union, Director of the National Science Foundation, and the President of the American Mathematical Society attended ceremonies, as did two winners of the Fields Medal, the equivalent of the Nobel Prize in mathematics, and numerous students from several schools. The Clay Mathematics Institute (CMI) was formed in September 1998, according to Director Arthur Jaffe. www-tech.mit.edu /V119/N26/26cmi.26n.html   (533 words)

String Figures and Knot Theory: Part V We have shown how knot theory may be used to determine the set of look-alikes of a given string figure. What remains to be established is a mathematical, as opposed to a mechanical, determination of the possible partial unravellings of a given string figure. If no non-empty proper subset of motifs of a string figure F is unravellable, then the only other possible look-alike of F is the reflection of F. (A distinct look-alike of opposite parity will only exist if F is a non-trivial two-dimensional figure.) website.lineone.net /~m.p/sf/sfmaths5.html   (591 words)

NDSU Club Math Page For example, the volume of the complement of the figure-eight knot is approximately 2.02988321282. Among knots with few crossings, the alternating behavior prevails; in fact, the simplest non-alternating knot has 8 crossings. This means that we can tell these knots apart by the (hyperbolic) volume of their complement. www.ndsu.nodak.edu /ndsu/coykenda/math/mathclub/calvo9.html   (234 words)

Knots Knots can be constructed with enormous amounts of complexity, which leads to the mathematical problem of classifying them. The main problem of knot theory is the classification of knots -- that is, the determintation of knot types and how to tell when two knots are equivalent. It is possible to associate a certain polynomial with a knot, in such a way that if the polynomials are different, then the knots are not equivalent. www.jcu.edu /math/vignettes/knots.htm   (855 words)

Hyperseeing, Hypersculptures and Space Curves The so-called figure eight knot is a knot with four crossings. In general, knots are classified according to the minimum number of crossings in a diagram of the knot. The object in Figure 6 is a black rectangle and a white triangle, each made of foamcore, and joined edge to edge using a toothpick piercing the styrofoam edges. www.mi.sanu.ac.yu /vismath/friedman   (3076 words)

thurston.html We prove the conjecture in the simplest case, the figure eight knot complement, and give heuristic reasons why the conjecture should be true in general; we also explain why you should not trust the heuristics. We present the Kashaev-Murakami-Murakami conjecture which relates the asymptotic growth rate of certain values of the colored Jones polynomial of a knot to the hyperbolic volume of its complement. We close with speculations about relations between hyperbolic geometry and q-hypergeometric functions in general. www-math.mit.edu /~combin/abstracts/feb01/thurston.html   (95 words)

Untitled file 'welcome.html' Note that even though the manifold m003 has the same volume as the figure eight knot complement, it has first homology Z + Z5, and is therefore not a knot complement. Note that because the figure eight knot complement has an orientation reversing symmetry each closed geodesic has a mate of the same length but opposite torsion. Type "+" for the sign of the determinant, "+" for the sign of the trace, and "RL" for the RL factorization, and you will see that you've created a manifold of the same volume (2.02...) as the figure eight knot complement and the manifold m003. www.geom.uiuc.edu /software/snappea   (4153 words)

Mathematics applications available Here is an example in which we compute the volume of the complement of the figure eight knot, and then do a Dehn surgery with parameters (5,1), yielding a complete hyperbolic manifold of second smallest known volume. Here are some mathematics applications you may want to use while using the department's computers. The program will then compute such things as the volume of the knot complement (if it has a hyperbolic structure) in the three-sphere with respect to a Riemannian metric of constant curvature -1, if one exists. www.math.uiuc.edu /consult/use_system/apps/mathsw.html   (1588 words)

DHD Multimedia Gallery: Mathematics: knot-figure-of-eight-black- (Mathematics: knot figure of eight black backdrop orange nylon rope) knot figure of eight blue backdrop 1 AJHD.jpg (Mathematics) Home > Virtual Collections > All Exhibits By Category > Mathematics > knot-figure-of-eight-black- gallery.hd.org /_c/maths/knot-figure-of-eight-black-backdrop-orange-nylon-rope-2-AJHD.jpg.html   (156 words)

Mathematics and Knots, Show 9 The figure eight knot (the only knot with crossing number 4) is achiral. Here we have mirror image versions of the figure eight knot. Use an extension cord to make one version with the cord plugged into itself to close the curve. math.usask.ca /~taylor/knots/WEB400/show9.html   (42 words)

DHD Multimedia Gallery: Not Found Brazil Ceara region Christ figure on cross in white alcove in yellow wall with greenery hanging from above 1 JBG.jpg (Earth Views) England Kent Sarre windmill tarred weatherboarded smock mill with post mill cap winded by eight vane fantail carrying four white patent sweeps 3 PAR.jpg (Earth Views) England Kent Sarre windmill tarred weatherboarded smock mill with post mill cap winded by eight vane fantail carrying four white patent sweeps 4 PAR.jpg (Earth Views) gallery.hd.org /_c/maths/knot-figure-of-eight-black-backdrop-orange-...   (280 words)

Sculpture by Helaman Ferguson This sculpture celebrates among other things the mathematical knowledge that the complement of the figure-eight knot is the quotient of hyperbolic three-space H (The figure-eight knot is the only possible arithmetic knot.) The knot complement is homeomorphic to the hyperbolic manifold H Helaman Ferguson: Mathematics in Stone and Bronze, a book by his wife Claire. www.mtholyoke.edu /acad/math/other/sculpture.htm   (255 words)

Pythagorean Theorem and its many proofs The first proof I merely pass on from the excellent discussion in the Project Mathematics series, based on Ptolemy's theorem on quadrilaterals inscribed in a circle: for such quadrilaterals, the sum of the products of the lengths of the opposite sides, taken in pairs equals the product of the lengths of the two diagonals. In right-angled triangles the figure on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle. For the same reason also, as BC is to CD, so is the figure on BC to that on CA; so that, in addition, as BC is to BD, DC, so is the figure on BC to the similar and similarly described figures on BA, AC. www.cut-the-knot.org /pythagoras/index.shtml   (6005 words)

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