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Topic: Knot group

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	Knot group - Wikipedia, the free encyclopedia
		In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space.
		The knot group of a knot K is defined as the fundamental group of the knot complement of K in R
		Two equivalent knots have isomorphic knot groups, so the knot group is a knot invariant and can be used to distinguish between inequivalent knots.
	en.wikipedia.org /wiki/Knot_group (129 words)

	Trefoil knot - Wikipedia, the free encyclopedia
		In knot theory, the trefoil knot is the simplest nontrivial knot.
		It is not a slice knot, meaning that it does not bound a smooth 2-dimensional disk in the 4-dimensional ball; one way to prove this is to note that its signature is not zero.
		The knot group of the trefoil is isomorphic to B
	en.wikipedia.org /wiki/Trefoil_knot (246 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		Now, the group presentations which result from a knot projection do not include all group presentations, so it is possible that one _could_ develop a procedure to decide unfailingly whether or not the corresponding fundamental group is trivial (or perhaps whether two of them are isomorphic).
		It is true (I think) that a string is unknotted iff its knot group is Z (the simplest possible), but it's also true that there are pairs of knots with the same groups but which are really _distinct_ knots.
		The algorithm is based on triangulating the exterior of the knot, and showing that a compressing disk for the knot exterior (if it were an unknot) would necessarily intersect the triangulation in specific way (it is a normal surface).
	www.math.niu.edu /~rusin/known-math/96/knothy (576 words)

	The Knot Group (Site not responding. Last check: 2007-10-13)
		The fundamental group of M is called the knot group of K. Obviously this group does not depend on any particular projection of a knot and is thus a knot invariant.
		To each strand of a knot he assigned a distinct generator, and to each crossing the generators assigned to the three strands that meet are related by CB=BA, where B is the generator for the overcrossing strand, while A and C the generators for the undercrossing strands.
		This group is a module of the group of the trefoil, but not of the group of the trivial knot.
	www.inst.bnl.gov /~wei/group.html (441 words)

	Event Symmetric Space-Time
		Knot theory is important in understanding the physics of particles with fractional statistics: anyons or parafermions.
		The symmetric group is the symmetry of fermions and bosons, while the braid group from knot theory plays the same role for anyons.
		Knotted loops have turned out to be important in the canonical approach to quantum gravity and it is natural to wonder if these loops are the same stuff as the strings of string theory, the other important approach to quantum gravity.
	www.weburbia.com /press/html/g09.htm (3600 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		An >important invariant is the fundamental group of the complement S3-K, which >we call the group of K'.
		This group is called the "fundamental group of X with basepoint " (or "the first homotopy group* of X with basepoint *").
		The group generated by x_1,...,x_n subject to the relations just written down is the knot group.
	www.math.niu.edu /~rusin/papers/known-math/98/whatis_pi1 (447 words)

	Some horoball diagrams of knot complements (Site not responding. Last check: 2007-10-13)
		The knot group acts transitively on the entire collection of horoballs, but there is a (parabolic) subgroup, isomorphic to Z+Z, which preserves the "infinite" horoball z=1, and which acts on the remaining horoballs via Euclidean translations.
		We can take a fundamental region for the Z+Z action to be a rectangle whose vertices are the points of tangency with z=0 of two adjacent full-size (pink-maroon) spheres on the left edge of the picture, together with the points of tangency of two adjacent full-size spheres on the right edge of the picture.
		The symmetries of the knot lift to symmetries of the horoball packing, which are quite easy to spot in this example: for example, the order 8 symmetry lifts to a glide reflection along a horizontal axis.
	www.math.utk.edu /~morwen/3d_pics/horoball.html (397 words)

	Citebase - Knot Group Epimorphisms (Site not responding. Last check: 2007-10-13)
		Any knot group is the image of the group of a prime knot by a homomorphism that preserves peripheral structure.
		As usual a knot is the image of a smooth embedding of a circle in S 3.
		Since any knot group is residually finite [H87] and finitely generated, it has the Hopfian property:any epimorphism from the group to itself is an isomorphism [M40].
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0405462 (3644 words)

	Topfest 99 -- Abstracts
		A special case of Scott's compact core theorem asserts that if G is a finitely generated 1-ended group acting freely and discretely on a contractible 3-dimensional manifold X, then G is the fundamental group of a compact 3-dimensional manifold Q with incompressible boundary and Q embeds in X/G as a deformation retract.
		More generally, in a closed, orientable, irreducible 3-manifold with cyclic fundamental group, knots which are round, in the sense that their exteriors are solid tori, can be characterized among all knots with irreducible exterior in terms of their essential surfaces or their boundary slopes.
		A subgroup H of a group G is said to be separable provided that H is the intersection of finite index subgroups of G. A group is called subgroup separable provided that every finitely generated subgroup is separable.
	www.math.cornell.edu /~festival/1999/99abstracts.html (1415 words)

	Knots on the Web (Peter Suber)
		Maintained by Jan Korpegård. Under each knot, Korpegård gives the knot's name in 10 languages, and asks readers of other languages to send him the names of the same knots in their languages; he even provides the form for submitting the names electronically.
		Decorative knotting, some for sale, by a man who calls himself the world's best (which may or may be true), and who thinks of himself as alone in his art (which is fortunately false).
		Because the knots for (say) square-rigged sailing ships, stevedores, steeplejacks, and draymen have been made obsolete by modern tecchnologies, the book is of immense historical value, even if these knots were not transferable to other applications.
	www.earlham.edu /~peters/knotlink.htm#sailing (7098 words)

	Cohopficity of Groups in the Study of Knots and 3-Manifolds, by Wilbur Whitten (Site not responding. Last check: 2007-10-13)
		The group of M is cohopfian if and only if the collection of components of the characteristic submanifold of M meeting the boundary of M is a disjoint union of collars [GW1].
		The group of a (tame) nontrivial knot K is cohopfian if and only if K is not a torus knot, a cable knot, or a composite knot [GW1].
		A nontrivial knot whose group properly imbeds in itself with finite index is a torus knot [GW1].
	at.yorku.ca /t/a/i/c/29.htm (351 words)

	Table of Contents for Reidemeister
		The first one is an introduction to knots and braids, including (a sketch of) the original proof that two knots are equivalent if and only if their projections are related by a finite sequence of the three so-called Reidemeister moves.
		The third chapter deals with knot groups: definition by generators and relations from a projection, invariance, equivalence with the fundamental group of the knot complement, calculation of the group of special families of knots.
		This landmark in the history of knot theory should have a choice place on the shelves of all knot theorists.
	www.harbornet.com /bcsassociates/toc_rei.html (337 words)

	Marine Knots
		The Surgeon's Knot is a modified form of the reef knot, and the extra turn taken in the first tie prevents slipping before the knot is completed.
		The Fisherman's Knot is probably the strongest known method of joining fine lines, such as fishing lines.
		This knot is used at the end of a rope to temporarily prevent the strands from unlaying.
	www.lehighgroup.com /KNOTS.HTM (231 words)

	Undergraduate Handbook (Site not responding. Last check: 2007-10-13)
		In this course, we study the geometry of surfaces such as the plane, the sphere, the Mobius strip, the torus, the Klein bottle, real projective space, and spheres with handles, and also of knots such as the square knot, the granny knot, the trefoil knot and the figure of eight knot.
		We also use the fundamental group, a group associated with shapes such as surfaces and knots (for the torus it is an infinite group, for real projective space a group with two elements, for the sphere it is the zero group), which again incorporates crucial information concerning the geometric configuration that is being discussed.
		Knots and the Wirtinger presentation of the knot group.
	www.math.mun.ca /includes/undergradhandbook/courses/pm3303.htm (428 words)

	Games -- Team Building and Confidence Building Games (Site not responding. Last check: 2007-10-13)
		The groups were timed to see which group was the fastest to assemble the puzzles.
		Group coordinates efforts to walk while standing on wooden trolleys (long boards with ropes to hang on to every few feet).
		One to teach a knot, Two to throw a Life Line and Three, which is the big one, is to teach Team Work within a Patrol.
	www.usscouts.org /games/game_t.html (2120 words)

	Summer 2001 REU Participants (Site not responding. Last check: 2007-10-13)
		I then showed that the property of n-coloration of a knot is also a basic property of the knot group, i.e.
		My project was to write a program that allowed a user to draw a knot with a mouse or other pointing device, and determine from the drawing if that knot could be reduced to the "UN-knot." This was written in Java and is available from my web site at http://www.cs.utah.edu/~rpalmer/knot/knot.jar.
		The paper describes how the colorability of a knot/link is determined via the crossing matrix of a knot, using special matrix forms in linear algebra; along with important consequences/properties of prime knots, composite knots, and links relating to nullity and multicolorability.
	www.math.utah.edu /reu/summer01reu.html (1671 words)

	[No title]
		Primary examples of discontinuous stitches utilize either the square knot or the surgeon’s knot square on the basis of reliability and versatility.
		The surgeon’s knot square, also called a friction knot, has the added advantage of allowing the surgeon to control the tension of the suture.
		Group A1—continuous simple Group A2—continuous but locked Group A3—discontinuous using square knot Group A4—discontinuous using surgeon’s knot square Variation in suture density (number of stitches across incision) Create two experimental group (N=5) of rubber samples that utilize a certain suture density.
	www.seas.upenn.edu /courses/belab/LabProjects/2004/be210s04w8.doc (1252 words)

	Citebase - Structure in the classical knot concordance group (Site not responding. Last check: 2007-10-13)
		We construct many examples of non-slice knots in 3-space that cannot be distinguished from slice knots by previously known invariants.
		By a recent result of Livingston, it is known that if a knot has a prime power branched cyclic cover that is not a homology sphere, then there is an infinite family of non-concordant knots having the same Seifert form as the knot.
		The classical abelian invariants of a knot are the Alexander module, which is the first homology group of the the unique infinite cyclic covering space of S
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0206059 (972 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		This project will involve such a study for knots in a closed, orientable 3-manifold where we are seeking to understand the topology of the manifold.
		For example, if the property that detects the trivial knot in closed, orientable, irreducible 3-manifolds with cyclic fundamental group is also realizable in such manifolds, then these manifolds are lens spaces; in particular, a simply connected one is the 3-sphere (The Poincare Conjecture).
		Jaco and Rubinstein have shown that under reasonable restrictions a 3-manifold admits a triangulation in which each edge is a knot (one vertex triangulation) and particular edges, ``thick edges," are candidates for realizing knots with the desired properties, depending on the initial restrictions on the 3-manifold.
	www.aimath.org /projects/lopez.html (366 words)

	Mathematics Geometry
		Suppose that f is a diffeomorphism of the 3-sphere to itself and C is a knot in the 3-sphere such that that every point of C is mapped to itself by f.
		We describe the relationship of the local structure of knot space to the thickness of the knot.
		What all of these have in common is a curiosity about the nature of polygonal knot space, about the spatial properties of polygonal knots, especially those that appear to express characteristics that are tied to physical manifestations of these knots, whether at the scale of DNA or solar storms.
	www.math.ucsb.edu /department/topology_interest.php (1083 words)

	Open Directory - Science: Math: Topology: Knot Theory (Site not responding. Last check: 2007-10-13)
		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.
		Knot Theory Invariants: The HOMFLY Polynomial - A brief article on the HOMFLY polynomial and how it is calculated.
		Knot Theory Online - This site is designed for mathematics students at the high school and college levels as an introduction to an area of mathematics seldom explored in the typical math classroom - the Theory of Knots.
	dmoz.org /Science/Math/Topology/Knot_Theory (751 words)

	topology seminars
		Abstract: If a knot, K, in thin position in S^3 has at least 1 thin level then Thompson has shown that there is a planar, meridional, essential surface in the complement of K. Bachman and Schleimer showed that this result also holds for knots in B^3.
		A knot is called doubly slice if it is the slice of some smooth unknotted 2-sphere in the 4-sphere.
		The main result of this talk is a characterization of Seifert-fibered orbifolds with infinite fundamental group by the presence of an infinite cyclic normal subgroup in their fundamental group.
	www.math.ucsb.edu /~mgscharl/seminar/0102seminar.htm (1269 words)

	Lesson Exchange: The Human Knot (all, Phys Ed)
		Ask a group of 6 or more people (even numbers works best) to form a circle.
		Each person should hold out their right hand and grab the right hand of the person cross from them as though the two were shaking hands.
		If the group has been struggling with a knot for a long time, offer "Knot First Aid." Let the students decide amongst the group, which grip needs first aid.
	www.teachers.net /lessons/posts/1330.html (211 words)

	Physical and Biological Computing Group: Simulated Knot Tying (Site not responding. Last check: 2007-10-13)
		In microsurgery simulation, laparoscopic surgery simulation, and other surgical simulations, realistic real-time simulation of rope to tie knots for surgical suturing is identified as a task.
		Our experiments consist of pulling a loosely knotted rope configuration tight and witnessing that the knot was maintained.
		The dynamics simulation must be able to cope adaptively as the configuration of the rope varies and must be approximately continuous over time.
	www.cs.rice.edu /CS/Robotics/robotics/simknottying.html (312 words)

	GMDX Group - Celtic Knot Information
		The Celtic Knot award Scheme promotes contact with Celtic nations around the World.
		In 2001, the GMDX Group mounted a highly successful DXpedition to the Falkland Islands.
		In 2004, the GMDX Group may undertake another world-class DXpedition.
	www.gmdx.org.uk /knotinfo.html (81 words)

	Links to low-dimensional topology: Knot Theory
		An introduction to knot theory which seems to be aimed at teachers of mathematics can be found at Los Alamos National Laboratory.
		An on knot theory appears in the November 1997 issue of American Scientist.
		If all you want are the knots and volumes, this Dvi file (or this Postscript file) is easier to work with.
	www.math.unl.edu /~mbrittenham2/ldt/knots.html (504 words)

	[XGAP] 4 Subgroup Lattices - Examples
		We again assume that you are familiar with the general ideas, mouse actions and menus, which were discussed in The Subgroup Lattice of the Dihedral Group of Order 8 and A Partial Subgroup Lattice of the Symmetric Group on 6 Points.
		This first determines the identification number of the symmetric group on 3 points within the small groups library, and then fetches this group as a polycyclic group.
		For groups of size less than 1000 this is often a good way to get a polycyclic presentation.
	www-groups.dcs.st-and.ac.uk /~gap/Manuals/pkg/xgap/htm/CHAP004.htm (5101 words)

	SOAR Winter 2002 Homework Ten (Site not responding. Last check: 2007-10-13)
		Find the knot group of a trefoil knot using the Wirtinger presentation, and simplify it as follows:
		Write down the knot group G. It should have three generators: call them a, b, and c.
		Recall from Homework 9 that the center of a group G is the set Z(G) of elements of G that commute with every other element of the group:
	www.math.toronto.edu /mathnet/SOAR2002/Winter/HTML/homework10.html (190 words)

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