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 Knot (mathematics) — the spatial arrangements that in principle could be assumed by a loop of string. In mathematics 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 Luminiferous aether (also known as 'ether'). He believed that an understanding and classification of all possible knots would explain why atoms spectroscopy light at only the discrete wavelengths that they do (i.e. explain what we now understand to depend on quantum energy levels). [http://www.kpbsd.k12.ak.us/kchs/JimDavis/CalculusWeb/Knot%20Theory%20History.htm] Scottish physicist Peter Tait spent many years listing unique knots under the belief that he was creating a table of elements. When ether was discredited through the Michelson-Morley experiment, vortex theory became completely obsolete, and knot theory fell out of scientific interest. Only in the past 100 years, with the rise of topology, have knots become a popular field of study. Today, knot theory is inextricably linked to String theory, DNA topology replication and recombination, and to areas of statistical mechanics. == An introduction to knot theory == Creating a knot is easy. Begin with a one-dimensional line segment, wrap it around itself arbitrarily, and then fuse its two free ends together to form a closed loop. One of the biggest unresolved problems in knot theory is to describe the different ways in which this may be done, or conversely to decide whether two such embeddings are different or the same.
Before we can do this, we must decide what it means for embeddings to be "the same". We consider two embeddings of a loop to be the same if we can get from one to the other by a series of slides and distortions of the string which do not tear it, and do not pass one segment of string through another. If no such sequence of moves exists, the embeddings are different knots.
A useful way to visualise knots and the allowed moves on them is to project the knot onto a plane - think of the knot casting a shadow on the wall. Now we can draw and manipulate pictures, instead of having to think in 3D. However, there is one more thing we must do - at each crossing we must indicate which section is "over" and which is "under". This is to prevent us from pushing one piece of string through another, which is against the rules. To avoid ambiguity, we must avoid having three arcs cross at the same crossing and also having two arcs meet without actually crossing (we would say that the knot is in general position with respect to the plane). Fortunately a small perturbation in either the original knot or the position of the plane is all that is needed to ensure this.
=== Reidemeister moves ===
In 1927, working with this diagrammatic form of knots, J.W. Alexander and G.B. Briggs, and independently Kurt Reidemeister, demonstrated that two knot diagrams belonging to the same knot can be related by a sequence of three kinds of moves on the diagram, shown right. These operations, now called the Reidemeister moves, are:
#Twist and untwist in either direction.
#Move one loop completely over another.
#Move a string completely over or under a crossing.
Knot invariant can be defined by demonstrating a property of a knot diagram which is not changed when we apply any of the Reidemeister moves. Some very important invariants can be defined in this way, including the Jones polynomial. ===Higher dimensions=== You can unknot any circle in four dimensions. There are two steps to this. First, "push" the circle into a 3-dimensional subspace. This is the hard, technical part which we will skip. Now imagine temperature to be a fourth dimension to the 3-dimensional space. Then you could make one section of a line cross through the other by simply warming it with your fingers. In general piecewise-linear n-spheres form knots only in ''n''+2 space, although one can have smoothly knotted n-spheres in ''n''+3 space. ===Adding knots=== Two knots can be added by breaking the circles and connecting the pairs of ends. Knots in 3-space form a commutative monoid with prime factorization. The trefoil knots are the simplest prime knots. Higher dimensional knots can be added by splicing the spheres. While you cannot form the unknot in three dimensions by adding two non-trivial knots, you can in higher dimensions. == See also == * list of knot theory topics * knot invariant * braid theory * topoisomerase * DNA topology * linking number * Borromean rings == Further reading == * ''The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots'', Colin Adams, 2001, ISBN 0716742195 * ''Knots: Mathematics With a Twist'', Alexei Sossinsky, 2002, ISBN 0674009444 * ''Knot Theory'', Vassily Manturov, 2004, ISBN 0415310016 * * Dror Bar-Natan, [http://www.math.toronto.edu/~drorbn/KAtlas/ The Knot Atlas]. * [http://www.cs.ubc.ca/nest/imager/contributions/scharein/KnotPlot.html KnotPlot] == Other resources == * [http://www.pims.math.ca/knotplot/download.html Software for Viewing Knots (Freeware)] Algebraic topology Geometric topology Knot theory
I moved the following todo list from the article (where it doesn't belong) to this Talk page. ''Still to come:'' * Gauß diagrams[http://arxiv.org/pdf/math.GT/9805078] * Signed Graph representation of knots * History of knot theory, including resurgence since Jones polynomial * Maybe try to explain Edward Witten's connection between knots and quantum gravity! —User:Herbee 13:17, 2004 May 16 (UTC) == broken link == [http://www.ma.hw.ac.uk/RSE/meetings_etc/ordmtgs/2001/reports/knots.htm] == You/Me == I'm not comfortable with this article using first and third person ('you'/'me'). The [http://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style#Avoid_self-referential_pronouns Wiki style manual] isn't so keen on it either. Is it all right with everyone if I dive in and do my best to avoid the 'bad' pronouns? User:Spamguy 22:13, Jun 4, 2005 (UTC) ==List of knot theory topics== Please help complete the list of knot theory topics by adding relevant articles on knots, braids, links, etc. User:Michael Hardy 20:10, 9 Jun 2005 (UTC)
Knot theory is a branch of topology that concerns itself with abstract properties of knot (mathematics) — the spatial arrangements that in principle could be assumed by a closed loop of string. Geometric topologyAlgebraic topology
See other meanings of words starting from letter:
Words begining with Knot_theory:
Knot_theory
Knot_theory
Knot_theory
Knot theory
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 Knot (mathematics) — the spatial arrangements that in principle could be assumed by a loop of string. In mathematics 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 Luminiferous aether (also known as 'ether'). He believed that an understanding and classification of all possible knots would explain why atoms spectroscopy light at only the discrete wavelengths that they do (i.e. explain what we now understand to depend on quantum energy levels). [http://www.kpbsd.k12.ak.us/kchs/JimDavis/CalculusWeb/Knot%20Theory%20History.htm] Scottish physicist Peter Tait spent many years listing unique knots under the belief that he was creating a table of elements. When ether was discredited through the Michelson-Morley experiment, vortex theory became completely obsolete, and knot theory fell out of scientific interest. Only in the past 100 years, with the rise of topology, have knots become a popular field of study. Today, knot theory is inextricably linked to String theory, DNA topology replication and recombination, and to areas of statistical mechanics. == An introduction to knot theory == Creating a knot is easy. Begin with a one-dimensional line segment, wrap it around itself arbitrarily, and then fuse its two free ends together to form a closed loop. One of the biggest unresolved problems in knot theory is to describe the different ways in which this may be done, or conversely to decide whether two such embeddings are different or the same.
equivalent to it |
#Move one loop completely over another.
#Move a string completely over or under a crossing.
Knot invariant can be defined by demonstrating a property of a knot diagram which is not changed when we apply any of the Reidemeister moves. Some very important invariants can be defined in this way, including the Jones polynomial. ===Higher dimensions=== You can unknot any circle in four dimensions. There are two steps to this. First, "push" the circle into a 3-dimensional subspace. This is the hard, technical part which we will skip. Now imagine temperature to be a fourth dimension to the 3-dimensional space. Then you could make one section of a line cross through the other by simply warming it with your fingers. In general piecewise-linear n-spheres form knots only in ''n''+2 space, although one can have smoothly knotted n-spheres in ''n''+3 space. ===Adding knots=== Two knots can be added by breaking the circles and connecting the pairs of ends. Knots in 3-space form a commutative monoid with prime factorization. The trefoil knots are the simplest prime knots. Higher dimensional knots can be added by splicing the spheres. While you cannot form the unknot in three dimensions by adding two non-trivial knots, you can in higher dimensions. == See also == * list of knot theory topics * knot invariant * braid theory * topoisomerase * DNA topology * linking number * Borromean rings == Further reading == * ''The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots'', Colin Adams, 2001, ISBN 0716742195 * ''Knots: Mathematics With a Twist'', Alexei Sossinsky, 2002, ISBN 0674009444 * ''Knot Theory'', Vassily Manturov, 2004, ISBN 0415310016 * * Dror Bar-Natan, [http://www.math.toronto.edu/~drorbn/KAtlas/ The Knot Atlas]. * [http://www.cs.ubc.ca/nest/imager/contributions/scharein/KnotPlot.html KnotPlot] == Other resources == * [http://www.pims.math.ca/knotplot/download.html Software for Viewing Knots (Freeware)] Algebraic topology Geometric topology Knot theory
Knot theory
I moved the following todo list from the article (where it doesn't belong) to this Talk page. ''Still to come:'' * Gauß diagrams[http://arxiv.org/pdf/math.GT/9805078] * Signed Graph representation of knots * History of knot theory, including resurgence since Jones polynomial * Maybe try to explain Edward Witten's connection between knots and quantum gravity! —User:Herbee 13:17, 2004 May 16 (UTC) == broken link == [http://www.ma.hw.ac.uk/RSE/meetings_etc/ordmtgs/2001/reports/knots.htm] == You/Me == I'm not comfortable with this article using first and third person ('you'/'me'). The [http://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style#Avoid_self-referential_pronouns Wiki style manual] isn't so keen on it either. Is it all right with everyone if I dive in and do my best to avoid the 'bad' pronouns? User:Spamguy 22:13, Jun 4, 2005 (UTC) ==List of knot theory topics== Please help complete the list of knot theory topics by adding relevant articles on knots, braids, links, etc. User:Michael Hardy 20:10, 9 Jun 2005 (UTC)
Knot theory
Knot theory is a branch of topology that concerns itself with abstract properties of knot (mathematics) — the spatial arrangements that in principle could be assumed by a closed loop of string. Geometric topologyAlgebraic topology
See other meanings of words starting from letter:
K
KA | KB | KC | KD | KE | KF | KG | KH | KI | KJ | KL | KM | KN | KO | KP | KR | KS | KT | KU | KW | KX | KY | KZ |Words begining with Knot_theory:
Knot_theory
Knot_theory
Knot_theory
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