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67-470: A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One tool used to answer such questions is a knot polynomial, which is computed from a diagram of the knot and can be shown to be an invariant of the knot , i.e. diagrams representing the same knot have the same polynomial . The converse may not be true. The HOMFLY polynomial is one such invariant and it generalizes two polynomials previously discovered,

134-429: A knot invariant , a "quantity" which is the same when computed from different descriptions of a knot. Important invariants include knot polynomials , knot groups , and hyperbolic invariants. The original motivation for the founders of knot theory was to create a table of knots and links , which are knots of several components entangled with each other. More than six billion knots and links have been tabulated since

201-424: A natural number . Both the m = n + 2 {\displaystyle m=n+2} and the m > n + 2 {\displaystyle m>n+2} cases are well studied, and so is the n > 1 {\displaystyle n>1} case. Two knots can be added by cutting both knots and joining the pairs of ends. The operation is called the knot sum , or sometimes

268-433: A "nearly" injective and continuous function K : [ 0 , 1 ] → R 3 {\displaystyle K\colon [0,1]\to \mathbb {R} ^{3}} , with the only "non-injectivity" being K ( 0 ) = K ( 1 ) {\displaystyle K(0)=K(1)} . Topologists consider knots and other entanglements such as links and braids to be equivalent if

335-528: A bit of sneakiness: which implies that C (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 are mirror images of each other (take

402-413: 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 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

469-466: A function H {\displaystyle H} is known as an ambient 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 R 3 {\displaystyle \mathbb {R} ^{3}} to itself

536-404: A knot theorist, an unknot is any embedded topological circle in the 3-sphere that is ambient isotopic (that is, deformable) to a geometrically round circle , the standard unknot . The unknot is the only knot that is the boundary of an embedded disk , which gives the characterization that only unknots have Seifert genus 0. Similarly, the unknot is the identity element with respect to

603-524: 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

670-571: A precursor to the Dowker 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

737-811: A series of breakthroughs transformed the subject. In the late 1970s, William Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem . Many knots were shown to be hyperbolic knots , enabling the use of geometry in defining new, powerful knot invariants . The discovery of the Jones polynomial by Vaughan Jones in 1984 ( Sossinsky 2002 , pp. 71–89), 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 then, utilizing sophisticated tools such as quantum groups and Floer homology . In

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804-411: 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

871-416: Is a continuous family of homeomorphisms { h t : R 3 → R 3   f o r   0 ≤ t ≤ 1 } {\displaystyle \{h_{t}:\mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}\ \mathrm {for} \ 0\leq t\leq 1\}} of space onto itself, such that the last one of them carries the first knot onto

938-401: Is a single S n {\displaystyle \mathbb {S} ^{n}} embedded in R m {\displaystyle \mathbb {R} ^{m}} . An n -link consists of k -copies of S n {\displaystyle \mathbb {S} ^{n}} embedded in R m {\displaystyle \mathbb {R} ^{m}} , where k is

1005-497: Is a smoothly knotted 3-sphere in R 6 {\displaystyle \mathbb {R} ^{6}} ( 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 k -sphere embedded in R n {\displaystyle \mathbb {R} ^{n}} with 2 n − 3 k − 3 > 0 {\displaystyle 2n-3k-3>0}

1072-414: Is an orientation-preserving homeomorphism h : R 3 → R 3 {\displaystyle h\colon \mathbb {R} ^{3}\to \mathbb {R} ^{3}} with h ( K 1 ) = K 2 {\displaystyle h(K_{1})=K_{2}} . What this definition of knot equivalence means is that two knots are equivalent when there

1139-515: 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 the relation is applied. gives the unknot and the Hopf link . Applying the relation to the Hopf link where indicated, gives a link deformable to one with 0 crossings (it is actually the unlink of two components) and an unknot. The unlink takes

1206-485: Is defined using skein relations : where L + , L − , L 0 {\displaystyle L_{+},L_{-},L_{0}} are links formed by crossing and smoothing changes on a local region of a link diagram, as indicated in the figure. The HOMFLY polynomial of a link L that is a split union of two links L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}}

1273-675: Is given by See the page on skein relation for an example of a computation using such relations. This polynomial can be obtained also using other skein relations: The Jones polynomial, V ( t ), and the Alexander polynomial, Δ ( t ) {\displaystyle \Delta (t)\,} can be computed in terms of the HOMFLY polynomial (the version in α {\displaystyle \alpha } and z {\displaystyle z} variables) as follows: Knot theory In topology , knot theory

1340-430: Is knotted if there is no homeomorphism of R 4 {\displaystyle \mathbb {R} ^{4}} onto itself taking the embedded 2-sphere to the standard "round" embedding of the 2-sphere. Suspended knots and spun knots are two typical families of such 2-sphere knots. The mathematical technique called "general position" implies that for a given n -sphere in m -dimensional Euclidean space, if m

1407-425: Is large enough (depending on n ), the sphere should be unknotted. In general, piecewise-linear n -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 ( 4 k − 1 ) {\displaystyle (4k-1)} -spheres in 6 k -dimensional space; e.g., there

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1474-405: Is over and which is under at each crossing. (These diagrams are called knot diagrams when they represent a knot and link diagrams when they represent a link .) Analogously, knotted surfaces in 4-space can be related to immersed surfaces in 3-space. A reduced diagram is a knot diagram in which there are no reducible crossings (also nugatory or removable crossings ), or in which all of

1541-419: 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]. Unknot In the mathematical theory of knots , the unknot , not knot , or trivial knot , is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To

1608-440: 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 t = 1 {\displaystyle t=1} (final) stage of the ambient isotopy must be an orientation-preserving homeomorphism carrying one knot to the other. The basic problem of knot theory,

1675-480: Is the minimal number of segments needed to represent a knot as a linkage, and a stuck unknot is a particular unknotted linkage that cannot be reconfigured into a flat convex polygon. Like crossing number, a linkage might need to be made more complex by subdividing its segments before it can be simplified. The Alexander–Conway polynomial and Jones polynomial of the unknot are trivial: No other knot with 10 or fewer crossings has trivial Alexander polynomial, but

1742-505: Is the study of mathematical knots . While inspired by knots 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 an embedding of a circle in 3-dimensional Euclidean space , E 3 {\displaystyle \mathbb {E} ^{3}} . Two mathematical knots are equivalent if one can be transformed into

1809-478: Is the study of slice knots and ribbon 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 a two-dimensional sphere ( S 2 {\displaystyle \mathbb {S} ^{2}} ) embedded in 4-dimensional Euclidean space ( R 4 {\displaystyle \mathbb {R} ^{4}} ). Such an embedding

1876-456: Is unknotted. The notion of a knot has further generalisations in mathematics, see: Knot (mathematics) , isotopy classification of embeddings . Every knot in the n -sphere S n {\displaystyle \mathbb {S} ^{n}} is the link of a real-algebraic set with isolated singularity in R n + 1 {\displaystyle \mathbb {R} ^{n+1}} ( Akbulut & King 1981 ). An n -knot

1943-534: The Alexander polynomial and the Jones polynomial , both of which can be obtained by appropriate substitutions from HOMFLY. The HOMFLY polynomial is also a quantum invariant . The name HOMFLY combines the initials of its co-discoverers: Jim Hoste , Adrian Ocneanu , Kenneth Millett , Peter J. Freyd , W. B. R. Lickorish , and David N. Yetter. The addition of PT recognizes independent work carried out by Józef H. Przytycki and Paweł Traczyk. The polynomial

2010-536: The Borromean rings have made repeated appearances in different cultures, often representing strength in unity. The Celtic monks who created the Book of Kells lavished entire pages with intricate Celtic knotwork . A mathematical theory of knots was first developed in 1771 by Alexandre-Théophile Vandermonde who explicitly noted the importance of topological features when discussing the properties of knots related to

2077-485: The Tait conjectures . This record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of topology . 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 the knot group and invariants from homology theory such as the Alexander polynomial . This would be the main approach to knot theory until

HOMFLY polynomial - Misplaced Pages Continue

2144-415: The connected sum or composition 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

2211-408: The knot sum operation. Deciding if a particular knot is the unknot was a major driving force behind knot invariants , since it was thought this approach would possibly give an efficient algorithm to recognize the unknot from some presentation such as a knot diagram . Unknot recognition is known to be in both NP and co-NP . It is known that knot Floer homology and Khovanov homology detect

2278-400: The recognition problem , is determining the equivalence of two knots. Algorithms exist to solve this problem, with the first given by Wolfgang 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 the unknot , called

2345-409: The unknotting problem , is of particular interest ( Hoste 2005 ). In February 2021 Marc Lackenby announced a new unknot recognition algorithm that runs in quasi-polynomial time . 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 is one-to-one except at

2412-516: The Alexander polynomial, the Alexander–Conway polynomial , is a polynomial in the variable z with integer coefficients ( Lickorish 1997 ). The Alexander–Conway polynomial is actually defined in terms of links , 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

2479-864: 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 other three-dimensional spaces and objects other than circles can be used; see knot (mathematics) . For example, a higher-dimensional knot is an n -dimensional sphere embedded in ( n +2)-dimensional Euclidean space. Archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as recording information and tying 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 (see Chinese knotting ). The endless knot appears in Tibetan Buddhism , while

2546-490: The chosen crossing's configuration. Then the Alexander–Conway polynomial, C ( z ) {\displaystyle C(z)} , is recursively defined according to the rules: The second rule is what is often referred to as a skein 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

2613-427: The computation above with the mirror image. The Jones polynomial can in fact distinguish between the left- and right-handed trefoil knots ( Lickorish 1997 ). William Thurston proved many knots are hyperbolic knots , meaning that the knot complement (i.e., the set of points of 3-space not on the knot) admits a geometric structure, in particular that of hyperbolic geometry . The hyperbolic structure depends only on

2680-446: The construction of quantum computers, through the model of topological 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 K {\displaystyle K} is a "simple closed curve" (see Curve ) — that is:

2747-466: The diagram's crossing number . While rope is generally not in the form of a closed loop, sometimes there is a canonical way to imagine the ends being joined together. From this point of view, many useful practical knots are actually the unknot, including those that can be tied in a bight . Every tame knot can be represented as a linkage , which is a collection of rigid line segments connected by universal joints at their endpoints. The stick number

HOMFLY polynomial - Misplaced Pages Continue

2814-404: The double points, called crossings , 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 an immersed plane curve with the additional data of which strand

2881-687: The duplicates called the Perko pair , which would only be noticed in 1974 by Kenneth 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 by Alain Caudron . [see Perko (1982), Primality of certain knots, Topology Proceedings] Less famous

2948-428: The geometry of position. Mathematical studies of knots began in the 19th century with Carl Friedrich Gauss , who defined the linking integral ( Silver 2006 ). In the 1860s, Lord Kelvin 's theory that atoms were knots in the aether led to Peter 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

3015-427: The knot can be pushed about smoothly, without intersecting itself, to coincide with another knot. The idea of knot 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 K 1 , K 2 {\displaystyle K_{1},K_{2}} are equivalent if there

3082-417: 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 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

3149-503: The last several decades of the 20th century, scientists became interested in studying physical knots in order to understand knotting phenomena in DNA and other polymers. Knot theory can be used to determine if a molecule is chiral (has a "handedness") or not ( Simon 1986 ). Tangles , strings with both ends fixed in place, have been effectively used in studying the action of topoisomerase on DNA ( Flapan 2000 ). Knot theory may be crucial in

3216-489: 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 one duplication in the Tait–Little tables; however he missed

3283-467: 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 a knot invariant that is a polynomial . Well-known examples include the Jones polynomial , the Alexander polynomial , and the Kauffman polynomial . A variant of

3350-512: The link has a preferred direction indicated by an arrow. For a given crossing of the diagram, let L + , L − , L 0 {\displaystyle L_{+},L_{-},L_{0}} be the oriented link diagrams resulting from changing the diagram as indicated in the figure: The original diagram might be either L + {\displaystyle L_{+}} or L − {\displaystyle L_{-}} , depending on

3417-702: The mapping taking x ∈ R 3 {\displaystyle x\in \mathbb {R} ^{3}} to H ( x , t ) ∈ R 3 {\displaystyle H(x,t)\in \mathbb {R} ^{3}} is a homeomorphism of R 3 {\displaystyle \mathbb {R} ^{3}} onto itself; b) H ( x , 0 ) = x {\displaystyle H(x,0)=x} for all x ∈ R 3 {\displaystyle x\in \mathbb {R} ^{3}} ; and c) H ( K 1 , 1 ) = K 2 {\displaystyle H(K_{1},1)=K_{2}} . Such

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3484-509: 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, 46 972 , 253 293 , 1 388 705 ... (sequence A002863 in the OEIS ). 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

3551-585: 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,

3618-405: 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 as oriented , 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

3685-405: The other via a deformation of R 3 {\displaystyle \mathbb {R} ^{3}} upon itself (known as an ambient 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

3752-415: 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

3819-502: The rectangle. The knot sum of oriented knots is commutative and associative . A knot is prime 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 is composite . There is a prime decomposition for knots, analogous to prime 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

3886-441: The red component. The balls in the picture are views of horoball 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

3953-436: The reducible crossings have been removed. A petal 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". In 1927, working with this diagrammatic form of knots, J. W. Alexander and Garland Baird Briggs , and independently Kurt Reidemeister , demonstrated that two knot diagrams belonging to

4020-459: The same knot can be related by a sequence of three kinds of moves on the diagram, shown below. These operations, now called the Reidemeister moves , are: 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

4087-412: 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 unknown complexity . In practice, knots are often distinguished using

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4154-456: The same value on two different knots, so by itself may be incapable of distinguishing all knots. An elementary invariant is tricolorability . "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

4221-533: The second knot. (In detail: Two knots K 1 {\displaystyle K_{1}} and K 2 {\displaystyle K_{2}} are equivalent if there exists a continuous mapping H : R 3 × [ 0 , 1 ] → R 3 {\displaystyle H:\mathbb {R} ^{3}\times [0,1]\rightarrow \mathbb {R} ^{3}} such that a) for each t ∈ [ 0 , 1 ] {\displaystyle t\in [0,1]}

4288-549: The so-called hard contacts. Traditionally, knots have been catalogued in terms of crossing 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

4355-409: 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

4422-450: 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. Knots can also be constructed using the circuit topology approach. This is done by combining basic units called soft contacts using five operations (Parallel, Series, Cross, Concerted, and Sub). The approach is applicable to open chains as well and can also be extended to include

4489-515: The unknot, but these are not known to be efficiently computable for this purpose. It is not known whether the Jones polynomial or finite type invariants can detect the unknot. It can be difficult to find a way to untangle string even though the fact it started out untangled proves the task is possible. Thistlethwaite and Ochiai provided many examples of diagrams of unknots that have no obvious way to simplify them, requiring one to temporarily increase

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