Intopology,knot theory is the study ofmathematical knots. While inspired byknots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is anembedding of acircle in 3-dimensionalEuclidean space,. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself (known as anambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.
Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram, in which any knot can be drawn in many different ways. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot.
A complete algorithmic solution to this problem exists, which has unknowncomplexity.[1] In practice, knots are often distinguished using aknot invariant, a "quantity" which is the same when computed from different descriptions of a knot. Important invariants includeknot polynomials,knot groups, and hyperbolic invariants.
The original motivation for the founders of knot theory was to create a table of knots andlinks, which are knots of several components entangled with each other. More than six billion knots and linkshave been tabulated since the beginnings of knot theory in the 19th century.
To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in otherthree-dimensional spaces and objects other than circles can be used; seeknot (mathematics). For example, a higher-dimensional knot is ann-dimensional sphere embedded in (n+2)-dimensional Euclidean space.
Archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such asrecording information andtying objects together, knots have interested humans for their aesthetics and spiritual symbolism. Knots appear in various forms of Chinese artwork dating from several centuries BC (seeChinese knotting). Theendless knot appears inTibetan Buddhism, while theBorromean rings have made repeated appearances in different cultures, often representing strength in unity. TheCeltic monks who created theBook of Kells lavished entire pages with intricateCeltic knotwork.
A mathematical theory of knots was first developed in 1771 byAlexandre-Théophile Vandermonde who explicitly noted the importance of topological features when discussing the properties of knots related to the geometry of position. Mathematical studies of knots began in the 19th century withCarl Friedrich Gauss, who defined thelinking integral (Silver 2006). In the 1860s,Lord Kelvin'stheory that atoms were knots in the aether led toPeter Guthrie Tait's creation of the first knot tables for complete classification. Tait, in 1885, published a table of knots with up to ten crossings, and what came to be known as theTait conjectures. This record motivated the early knot theorists, but knot theory eventually became part of the emerging subject oftopology.
These topologists in the early part of the 20th century—Max Dehn,J. W. Alexander, and others—studied knots from the point of view of theknot group and invariants fromhomology theory such as theAlexander polynomial. This would be the main approach to knot theory until a series of breakthroughs transformed the subject.
In the late 1970s,William Thurston introducedhyperbolic geometry into the study of knots with thehyperbolization theorem. Many knots were shown to behyperbolic knots, enabling the use of geometry in defining new, powerfulknot invariants. The discovery of theJones polynomial byVaughan Jones in 1984 (Sossinsky 2002, pp. 71–89), and subsequent contributions fromEdward Witten,Maxim Kontsevich, and others, revealed deep connections between knot theory and mathematical methods instatistical mechanics andquantum field theory. A plethora of knot invariants have been invented since then, utilizing sophisticated tools such asquantum groups andFloer homology.
In the last several decades of the 20th century, scientists became interested in studyingphysical knots in order to understand knotting phenomena inDNA and other polymers. Knot theory can be used to determine if a molecule ischiral (has a "handedness") or not (Simon 1986).Tangles, strings with both ends fixed in place, have been effectively used in studying the action oftopoisomerase on DNA (Flapan 2000). Knot theory may be crucial in the construction of quantum computers, through the model oftopological quantum computation (Collins 2006).
A knot is created by beginning with a one-dimensional line segment, wrapping it around itself arbitrarily, and then fusing its two free ends together to form a closed loop (Adams 2004) (Sossinsky 2002). Simply, we can say a knot is a "simple closed curve" (seeCurve) — that is: a "nearly"injective andcontinuous function, with the only "non-injectivity" being. Topologists consider knots and other entanglements such aslinks andbraids to be equivalent if the knot can be pushed about smoothly, without intersecting itself, to coincide with another knot.
The idea ofknot equivalence is to give a precise definition of when two knots should be considered the same even when positioned quite differently in space. A formal mathematical definition is that two knots are equivalent if there is anorientation-preservinghomeomorphism with.
What this definition of knot equivalence means is that two knots are equivalent when there is a continuous family of homeomorphisms of space onto itself, such that the last one of them carries the first knot onto the second knot. (In detail: Two knots and areequivalent if there exists a continuous mapping such that a) for each the mapping taking to is a homeomorphism of onto itself; b) for all; and c). Such a function is known as anambient isotopy.)
These two notions of knot equivalence agree exactly about which knots are equivalent: Two knots that are equivalent under the orientation-preserving homeomorphism definition are also equivalent under the ambient isotopy definition, because any orientation-preserving homeomorphisms of to itself is the final stage of an ambient isotopy starting from the identity. Conversely, two knots equivalent under the ambient isotopy definition are also equivalent under the orientation-preserving homeomorphism definition, because the (final) stage of the ambient isotopy must be an orientation-preserving homeomorphism carrying one knot to the other.
The basic problem of knot theory, therecognition problem, is determining the equivalence of two knots.Algorithms exist to solve this problem, with the first given byWolfgang Haken in the late 1960s (Hass 1998). Nonetheless, these algorithms can be extremely time-consuming, and a major issue in the theory is to understand how hard this problem really is (Hass 1998). The special case of recognizing theunknot, called theunknotting problem, is of particular interest (Hoste 2005). In February 2021Marc Lackenby announced a new unknot recognition algorithm that runs inquasi-polynomial time.[2]
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 change in the direction of projection will ensure that it isone-to-one except at the double points, calledcrossings, where the "shadow" of the knot crosses itself once transversely (Rolfsen 1976). At each crossing, to be able to recreate the original knot, the over-strand must be distinguished from the under-strand. This is often done by creating a break in the strand going underneath. The resulting diagram is animmersed plane curve with the additional data of which strand is over and which is under at each crossing. (These diagrams are calledknot diagrams when they represent aknot andlink diagrams when they represent alink.) Analogously, knotted surfaces in 4-space can be related toimmersed surfaces in 3-space.
Areduced diagram is a knot diagram in which there are noreducible crossings (alsonugatory orremovable crossings), or in which all of the reducible crossings have been removed.[3][4] Apetal projection is a type of projection in which, instead of forming double points, all strands of the knot meet at a single crossing point, connected to it by loops forming non-nested "petals".[5]
In 1927, working with this diagrammatic form of knots,J. W. Alexander andGarland Baird Briggs, and independentlyKurt 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 below. These operations, now called theReidemeister moves, are:
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| Type I | Type II |
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 | |
| Type III | |
The proof that diagrams of equivalent knots are connected by Reidemeister moves relies on an analysis of what happens under the planar projection of the movement taking one knot to another. The movement can be arranged so that almost all of the time the projection will be a knot diagram, except at finitely many times when an "event" or "catastrophe" occurs, such as when more than two strands cross at a point or multiple strands become tangent at a point. A close inspection will show that complicated events can be eliminated, leaving only the simplest events: (1) a "kink" forming or being straightened out; (2) two strands becoming tangent at a point and passing through; and (3) three strands crossing at a point. These are precisely the Reidemeister moves (Sossinsky 2002, ch. 3) (Lickorish 1997, ch. 1).
A knot invariant is a "quantity" that is the same for equivalent knots (Adams 2004) (Lickorish 1997) (Rolfsen 1976). For example, if the invariant is computed from a knot diagram, it should give the same value for two knot diagrams representing equivalent knots. An invariant may take the same value on two different knots, so by itself may be incapable of distinguishing all knots. An elementary invariant istricolorability.
"Classical" knot invariants include theknot group, which is thefundamental group of theknot complement, and theAlexander 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,Vassiliev invariants and hyperbolic invariants were discovered. These aforementioned invariants are only the tip of the iceberg of modern knot theory.
A knot polynomial is aknot invariant that is apolynomial. Well-known examples include theJones polynomial, theAlexander polynomial, and theKauffman polynomial. A variant of the Alexander polynomial, theAlexander–Conway polynomial, is a polynomial in the variablez withinteger coefficients (Lickorish 1997).
The Alexander–Conway polynomial is actually defined in terms oflinks, which consist of one or more knots entangled with each other. The concepts explained above for knots, e.g. diagrams and Reidemeister moves, also hold for links.
Consider an oriented link diagram,i.e. one in which every component of the link has a preferred direction indicated by an arrow. For a given crossing of the diagram, let be the oriented link diagrams resulting from changing the diagram as indicated in the figure:
.svg)
The original diagram might be either or, depending on the chosen crossing's configuration. Then the Alexander–Conway polynomial,, is recursively defined according to the rules:
The second rule is what is often referred to as askein relation. To check that these rules give an invariant of an oriented link, 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 following is an example of a typical computation using a skein relation. It computes the Alexander–Conway polynomial of thetrefoil knot. The yellow patches indicate where the relation is applied.
) = C(
) + zC(
)gives the unknot and theHopf link. Applying the relation to the Hopf link where indicated,
) =C(
) +zC(
)gives a link deformable to one with 0 crossings (it is actually theunlink of two components) and an unknot. The unlink takes a bit of sneakiness:
) =C(
) +zC(
)which implies thatC(unlink of two components) = 0, since the first two polynomials are of the unknot and thus equal.
Putting all this together will show:
Since the Alexander–Conway polynomial is a knot invariant, this shows that the trefoil is not equivalent to the unknot. So the trefoil really is "knotted".
Actually, there are two trefoil knots, called the right and left-handed trefoils, which aremirror 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, meaning that they are not amphichiral. This was shown byMax Dehn, before the invention of knot polynomials, usinggroup 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. TheJones polynomial can in fact distinguish between the left- and right-handed trefoil knots (Lickorish 1997).
William Thurston proved many knots arehyperbolic knots, meaning that theknot complement (i.e., the set of points of 3-space not on the knot) admits a geometric structure, in particular that ofhyperbolic geometry. The hyperbolic structure depends only on the knot so any quantity computed from the hyperbolic structure is then a knot invariant (Adams 2004).
Geometry lets us visualize what the inside of a knot or link complement looks like by imagining light rays as traveling along thegeodesics of the geometry. An example is provided by the picture of the complement of theBorromean rings. The inhabitant of this link complement is viewing the space from near the red component. The balls in the picture are views ofhoroball neighborhoods of the link. By thickening the link in a standard way, the horoball neighborhoods of the link components are obtained. Even though the boundary of a neighborhood is a torus, when viewed from inside the link complement, it looks like a sphere. Each link component shows up as infinitely many spheres (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 and shows how to extend the pattern of spheres infinitely.
This pattern, the horoball pattern, is itself a useful invariant. Other hyperbolic invariants include the shape of the fundamental parallelogram, 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).
A knot in three dimensions can be untied when placed in four-dimensional space. This is done by 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. Analogies for the plane would be lifting a string up off the surface, or removing a dot from inside a circle.
In fact, in four dimensions, any non-intersecting closed loop of one-dimensional string is equivalent to an unknot. First "push" the loop into a three-dimensional subspace, which is always possible, though technical to explain.
Four-dimensional space occurs in classical knot theory, however, and an important topic is the study ofslice knots andribbon knots. A notorious open problem asks whether every slice knot is also ribbon.
Since a knot can be considered topologically a 1-dimensional sphere, the next generalization is to consider atwo-dimensional sphere () embedded in 4-dimensional Euclidean space (). Such an embedding is knotted if there is no homeomorphism of onto itself taking the embedded 2-sphere to the standard "round" embedding of the 2-sphere.Suspended knots andspun knots are two typical families of such 2-sphere knots.
The mathematical technique called "general position" implies that for a givenn-sphere inm-dimensional Euclidean space, ifm is large enough (depending onn), the sphere should be unknotted. In general,piecewise-linearn-spheres form knots only in (n + 2)-dimensional space (Zeeman 1963), although this is no longer a requirement for smoothly knotted spheres. In fact, there are smoothly knotted-spheres in 6k-dimensional space; e.g., there is a smoothly knotted 3-sphere in (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 smoothk-sphere embedded in with is unknotted. The notion of a knot has further generalisations in mathematics, see:Knot (mathematics),isotopy classification of embeddings.
Every knot in then-sphere is the link of areal-algebraic set with isolated singularity in (Akbulut & King 1981).
Ann-knot is a single embedded in. Ann-link consists ofk-copies of embedded in, wherek is anatural number. Both the and the cases are well studied, and so is the case.[6][7]
Two knots can be added by cutting both knots and joining the pairs of ends. The operation is called theknot sum, or sometimes theconnected sum orcomposition of two knots. This can be formally defined as follows (Adams 2004): 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 a sum of the original knots. Depending on how this is done, two different knots (but no more) may result. This ambiguity in the sum can be eliminated regarding the knots asoriented, i.e. having a preferred direction of travel along the knot, and requiring the arcs of the knots in the sum are oriented consistently with the oriented boundary of the rectangle.
The knot sum of oriented knots iscommutative andassociative. Aknot isprime if it is non-trivial and cannot be written as the knot sum of two non-trivial knots. A knot that can be written as such a sum iscomposite. There is a prime decomposition for knots, analogous toprime and composite numbers (Schubert 1949). For oriented knots, this decomposition is also unique. Higher-dimensional knots can also be added but there are some differences. While you cannot form the unknot in three dimensions by adding two non-trivial knots, you can in higher dimensions, at least when one considerssmooth knots in codimension at least 3.
Knots can also be constructed using thecircuit topology approach. This is done by combining basic units called soft contacts using five operations (Parallel, Series, Cross, Concerted, and Sub).[8][9] The approach is applicable to open chains as well and can also be extended to include the so-called hard contacts.
Traditionally, knots have been catalogued in terms ofcrossing number. Knot tables generally include only prime knots, and only one entry for a knot and its mirror image (even if they are different) (Hoste, Thistlethwaite & Weeks 1998). The number of nontrivial knots of a given crossing number increases rapidly, making tabulation computationally difficult (Hoste 2005, p. 20). Tabulation efforts have succeeded in enumerating over 6 billion knots and links (Hoste 2005, p. 28). The sequence of the number of prime knots of a given crossing number, up to crossing number 16, is 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988,46972,253293,1388705... (sequenceA002863 in theOEIS). While exponential upper and lower bounds for this sequence are known, it has not been proven that this sequence is strictly increasing (Adams 2004).
The first knot tables by Tait, Little, and Kirkman used knot diagrams, although Tait also used a precursor to theDowker notation. Different notations have been invented for knots which allow more efficient tabulation (Hoste 2005).
The early tables attempted to list all knots of at most 10 crossings, and all alternating knots of 11 crossings (Hoste, Thistlethwaite & Weeks 1998). 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 byJohn Horton Conway, who not only developed a new notation but also theAlexander–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 one duplication in the Tait–Little tables; however he missed the duplicates called thePerko pair, which would only be noticed in 1974 byKenneth Perko (Perko 1974). This famous error would propagate when Dale Rolfsen added a knot table in his influential text, based on Conway's work. Conway's 1970 paper on knot theory also contains a typographical duplication on its non-alternating 11-crossing knots page and omits 4 examples — 2 previously listed in D. Lombardero's 1968 Princeton senior thesis and 2 more subsequently discovered byAlain Caudron. [see Perko (1982), Primality of certain knots, Topology Proceedings] Less famous is the duplicate in his 10 crossing link table: 2.-2.-20.20 is the mirror of 8*-20:-20. [See Perko (2016), Historical highlights of non-cyclic knot theory, J. Knot Theory Ramifications].
In the late 1990s Hoste, Thistlethwaite, and Weeks tabulated all the knots through 16 crossings (Hoste, Thistlethwaite & Weeks 1998). In 2003 Rankin, Flint, and Schermann, tabulated thealternating knots through 22 crossings (Hoste 2005). In 2020 Burton tabulated allprime knots with up to 19 crossings (Burton 2020).
This is the most traditional notation, due to the 1927 paper ofJames W. Alexander andGarland B. Briggs and later extended byDale Rolfsen in his knot table (see image above andList of prime knots). 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 (though in each number of crossings thetwist knot comes after thetorus knot). Links are written by the crossing number with a superscript to denote the number of components and a subscript to denote its order within the links with the same number of components and crossings. Thus the trefoil knot is notated 31 and the Hopf link is 22
1. Alexander–Briggs names in the range 10162 to 10166 are ambiguous, due to the discovery of thePerko pair inCharles Newton Little's original and subsequent knot tables, and differences in approach to correcting this error in knot tables and other publications created after this point.[10]
TheDowker–Thistlethwaite notation, also called the Dowker 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 this 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–Thistlethwaite 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–Thistlethwaite notation.
TheConway notation for knots and links, named afterJohn Horton Conway, is based on the theory oftangles (Conway 1970). The advantage of this notation is that it reflects some properties of the knot or link.
The notation describes how to construct a particular link diagram of the link. Start with abasic polyhedron, a 4-valent connected planar graph with nodigon 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 polyhedra. For example, 10** denotes the second 10-vertex polyhedron on Conway's list.
Each vertex then has analgebraic 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.
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 arational tangle. One inserts this tangle at the vertex of the basic polyhedron 1*.
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 the ones are omitted 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.
Gauss code, similar to the Dowker–Thistlethwaite notation, represents a knot with a sequence of integers. However, rather than every crossing being represented by two different numbers, crossings are labeled with only one number. When the crossing is an overcrossing, a positive number is listed. At an undercrossing, a negative number. For example, the trefoil knot in Gauss code can be given as: 1,−2,3,−1,2,−3
Gauss code is limited in its ability to identify knots. This problem is partially addressed with by theextended Gauss code.
There are a number of introductions to knot theory. A classical introduction for graduate students or advanced undergraduates is (Rolfsen 1976). Other good texts from the references are (Adams 2004) 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. (Cromwell 2004) is suitable for undergraduates who know point-set topology; knowledge of algebraic topology is not required.
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