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Topic: Prime knot

	Knot theory - Wikipedia, the free encyclopedia
		Knot theory concerns itself with abstract properties of theoretical knots — the spatial arrangements that in principle could be assumed by a loop of string.
		Knot theory originated in an idea of Lord Kelvin's (1867), that atoms were knots of swirling vortices in the æther.
		Knots in 3-space form a commutative monoid with prime factorization, which allows us to define what is meant by a prime knot.
	en.wikipedia.org /wiki/Knot_theory (1165 words)

	Prime Knots
		A prime knot is one that is not the sum of simpler knots.
		These prime knots are determined by the original knot, but in practice it can be very difficult to work out what these prime knots will be.
		Here the prime knots are the simple elements and the fact that any knot can be expressed uniquely (up to order) as a sum of prime knots is clearly an important fact about knots.
	www.popmath.org.uk /exhib/pagesexhib/prime.html (421 words)

	Mathematical knots (Site not responding. Last check: 2007-10-09)
		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.
		Knots that are equivalent to the unit circle are considered to be unknotted or trivial.
		There are two knots with a crossing number of five, three with a crossing number of six, and seven knots with a crossing number of seven.
	www.cs.ubc.ca /nest/imager/contributions/scharein/knot-theory/knot-theory.html (928 words)

	Knots and Links (Site not responding. Last check: 2007-10-09)
		Some say he struck it with his sword, cut the knot, and said it was now untied-but Aristobulus says that he took out the pole-pin, a bolt driven right through the pole, holding the knot together, and so removed the yoke from the pole.
		Knots were not treated as mathematical objects until the eighteenth and nineteenth centuries, when mathematicians including Alexandre-Theophile Vandermonde (1735-1796) and Carl Friedrich Gauss (1777-1855) introduced knots as subjects for a "geometry of position" (a concept first proposed by Leibniz in 1679).
		Composite knots are not given minimal crossing or unknotting numbers, as they can be better described in terms of their component, or prime, knots.
	lucien.blight.com /~cr/knots.html (1505 words)

	Bowline (Site not responding. Last check: 2007-10-09)
		A chant used by many to remember this knot is "The rabbit comes out of the hole, round the tree, and back down the hole again", where the hole is the small loop, and the rabbit is the running end of the rope.
		In the same way that a Left Handed Sheet bend is a Sheet bend that has the running end of the rope coming out of the wrong side of the knot, a cowboy bowline is a bowline that also has the running end of the rope coming out of the wrong side of the knot.
		So the knot that secured it was literally a bow line knot, but it has since become diminished and its pronunciation altered.
	www.deepcreekyachtclub.com /WebPage/bowline.html (444 words)

	PAKG - Prime Alternating Knot Generator
		The knots themselves may be viewed at http://knotilus.math.uwo.ca.
		PAKG generates the complete collection of prime alternating knots at the given crossing size by applying various operators to the complete collection of prime alternating knots at the preceding crossing size.
		An h-Dowker code for a prime alternating knot is the Dowker code for the configuration of the knot that is obtained by taking the zero position of each group in the MA for the knot (and then incorporating any non-flyping information as described above).
	www.math.uwo.ca /~srankin/papers/knots/pakg.html (1959 words)

	Ideas, Concepts, and Definitions (Site not responding. Last check: 2007-10-09)
		Knots can be combined in a process called knot addition.
		It is interesting to look at a knot and ask, "Are there 2 simpler knots that can be added together to produce this knot?" If the answer is "no", then the knot is a prime knot.
		When thinking about prime knots and knot addition, the zero knot plays a role very much like the number 1 plays when you think about prime numbers and multiplication.
	www.c3.lanl.gov /mega-math/gloss/primes/prknot.html (160 words)

	Prime Knot Tabulation
		If a knot is not composite (excluding composition with the unknot, which always results in the original knot) it is called a prime knot.
		Any knot where odd numbers are paired with odd numbers or even numbers are matched with even numbers results in an undrawable knot.
		All knots with identical notation will transform into this q, and so we can compare to see if the knot we are considering has already been written.
	members.aol.com /jdyer41/knotfinal.html (3053 words)

	Motivate : Stephen's talk
		An alternating knot is one having an alternating projection: as you walk around the knot the crossings alternate between under and over.
		A prime knot is one which cannot be represented as the join of two knots (unless one of them is the unknot).
		After that, knot theory was relatively quiet until Vaughan Jones discovered a completely new polynomial invariant in 1985; this was the catalyst for a burst of ideas and results.
	www.motivate.maths.org /conferences/conf28/c_28_talk.shtml (1174 words)

	The Knots Exhibition - Analogy
		Note here that we are using a particular example to illustrate a general law, the commutative law for addition of knots.
		We are also able to illustrate the deep fact that knots have a decomposition into a sum of prime knots, and this decomposition is 'unique up to order'.
		It so happened that the lecturer had not previously formalised the question for himself, so had really to think in order to be clear that all torus knots are prime, and that there are an infinite number of them.
	www.popmath.org.uk /exhib/pagesexhib/analogy.html (625 words)

	[No title]
		If a knot is tied on a string or a tube (let us remember: neccesserily closed at the end of the tying procedure), the condition of self--avoidance found within the formal definition is surely fulfilled.
		As easy to check, tying two nontrivial knots on the same piece of rope creates a new knot, which is not simpler than the factor knots; they never anihilate each other to result in a trivial knot.
		the writhe value of the composite knot is, (with an unexpected accuracy) equal to the sum of the writhe of the factor knots [9].
	www.man.poznan.pl /cmst/papers/4/art_3/vol4art3.html (2942 words)

	NEW KNOT TABLES
		After Redmeister [4], all knot tables that can be found in knot theory books are simple copies of the first: sometimes, some projection is slightly changed, or turned upside down, and that's all.
		A prime knot or link with singular digons, expressed by a Conway symbol, is called generating, and a knot or link without digons is called a basic polyhedron [14,16,17].
		Because the complete concept of new knot tables is based on the notion of generating knots and links and families originating from them, one of the possible future aims can be a search for new knot and link invariants that will be the invariants of families.
	www.mi.sanu.ac.yu /~jablans/knotab (1879 words)

	The CTK Exchange Forums (Site not responding. Last check: 2007-10-09)
		+ 1 is a prime (by subtracting 1).
		For Fermat primes greater than 3, the only double of a Fermat prime followed by a prime is 10 followed by 11.
		Double of the prime 47 plus 1 (n = 95) - an odd number - is still losing, because it is not a prime.
	www.cut-the-knot.com /htdocs/dcforum/DCForumID4/417.shtml (1456 words)

	Untangling the Mathematics of Knots (Site not responding. Last check: 2007-10-09)
		When you receive a knotted rope from another group of students, try to figure out what the two knots were that the knot you received was made from.
		prime knots--knots that cannot be decomposed into two simpler knots (unless, of course, one of the two knots is the zero knot).
		Whether these knots are prime knots, (and the conjecture that none of these knots are equivalent) has not been conclusively proved, however.
	www.c3.lanl.gov /mega-math/workbk/knot/knbuild.html (713 words)

	Problems in Knot Theory (Site not responding. Last check: 2007-10-09)
		Remark: As observed by Oliver Dasbach, the roots of the Jones polynomial of the figure 8 knot all have norm 1.
		A conjecture would be that, if K is a prime knot, L(K) is homotopy equivalent to the circle iff K is non-trivial, with the fundamental group generated by the obvious loop in L(K) shown in the above picture.
		A paper of Allen Hatcher "Spaces of knots" seems to be related with this problem.
	math.ucr.edu /~xl/knotprob/knotprob.html (909 words)

	Amazon.com: The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots: Books: Colin Conrad Adams (Site not responding. Last check: 2007-10-09)
		Knot theory has been a branch of mathematics that has been around for over a century, and now is finding applications in mnay areas, some of these being electrical circuit analysis, genetics, dynamical systems, and cryptography.
		Knot theory now is an established branch of mathematics, and it involves the use of tools from topology, analysis, and algebra.
		The assignment of polynomials to knots goes back to the early 20th century, but it took the work of Vaughan Jones and his use of ideas from operator theory and statistical mechanics to provide polynomial invariants of knots that were much finer than the Alexander polynomial of the 1930s.
	www.amazon.com /exec/obidos/tg/detail/-/071672393X?v=glance (2296 words)

	Citebase - Knot Group Epimorphisms (Site not responding. Last check: 2007-10-09)
		Any knot group is the image of the group of a prime knot by a homomorphism that preserves peripheral structure.
		Two knots are equivalent if they have the same knot type, that is, there exists an autohomeomorphism of S 3 taking one knot to the other.
		The satellite knot k is the image of a diffeomorphism g : V → V ⊂ S 3, where ˜ ˆ ˜ ˆ V is a standard solid torus containing k, and V is a tubular neighborhood of a knot k, ˜ ˜ the companion knot.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0405462 (5351 words)

	Knots and Links
		To explain differing chemical properties, Kelvin suggested that the vortex loops were knotted in various ways, so that a hydrogen atom might have the structure of a trefoil knot, for instance, while an oxygen atom might have the structure of a figure of eight knot.
		When mathematicians consider a knot they think of it as a closed loop, with the two ends joined so that the knot cannot be untied without cutting the loop.
		If a knot is laid out flat, the crossing number is the minimum number of times that the knot crosses over itself for all possible arrangements of the knot.
	www.btinternet.com /~connectionsinspace/Form_and_Structure/Knots_and_Links/body_knots_and_links.html (629 words)

	Buck Research Detail
		For knots tied in real stuff, such as molecules or rope, the filament has a non-zero cross-section, and so some length is required to complete a knot pattern.
		We showed that, depending on the knotting or tangling process, the crossing number varies at least with rope length to the P, where P varies between 1 and 4/3, and the 4/3 is a sharp upper bound.
		It is natural to suppose that rope-length would be additive on factors, that is, that the rope-length required to tie the sum of two prime knots is approximately the sum of the rope-lengths of the factors.
	www.anselm.edu /academic/mathematics/researchlong.html (8314 words)

	Bowline (Site not responding. Last check: 2007-10-09)
		The Bowline (BOH'-LIN) is considered the boaters' prime knot.
		The bowline is used to tie sheets and halyards (control lines) to sails, to temporarily tie a rode (anchor line) to an anchor, or, using one knot on each line, tie together 2 lines of different diameters.
		You know you have it correct when the bitter end is in the inside of the loop.
	www.dirauxwest.org /knots/bowline.htm (184 words)

	Knot Theory Online - The Web Site for Learning More about Mathematical Knot Theory
		A Brief History of Knot Theory - From the mathematical surge of interest in knots a little over a century ago to the recent and exciting application of Knot Theory to DNA and synthetic chemistry, you can get an overview of why knots are such a fascination for scientists and mathematicians alike.
		Advanced Knot Theory Topics - Once you understand the concepts in the introduction to knots, this page expands your knowledge with connected sums, composite and prime knots, stick knots, wild knots, and even has a section on coloring knots and links and why coloring is such an important topic to mathematicians.
		We are especially interested in hearing from teachers who have used knots in the classroom and students who have been motivated to study knots either in the classroom or on their own.
	www.freelearning.com /knots (563 words)

	On Composite Twisted Unknots (Site not responding. Last check: 2007-10-09)
		A twisting of a knot K is parametrized by a choice of oriented twisting torus V and number of full twists d (=delta in paper).
		If a composite knot is given two (non-trivial) full twists and the result is composite, we began with a granny knot and obtained the granny knot of opposite handedness.
		The granny knot case is due to Motegi and Hayashi, and required very special attention in the proof (an additional strand had to be woven through the induction.
	comp.uark.edu /~strauss/papers/ctu.html (600 words)

	Knot Theory Online - The Web Site for Learning More about Mathematical Knot Theory
		If a knot is not composite, meaning it cannot be expressed as the connected sum of other knots, we call it a prime knot.
		The trefoil knot (drawn as a stick knot to the right) has a stick number of 6, meaning we need at least 6 sticks to form the trefoil.
		Since the trefoil is tri-colorable, and the unknot is not tri-colorable, we therefore conclude that the trefoil knot is not the unknot.
	www.freelearning.com /knots/advanced.htm (1419 words)

	Mike Sullivan's Papers and Publications (Site not responding. Last check: 2007-10-09)
		Here, we study the properties of various templates, especially whether or not there is a bound on the number of prime factors of the knot types of the periodic orbits.
		Birman and Williams conjectured in 1983 that the knot types of the periodic orbits of this flow could have at most two prime factors.
		Williams showed that all knots in the Lorenz template are prime.
	galileo.math.siu.edu /~msulliva/Preprints (1293 words)

	Preprint Page for Stuart Rankin
		In the first paper in the list, we introduce four operators on knots and show that, when used according to very simple rules on the prime alternating knots of n crossings, the set of all prime alternating knots of n+1 crossings is obtained.
		The master array can be used to construct an ideal knot configuration for a prime alternating knot, by which we mean that two prime alternating knots, each in their ideal configuration, are equivalent if and only if they are identical.
		It is shown that one may choose any prime alternating link diagram of a given minimal crossing size and by applications of just two operators (namely T and OTS) to the selected seed link, one obtains all prime alternating link diagrams of the desired minimal crossing size.
	www.math.uwo.ca /~srankin/knotprint.html (1080 words)

	Amazon.com: Knot Theory (Carus Mathematical Monographs): Books: Charles Livingston,William Watkins (Site not responding. Last check: 2007-10-09)
		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.
		This book is an excellent introduction to knot theory for the serious, motivated undergraduate students, beginning graduate students,mathematicains in other disciplines, or mathematically oriented scientists who want to learn some knot theory.
		Many different aspects of knot theory are touched on, including some of the polynomial invariants, knot groups, Alexander polynomial and related abelian invariants, as well as some of the more geometric invariants.
	www.amazon.com /exec/obidos/tg/detail/-/0883850273?v=glance (972 words)

	Trefoil knot - TheBestLinks.com - Knot, Mirror image, Prime knot, Stub, ... (Site not responding. Last check: 2007-10-09)
		Trefoil knot - TheBestLinks.com - Knot, Mirror image, Prime knot, Stub,...
		Trefoil knot, Knot, Mirror image, Prime knot, Stub, Crossing, Torus knot, Braid...
		The trefoil knot is the only prime knot with three crossings.
	www.thebestlinks.com /Trefoil_knot.html (124 words)

	Mathematics Tasmania Colloquia Page for 1997
		Topologists study knots by considering the ambient space remaining after the knot is removed, in the popular parlance, what is not knot.
		But prime knots, such as the trefoil, figure-eight, or stevedore, are in fact determined by their groups, a fact established by Gordon and Luecke in the mid 1980's.
		Thus, in principal, one must be able to recover a prime knot, or at least a complete topological description of its exterior, from a presentation of its group.
	www.maths.utas.edu.au /HomePage/Coll97.html (1653 words)

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