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Encyclopedia > Knot theory
Trefoil knot, the simplest non-trivial knot.
Trefoil knot, the simplest non-trivial knot.

In mathematics, knot theory is the branch of topology that studies mathematical knots, which are defined as embeddings of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations (isotopies). This is basically equivalent to a conventional knot with the ends of the string joined together to prevent it from becoming undone. Diagram of a right trefoil knot. ... Diagram of a right trefoil knot. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... A trefoil knot. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... Circle illustration This article is about the shape and mathematical concept of circle. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... The two bold paths shown above are homotopic relative to their endpoints. ... Some knots: 1. ...


Knots can be described in various ways, but the most common method is by planar diagrams. Given a method of description, a knot will have many descriptions, e.g., many diagrams, representing it. A fundamental problem in knot theory is determining when two descriptions represent the same knot. One way of distinguishing knots is by using a knot invariant, a "quantity" which remains the same even with different descriptions of a knot. Two knots can be shown to be different if an invariant takes different values on them; however, an invariant may take the same value on different knots. A knot invariant is a useful tool in knot theory. ...


The concept of a knot has been extended to higher dimensions by considering n-dimensional spheres in m-dimensional Euclidean space. This was investigated most actively in the period 1960-1980, when a number of breakthroughs were made. In recent years, low dimensional phenomena has garnered the most interest. For other uses, see sphere (disambiguation). ... In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. ...


Research in knot theory began with the creation of knot tables and the systematic tabulation of knots. While tabulation remains an important task, today's researchers have a wide variety of backgrounds and goals. Classical knot theory, as initiated by Max Dehn, J. W. Alexander, and others, concerns primarily knot invariants such as the knot group or those coming from homology theory, such as the Alexander polynomial. Max Dehn (November 13, 1878 – June 27, 1952) was a German mathematician. ... J. W. Alexander James Waddell Alexander II (September 19, 1888 – September 23, 1971) was an important topologist of the pre-WWII era and part of an influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others. ... In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. ... In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. ... In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. ...


Vaughan Jones' discovery of the Jones polynomial in 1984 and subsequent contributions from Edward Witten, Maxim Kontsevich, and others, revealed deep connections between knot theory and mathematical methods in statistical mechanics and quantum field theory. A plethora of knot invariants have been invented since, including the quantum invariants and finite type invariants. These have been shown to be connected to more general invariants of 3-manifolds. Vaughan Frederick Randal Jones (born 31 December 1952) is a New Zealand mathematician, known for his work on von Neumann algebras, knot polynomials and conformal field theory. ... Edward Witten (born August 26, 1951) is an American mathematical physicist, Fields Medalist, and professor at the Institute for Advanced Study. ... Maxim Kontsevich (Russian: Максим Концевич) (born August 25, 1964) is a Russian mathematician. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... A knot invariant is a useful tool in knot theory. ... In mathematics, a 3-manifold is a 3-dimensional manifold. ...


In the last 30 years, knot theory has also become a tool in applied mathematics. Chemists and biologists use knot theory to understand, for example, chirality of molecules and the actions of enzymes on DNA. Chirality is a manga by Satoshi Urushihara Chirality (Greek handedness, derived from the word stem χειρ~, ch[e]ir~ - hand~) is an asymmetry property important in several branches of science. ... In science, a molecule is a group of atoms in a definite arrangement held together by chemical bonds. ... Ribbon diagram of the enzyme TIM, surrounded by the space-filling model of the protein. ... The structure of part of a DNA double helix Deoxyribonucleic acid (DNA) is a nucleic acid that contains the genetic instructions for the development and function of living organisms. ...

A more complicated knot
A more complicated knot

Contents

Image File history File links Knot_8sb19. ... Image File history File links Knot_8sb19. ...

History

Knots were studied by Carl Friedrich Gauss, who developed the Gauss linking integral for computing the linking number of two knots. His student Johann Benedict Listing, after whom Listing's knot is named, furthered their study. The early, significant stimulus in knot theory would arrive later with Sir William Thomson (Lord Kelvin) and his theory of vortex atoms.   (30 April 1777 – 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ... In mathematics, in the area of knot theory, the linking coefficient is a knot invariant that assigns an integer to a pair of closed curves. ... In mathematics, the linking number is a simple invariant for links (i. ... Johann Benedict Listing born July 25, 1808, died December 24, 1882 was a German mathematician, born in Frankfurt, Germany, and died in Göttingen, Germany. ... 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. ... William Thomson, 1st Baron Kelvin, OM, GCVO, PC, PRS, FRSE, (26 June 1824 – 17 December 1907) was a mathematical physicist, engineer, and outstanding leader in the physical sciences of the 19th century. ...


In 1867 after observing Scottish physicist Peter Tait's experiments involving smoke rings, Thomson came to the idea that atoms were knots of swirling vortices in the æther. Chemical elements would thus correspond to knots and links. Tait's experiments were inspired by a paper of Helmholtz's on vortex-rings in incompressible fluids. Thomson and Tait believed that an understanding and classification of all possible knots would explain why atoms absorb and emit light at only the discrete wavelengths that they do. For example, Thomson thought that sodium could be the Hopf link due to its two lines of spectra. Motto: (Latin) No one provokes me with impunity(English) Wha daur meddle wi me? (Scots)[1] Anthem: Multiple unofficial anthems Capital Edinburgh Largest city Glasgow Official languages English, Gaelic, Scots[2] Government  - Queen Queen Elizabeth II  - Prime Minister Tony Blair MP  - First Minister Jack McConnell MSP Unification    - by Kenneth I... ... Peter Tait Peter Guthrie Tait (April 28, 1831 - July 4, 1901) was a Scottish mathematical physicist. ... The luminiferous aether: it was hypothesised that the Earth moves through a medium of aether that carries light In the late 19th century luminiferous aether (light-bearing aether) was the term used to describe a medium for the propagation of light. ... Extremely high resolution spectrum of the Sun showing thousands of elemental absorption lines (fraunhofer lines) Spectroscopy is the study of matter and its properties by investigating light, sound, or particles that are emitted, absorbed or scattered by the matter under investigation. ... The wavelength is the distance between repeating units of a wave pattern. ... Skein relation for the Hopf link. ...


Tait subsequently began listing unique knots in the belief that he was creating a table of elements. He formulated what are now known as the Tait conjectures on alternating knots. The conjectures spurred research in knot theory, and were finally resolved in the 1990s. Tait's knot tables were subsequently improved upon by C. N. Little and T. P. Kirkman. One of three non-alternating knots with crossing number 8 In knot theory, a link diagram is alternating if the crossings alternate under, over, under, over, as you travel along each component of the link. ... One of three non-alternating knots with crossing number 8 In knot theory, a knot diagram is alternating if the crossings alternate under, over, under, over, as you travel along the strand. ...


James Clerk Maxwell, a colleague and friend of Thomson's and Tait's, also gained a strong interest of knots. Maxwell studied Listing's work on knots. He re-interpreted Gauss' linking integral in terms of electromagnetic theory. In his formulation, the integral represented the work done by a charged particle moving along one component of the link under the influence of the magnetic field generated an electric current along the other component. Maxwell also continued the study of smoke rings by considering three interacting rings. James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and theoretical physicist. ...


When the luminiferous æther was not detected in the Michelson-Morley experiment, vortex theory became completely obsolete, and knot theory ceased to be of great scientific interest. Modern physics demonstrates that the discrete wavelengths depend on quantum energy levels. The Michelson-Morley experiment, one of the most important and famous experiments in the history of physics, was performed in 1887 by Albert Michelson and Edward Morley at what is now Case Western Reserve University, and is considered by some to be the first strong evidence against the theory of... In physics, an energy level is a quantified stable energy, which a physical system can have; the term is most commonly used in reference to the electron configuration of electrons, in atoms or molecules. ...


Following the development of topology in the early 20th century spearheaded by Henri Poincare, topologists such as Max Dehn, J. W. Alexander, and Kurt Reidemeister, investigated knots. Out of this sprang the Reidemeister moves and the Alexander polynomial. Dehn also developed Dehn surgery, which related knots to the general theory of 3-manifolds and formulated the Dehn problems in group theory, such as the word problem. Early pioneers in the first half of the 20th century include Ralph Fox, who popularized the subject. In this early period, knot theory primarily consisted of study of the knot group and homological invariants of the knot complement. A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... Henri Poincaré, photograph from the frontispiece of the 1913 edition of Last Thoughts Jules Henri Poincaré (April 29, 1854 – July 17, 1912) was one of Frances greatest mathematicians, theoretical scientists and a philosopher of science. ... Max Dehn (November 13, 1878 – June 27, 1952) was a German mathematician. ... J. W. Alexander James Waddell Alexander II (September 19, 1888 – September 23, 1971) was an important topologist of the pre-WWII era and part of an influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others. ... Kurt Werner Friedrich Reidemeister (October 13, 1893 - July 8, 1971) was a mathematician born in Brunswick, Germany. ... Trefoil knot, the simplest non-trivial knot. ... In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. ... A Dehn surgery is a specific construction used to modify 3-manifolds with at least one torus boundary component, e. ... Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics education, a word problem is a mathematical question written without relying heavily on mathematics notation. ... Ralph H. Fox was an American mathematician. ... In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. ... In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. ... In mathematics, the knot complement of a tame knot K is the set-theoretic complement of the interior of the embedding of a solid torus into the 3-sphere. ...


A few major discoveries in the late 20th century greatly revived knot theory. The first was Thurston's hyperbolization theorem which introduced the theory of hyperbolic 3-manifolds into knot theory and made it of prime importance. Thurston's work also led, after much expansion by others, to the effective use of tools from representation theory and algebraic geometry. Important results followed, including the Gordon-Luecke theorem, which showed that knots were determined (up to mirror-reflection) by their complements, and the Smith conjecture. The geometrization conjecture, also known as Thurstons geometrization conjecture, concerns the geometric structure of compact 3-manifolds. ... A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannnian metric of constant sectional curvature -1. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, the Gordon-Luecke theorem on knot complements states that every homeomorphism between two complements of knots in the 3-sphere extends to give a self-homeomorphism of the 3-sphere. ... In mathematics, the Smith conjecture was a problem open for many years, and proved at the end of the 1970s. ...


Interest in knot theory grew significantly after Vaughan Jones' discovery of the Jones polynomial. This led to other knot polynomials such as the bracket polynomial, HOMFLY polynomial, and Kauffman polynomial. Jones was awarded the the highest honor in mathematics, the Fields medal, in 1990 for this work. In 1988 Edward Witten proposed a new framework for the Jones polynomial, utilizing existing ideas from mathematical physics, such as Feynman path integrals, and introducing new notions such as topological quantum field theory. Witten also received the Field medal in 1994 partly for this work. Witten's description of the Jones polynomial implied related invariants for 3-manifolds. Different attempts to put Witten's work on rigorous mathematical foundations created a number of interesting developments, such as the Witten-Reshetikhin-Turaev invariants and various so-called "quantum invariants". Vaughan Frederick Randal Jones (born 31 December 1952) is a New Zealand mathematician, known for his work on von Neumann algebras, knot polynomials and conformal field theory. ... This article needs cleanup. ... In the mathematical field of knot theory, the bracket polynomial (also known as the Kauffman bracket) is a polynomial invariant of framed links. ... In the mathematical field of knot theory, the HOMFLY polynomial, sometimes called the HOMFLY-PT polynomial or the generalized Jones polynomial, is a 2-variable knot polynomial, i. ... The Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman. ... The Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. ... Edward Witten (born August 26, 1951) is an American mathematical physicist, Fields Medalist, and professor at the Institute for Advanced Study. ... Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories1. ... This article is about a formulation of quantum mechanics. ... A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. ... In mathematics, a 3-manifold is a 3-dimensional manifold. ... In physics, Chern-Simons theory is a 3-dimensional topological quantum field theory of Schwarz type. ...


The last several decades of the 20th century, scientists and mathematicians began finding applications of knot theory to problems in biology and chemistry. Knot theory can be used to determine if a molecule is chiral (has a "handedness") or not. Chemical compounds of different handedness can have drastically differing properties. More generally, knot theoretic methods have been used in studying topoisomers, topologically different arrangements of the same chemical formula. The closely related theory of tangles have been effectively used in studying the action of certain enzymes on DNA. (Flapan 2000) This article or section does not adequately cite its references or sources. ... This article or section includes a list of works cited but its sources remain unclear because it lacks in-text citations. ... Chirality refers to several phenomena, all having to do with objects that differ from their mirror image. ... Topoisomers or topological isomers are molecules with the same chemical formula but different topologies. ... In mathematics, an n-tangle is a proper embedding of the disjoint union of n arcs into a 3-ball. ...


Knot equivalence

A knot is created by beginning with a one-dimensional line segment, wrap it around itself arbitrarily, and then fuse its two free ends together to form a closed loop. Some movements, such as small perturbations of the knot embeds the loop differently in three dimensional space, but intuitively, this embedding should really be considered the "same" as the first. The idea of knot equivalence is to make precise when two embeddings should be considered the same. :For other senses of this word, see dimension (disambiguation). ...

Two unknots
The unknot, and a knot
equivalent to it

When mathematical topologists consider knots and other entanglements such as links and braids, they describe how the knot is positioned in the space around it, called the ambient space. If the knot is moved smoothly, without cutting or passing a segment through another, to coincide with another knot, the two knots are considered equivalent. Image File history File links No higher resolution available. ... The Borromean rings, a link with three components each equivalent to the unknot. ... A braid Step by step creation of a basic braid using three strings To braid is to interweave or twine three or more separate strands of one or more materials in a diagonally overlapping pattern. ... The ambient space, in mathematics, is the space surrounding a mathematical object. ...


The basic problem of knot theory, the recognition problem, can thus be stated as: given two knots, determine whether or not they are equivalent or not. Algorithms exist to solve this problem, with the first given by Wolfgang Haken. Nonetheless, these algorithms use significantly many steps, and a major issue in the theory is to understand how hard this problem really is.[1] The special case of recognizing the unknot, called the unknotting problem, is of particular interest. In mathematics, computing, linguistics, and related disciplines, an algorithm is a procedure (a finite set of well-defined instructions) for accomplishing some task which, given an initial state, will terminate in a defined end-state. ... Wolfgang Haken (born June 21, 1928) is a mathematician who specialized in topology, in particular 3-manifolds. ... The unknot, and a knot equivalent to it The unknot is a loop of rope without a knot in it (in knot theory, ropes have no ends; they are loops). ... In mathematics, the unknotting problem is the problem of algorithmically recognizing the unknot, given some input, e. ...

Is this unknotted?
Is this unknotted?

Image File history File links Size of this preview: 763 × 600 pixel Image in higher resolution (868 × 682 pixel, file size: 36 KB, MIME type: image/png) I made this using Adobe Illustrator. ... Image File history File links Size of this preview: 763 × 600 pixel Image in higher resolution (868 × 682 pixel, file size: 36 KB, MIME type: image/png) I made this using Adobe Illustrator. ...

Knot diagrams

A useful way to visualise and manipulate knots is to project the knot onto a plane - think of the knot casting a shadow on the wall. A small perturbation in the choice of projection place will ensure that the projection is one to one except at the double points, called crossings, where the "shadow" of the knot crosses itself once transversely (Rolfsen 1976). At each crossing we must indicate which section is "over" and which is "under", so as to be able to recreate the original knot. This is often done by creating a break in the understrand.


Reidemeister moves

Main article: Reidemeister move
The Reidemeister moves
The 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: Trefoil knot, the simplest non-trivial knot. ... Image File history File links Reidemeitster. ... Image File history File links Reidemeitster. ... J. W. Alexander James Waddell Alexander II (September 19, 1888 – September 23, 1971) was an important topologist of the pre-WWII era and part of an influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others. ... Kurt Werner Friedrich Reidemeister (October 13, 1893 - July 8, 1971) was a mathematician born in Brunswick, Germany. ...

  1. Twist and untwist in either direction.
  2. Move one loop completely over another.
  3. Move a string completely over or under a crossing.

Knot invariants

Main article: knot invariant

A knot invariant is a "quantity" that is the same for equivalent knots (Adams 2001, Lickorish 1997, Rolfsen 1976). An invariant may take the same value on two different knots, so by itself may be incapable of distinguishing all knots. An elementary invariant is tricolorability. A knot invariant is a useful tool in knot theory. ... The Tricolorability of a knot refers to the ability of a knot to be colored with three colors according to two rules. ...


"Classical" knot invariants include the knot group, which is the fundamental group of the knot complement, and the Alexander polynomial, which can be computed from the Alexander invariant, a module constructed from the infinite cyclic cover of the knot complement (Lickorish 1997, Rolfsen 1976). In the late 20th century, invariants such as "quantum" knot polynomials and hyperbolic invariants were discovered. These aforementioned invariants are only the tip of the iceberg of modern knot theory. In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... In mathematics, the knot complement of a tame knot K is the set-theoretic complement of the interior of the embedding of a solid torus into the 3-sphere. ... In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. ...


Knot polynomials

Main article: knot polynomial

A knot polynomial is a knot invariant that is a polynomial. Well-known examples include the Jones and Alexander polynomials. A variant of the Alexander polynomial, the Alexander-Conway polynomial, is a polynomial in the variable z with integer coefficients (Lickorish 1997). This article needs to be cleaned up to conform to a higher standard of quality. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... This article needs cleanup. ... In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. ... In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. ... The integers are commonly denoted by the above symbol. ...


Suppose we are given a link diagram which is oriented, i.e. every component of the link has a preferred direction indicated by an arrow. Also suppose L + ,L ,L0 are oriented link diagrams resulting from changing the diagram at a specified crossing of the diagram, as indicated in the figure:

Then the Alexander-Conway polynomial, C(z), is recursively defined according to the rules: Image File history File links Download high-resolution version (2113x1127, 53 KB) I created this using Adobe Illustrator. ...

  • C(O) = 1 (where O is any diagram of the unknot)
  • C(L + ) = C(L ) + zC(L0)

The second rule is what is often referred to as a skein relation. To check that these rules give an invariant, one should determine that the polynomial does not change under the three Reidemeister moves. Many important knot polynomials can be defined in this way. The unknot, and a knot equivalent to it The unknot is a loop of rope without a knot in it (in knot theory, ropes have no ends; they are loops). ... Skein relations are a piece of knot theory usually used to recursively define knot polynomials using knot diagrams as bookkeeping (compare Stückelberg-Feynman diagrams). ...


The following is an example of a typical computation using a skein relation. It computes the Alexander-Conway polynomial of the trefoil knot. The yellow patches indicate where we applied the relation. Categories: Stub | Knot theory ...

C( image:skein-relation-trefoil-plus-sm.png)=C( image:skein-relation-trefoil-minus-sm.png) + z C( image:skein-relation-trefoil-zero-sm.png)

gives the unknot and the Hopf link. Applying the relation to the Hopf link where indicated, Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Skein relation for the Hopf link. ...

C( image:skein-relation-link22-plus-sm.png) = C( image:skein-relation-link22-minus-sm.png) + z C( image:skein-relation-link22-zero-sm.png)

gives a link deformable to one with 0 crossings (it is actually the unlink of two components) and an unknot. The unlink takes a bit of sneakiness: Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In the mathematical field of knot theory, the unlink is a link that is equivalent (under ambient isotopy) to finitely many disjoint circles in the plane. ...

C( image:skein-relation-link20-plus-sm.png) = C( image:skein-relation-link20-minus-sm.png)+ z C( image:skein-relation-link20-zero-sm.png)

which implies that C(unlink of two components) = 0, since the first two polynomials are of the unknot and thus equal. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...


Putting all this together will show:

C(trefoil) = 1 + z (0 + z) = 1 + z2

Note that if we believe that the Alexander-Conway polynomial is actually a knot invariant, this shows that the trefoil is not equivalent to the unknot. So there is really a knot that is "knotted".


Actually, there are two trefoil knots, called the right and left-handed trefoils, which are mirror images of each other (take a diagram of the trefoil given above and change each crossing to the other way to get the mirror image). These are not equivalent to each other! This was shown by Max Dehn, before the invention of knot polynomials, using group theoretical methods (Dehn, 1914). But the Alexander-Conway polynomial of each kind of trefoil will be the same, as can be seen by going through the computation above with the mirror image. The Jones polynomial can in fact distinguish between the left and right handed trefoil knots (Lickorish 1997). Max Dehn (November 13, 1878 – June 27, 1952) was a German mathematician. ...


Hyperbolic invariants

William Thurston proved many knots are hyperbolic knots, meaning that the knot complement, i.e. the points of 3-space not on the knot, admit a geometric structure, in particular that of hyperbolic geometry. The hyperbolic structure depends only on the knot so any quantity computed from the hyperbolic structure is then a knot invariant. (Adams 2001) William Thurston William Paul Thurston (born October 30, 1946) is an American mathematician. ... In mathematics, a hyperbolic link is a link in the 3-sphere with a complement that has a Riemannian metric of constant negative curvature, i. ... In mathematics, the knot complement of a tame knot K is the set-theoretic complement of the interior of the embedding of a solid torus into the 3-sphere. ... Lines through a given point P and hyperparallel to line l. ...


Geometry lets us visualize what the inside of a knot or link complement looks like by imagining light rays as traveling along the geodesics of the geometry. An example is provided by the picture of the complement of the Borromean rings. The inhabitant of this link complement is viewing the space from near the red component. The balls in the picture are views of horoball neighborhoods of the link. By thickening the link in a standard way, we obtain what are called horoball neighborhoods of the link components. Even though the boundary of a neighborhoods is a torus, when viewed by inside the link complement, it looks like a sphere, called a horoball. Each link component shows up as infinitely many horoballs (of one color) as there are infinitely many light rays from the observer to the link component. The fundamental parallelogram (which is indicated in the picture), tiles both vertically and horizontally. In mathematics, the Borromean rings consist of three topological circles which are linked despite the fact that no two of them are linked, i. ...


The pattern of horoballs is itself a useful invariant. Other hyperbolic invariants include the shape of the fundamental paralleogram, length of shortest geodesic, and volume. Modern knot and link tabulation efforts have utilized these invariants effectively. Fast computers and clever methods of obtaining these invariants make calculating these invariants, in practice, a simple task. (Adams, Hildebrand, & Weeks, 1991)


Higher dimensions

In four dimensions, any closed loop of one-dimensional string is equivalent to an unknot. We can achieve the necessary deformation in two steps. The first step is to "push" the loop into a three-dimensional subspace, which is always possible, though technical to explain. The second step is changing crossings. Suppose one strand is behind another as seen from a chosen point. Lift it into the fourth dimension, so there is no obstacle (the front strand having no component there); then slide it forward, and drop it back, now in front. An analogy for the plane would be lifting a string up off the surface.


Since a knot can be considered topologically a 1-dimensional sphere, the next generalization is to consider a two dimensional sphere embedded in a four dimensional sphere. Such an embedding is unknotted if there is a homeomorphism of the 4-sphere onto itself taking the 2-sphere to a standard "round" 2-sphere. Suspended knots and spun knots are two typical families of such 2-sphere knots. For other uses, see sphere (disambiguation). ...


The mathematical technique called "general position" implies that for a given n-sphere in the m-sphere, if m is large enough (depending on n), the sphere should be unknotted. In general, piecewise-linear n-spheres form knots only in (n+2)-space (Zeeman 1963), although this is no longer a requirement for smoothly knotted spheres. In fact, there are smoothly knotted 4k-1-spheres in 6k-space, e.g. there is a smoothly knotted 3-sphere in the 6-sphere (Haefliger 1962, Levine 1965). Thus the codimension of a smooth knot can be arbitrarily large when not fixing the dimension of the knotted sphere; however, any smooth 4k-1-sphere in an n-sphere with n > 6k is unknotted. In mathematics, a piecewise linear function , where V is a vector space and is a subset of a vector space, is any function with the property that can be decomposed into finitely many convex polytopes, such that f is equal to a linear function on each of these polytopes. ... For other uses, see sphere (disambiguation). ...


Adding knots

Main article: knot sum

Two knots can be added by cutting both knots and joining the pairs of ends. This can be formally defined as follows (Adams 2001): consider a planar projection of each knot and suppose these projections are disjoint. Find a rectangle in the plane where one pair of opposite sides are arcs along each knot while the rest of the rectangle is disjoint from the knots. Form a new knot by deleting the first pair of opposite sides and adjoining the other pair of opposite sides. The resulting knot is the sum of the original knots. Trefoil knot, the simplest non-trivial knot. ...


This operation is called the knot sum, or sometimes the connected sum or composition of two knots. The knot sum is commutative and associative. There is also a prime decomposition for a knot which allows us to define a prime or composite knot, analogous to prime and composite numbers. The trefoil knot is the simplest prime knot. Higher dimensional knots can be added by splicing the n-spheres. While you cannot form the unknot in three dimensions by adding two non-trivial knots, you can in higher dimensions, at least when one considers smooth knots in codimension at least 3. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, associativity is a property that a binary operation can have. ... In knot theory, a prime knot is a knot which is, in a certain sense, indecomposable. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In knot theory, a prime knot is a knot which is, in a certain sense, indecomposable. ...


Tabulating knots

Traditionally, knots have been catalogued in terms of crossing number. The number of nontrivial knots of a given crossing number increases rapidly, making tabulation computationally difficult. Knot tables generally include only prime knots and only one entry for a knot and its mirror image (even if they are different). The sequence of the number of prime knots of a given crossing number, up to crossing number 16, is 0, 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, 46972, 253293, 1388705... .[2] While exponential upper and lower bounds for this sequence are known, it has not been proven that this sequence is strictly increasing. (Adams 2001) In mathematics, crossing numbers arise in two related contexts: in knot theory and in graph theory. ...


The first knot tables by Tait, Little, and Kirkman used knot diagrams, although Tait also used a precursor to the Dowker notation. Different notations have been invented for knots which allow more efficient tabulation. A knot diagram with crossings labelled for a Dowker sequence In the mathematical field of knot theory, the Dowker notation, also called the Dowker-Thistlethwaite notation or code, for a knot is a sequence of even integers. ...


The early tables attempted to list all knots of at most 10 crossings, and all alternating knots of 11 crossings. The development of knot theory due to Alexander, Reidemeister, Seifert, and others eased the task of verification and tables of knots up to and including 9 crossings were published by Alexander-Briggs and Reidemeister in the late 1920s.


The first major verification of this work was done in the 1960s by John Horton Conway, who not only developed a new notation but also the Alexander-Conway polynomial. (Conway 1970, Doll-Hoste 1991) This verified the list of knots of at most 11 crossings and a new list of links up to 10 crossings. Conway found a number of omissions but only duplication in the Tait-Little tables; however he missed the duplicates called the Perko pair, which would only be noticed in 1974 by Kenneth Perko. (Perko 1974) This famous error would propogate when Dale Rolfsen added a knot table in his influential text, based on Conway's work. John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. ... In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. ...


Alexander-Briggs notation

This is the most traditional notation, due to the 1927 paper of J. W. Alexander and G. Briggs and later extended by Dale Rolfsen in his knot table. The notation simply organizes knots by their crossing number. One writes the crossing number with a subscript to denote its order amongst all knots with that crossing number. This order is arbitrary and so has no special significance.


The Dowker notation

Main article: Dowker notation
A knot diagram with crossings labelled for a Dowker sequence
A knot diagram with crossings labelled for a Dowker sequence

The Dowker notation, also called the Dowker-Thistlethwaite notation or code, for a knot is a finite sequence of even integers. The numbers are generated by following the knot and marking the crossings with consecutive integers. Since each crossing is visited twice, this creates a pairing of even integers with odd integers. An appropriate sign is given to indicate over and undercrossing. For example, in the figure the knot diagram has crossings labelled with the pairs (1,6) (3,-12) (5,2) (7,8) (9,-4) and (11,-10). The Dowker notation for this labelling is the sequence: 6 -12 2 8 -4 -10. A knot diagram has more than one possible Dowker notation, and there is a well-understood ambiguity when reconstructing a knot from a Dowker notation. A knot diagram with crossings labelled for a Dowker sequence In the mathematical field of knot theory, the Dowker notation, also called the Dowker-Thistlethwaite notation or code, for a knot is a sequence of even integers. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...


Conway notation

Main article: Conway notation (knot theory)

The Conway notation for knots and links, named after John Horton Conway, is based on the the theory of tangles. (Conway, 1970) The advantage of this notation is that it reflects some properties of the knot or link. John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. ... In mathematics, an n-tangle is a proper embedding of the disjoint union of n arcs into a 3-ball. ...


The notation describes how to construct a particular link diagram of the link. Start with a basic polyhedron, a 4-valent connected planar graph with no digon regions. Such a polyhedron is denoted first by the number of vertices then a number of asterisks which determine the polyhedron's position on a list of basic polyhedron. For example, 10** denotes the second 10-vertex polyhedron on Conway's list. In geometry a digon is a polygon with two sides and two vertices. ...


Each vertex then has an algebraic tangle substituted into it (each vertex is oriented so there is no arbitrary choice in substitution). Each such tangle has a notation consisting of numbers and + or - signs. In mathematics, an n-tangle is a proper embedding of the disjoint union of n arcs into a 3-ball. ...


An example is 1*2 -3 2. The 1* denotes the only 1-vertex basic polyhedron. The 2 -3 2 is a sequence describing the continued fraction associated to a rational tangle. One inserts this tangle at the vertex of the basic polyhedron 1*. In mathematics, an n-tangle is a proper embedding of the disjoint union of n arcs into a 3-ball. ...


A more complicated example is 8*3.1.2 0.1.1.1.1.1 Here again 8* refers to a basic polyhedron with 8 vertices. The periods separate the notation for each tangle.


Any link admits such a description, and it is clear this is a very compact notation even for very large crossing number. There are some further shorthands usually used. The last example is usually written 8*3:2 0, where we omitted the ones and kept the number of dots excepting the dots at the end. For an algebraic knot such as in the first example, 1* is often omitted.


Conway's pioneering paper on the subject lists up to 10-vertex basic polyhedra of which he uses to tabulate links, which have become standard for those links. For a further listing of higher vertex polyhedra, there are nonstandard choices available.


See also

This list contains articles related to the mathematical theory of knots, links, and braids. ... In topology, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalisations. ... Representation of a khipu Khipu, or quipu, were recording devices used during the Inca Empire and its predecessor societies in the Andean region. ... In mathematics, contact geometry is the study of completely nonintegrable hyperplane fields on manifolds. ... In chemistry, a molecular knot (knotane) is a molecule whose topology forms a knot. ... Topoisomerases (type I: EC 5. ...

References

  • Colin Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, 2001, ISBN 0-7167-4219-5
  • Adams, Colin; Hildebrand, Martin; Weeks, Jeffrey; Hyperbolic invariants of knots and links. Trans. Amer. Math. Soc. 326 (1991), no. 1, 1--56.
  • John Horton Conway, An enumeration of knots and links, and some of their algebraic properties. 1970 Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) pp. 329--358 Pergamon, Oxford
  • Max Dehn, Die beiden Kleeblattschlingen, Math. Ann. 75 (1914), 402-413.
  • Helmut Doll and Jim Hoste, A tabulation of oriented links. With microfiche supplement. Math. Comp. 57 (1991), no. 196, 747--761.
  • Erica Flapan, When topology meets chemistry: A topological look at molecular chirality. Outlooks. Cambridge University Press, Cambridge; Mathematical Association of America, Washington, DC, 2000. xiv+241 pp. ISBN 0-521-66254-0; ISBN 0-521-66482-9
  • André Haefliger, Knotted (4k-1)-spheres in 6k-space. Ann. of Math. (2) 75 1962 452--466.
  • Jerome Levine, A classification of differentiable knots. Ann. of Math. (2) 82 1965 15--50.
  • W.B. Raymond Lickorish, An Introduction to Knot Theory, Graduate Texts in Mathematics, Springer, 1997, ISBN 0-387-98254-X
  • Kenneth A. Perko Jr., On the classification of knots. Proc. Amer. Math. Soc. 45 (1974), 262--266.
  • Dale Rolfsen, Knots and Links, 1976, ISBN 0-914098-16-0
  • Silver, Dan, Scottish physics and knot theory's odd origins (expanded version of Silver, "Knot theory's odd origins," American Scientist, 94, No. 2, 158-165)
  • E. C. Zeeman, Unknotting combinatorial balls. Ann. of Math. (2) 78 1963 501--526.

2001 (MMI) was a common year starting on Monday of the Gregorian calendar. ... John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. ... Jerome Paul Levine (May 4, 1937–April 8, 2006) was a mathematician who contributed to the understanding of knot theory. ... The Annals of Mathematics (ISSN 0003-486X), often just called Annals, is a bimonthly mathematics research journal published by Princeton University and the Institute for Advanced Study. ... 1976 (MCMLXXVI) was a leap year starting on Thursday. ... Sir Erik Christopher Zeeman (born February 4, 1925), is a mathematician known for work in geometric topology and singularity theory. ... The Annals of Mathematics (ISSN 0003-486X), often just called Annals, is a bimonthly mathematics research journal published by Princeton University and the Institute for Advanced Study. ...

Further reading

There are a number of introductions to knot theory. A classical introduction for graduate students or advanced undergraduates is Rolfsen (1976), given in the references. Other good texts from the references are Adams (2001) and Lickorish (1997). Adams is informal and accessible for the most part to high schoolers. Lickorish is a rigorous introduction for graduate students, covering a nice mix of classical and modern topics.

  • Richard H. Crowell and Ralph Fox,Introduction to Knot Theory, 1977, ISBN 0-387-90272-4
  • Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, 1985, Walter de Gruyter, ISBN 3-11-008675-1
  • Louis H. Kauffman, On Knots, 1987, ISBN 0-691-08435-1

Ralph H. Fox was an American mathematician. ... For the album by Ash, see 1977 (album). ... Heiner Zieschang (* 12 November 1936; † 5 April 2004) was a german mathematician. ... Louis Kauffman, topologist Louis Kauffman is a topologist, whose work is primarily in knot theory and connections with statistical mechanics, quantum theory, algebra, combinatorics and foundations. ... 1987 (MCMLXXXVII) was a common year starting on Thursday of the Gregorian calendar. ...

External links

History

  • Thomson, Sir William (Lord Kelvin), On Vertex Atoms, Proceedings of the Royal Society of Edinburgh, Vol. VI, 1867, pp. 94-105.
  • Silliman, Robert H., William Thomson: Smoke Rings and Nineteenth-Century Atomism, Isis, Vol. 54, No. 4. (Dec., 1963), pp. 461-474. JSTOR link
  • Movie of a modern recreation of Tait's smoke ring experiment

Knot tables and software

  • KnotInfo: Table of Knot Invariants and Knot Theory Resources
  • The wiki Knot Atlas - detailed info on individual knots in knot tables
  • KnotPlot - software to investigate geometric properties of knots

  Results from FactBites:
 
Knot theory - definition of Knot theory in Encyclopedia (831 words)
Knot theory is a branch of topology that was inspired by observations, as the name suggests, of knots.
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 (also known as 'ether').
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