262:
3378:, 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 DowkerâThistlethwaite 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 DowkerâThistlethwaite notation.
20:
3089:): 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 as
1409:
1595:
3367:
2277:
2372:
1578:
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 (
2289:
212:
5772:
1781:
1564:
2356:
36:
452:
3066:
1547:
1535:
5784:
3154:
3462:, similar to the DowkerâThistlethwaite 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,â2,3,â1,2,â3
443:
3264:). 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 â 2 previously listed in D. Lombardero's 1968 Princeton senior thesis and 2 more subsequently discovered by
2420:
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
3440:
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.
2403:
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
1577:
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
1625:). 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 is
2407:
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
140:
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
1300:
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
3123:). 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 considers
3315:). 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 the
2404:
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.
3134:
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 the so-called hard contacts.
3444:
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.
3419:
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.
862:
3233:). 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 AlexanderâBriggs and Reidemeister in the late 1920s.
1330:
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
3323:). 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 3
3252:). 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 TaitâLittle tables; however he missed the duplicates called the
2424:
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.
2303:
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 by
1432:). 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
2027:. 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.
988:
3561:
Adams, Colin; Crawford, Thomas; DeMeo, Benjamin; Landry, Michael; Lin, Alex Tong; Montee, MurphyKate; Park, Seojung; Venkatesh, Saraswathi; Yhee, Farrah (2015), "Knot projections with a single multi-crossing",
715:
1420:
A useful way to visualise and manipulate knots is to project the knot onto a planeâthink of the knot casting a shadow on the wall. A small change in the direction of projection will ensure that it is
276:
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 with
2265:
2054:
4946:
Menasco and
Thistlethwaite's handbook surveys a mix of topics relevant to current research trends in a manner accessible to advanced undergraduates but of interest to professional researchers.
2142:
2098:
2067:
2044:
2155:
2132:
2111:
2088:
2018:
560:
4790:). 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. (
1111:
4983:
This is an online version of an exhibition developed for the 1989 Royal
Society "PopMath RoadShow". Its aim was to use knots to present methods of mathematics to the general public.
230:
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
59:
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 "
3093:, 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.
1505:, 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 the
1216:
1061:
621:
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
1479:
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".
2816:
1775:
2951:
2922:
2885:
2856:
2777:
2687:
2646:
2533:
2504:
2475:
1328:
1271:
1140:
131:
102:
460:
On the left, the unknot, and a knot equivalent to it. It can be more difficult to determine whether complex knots, such as the one on the right, are equivalent to the unknot.
3427:
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.
764:
1540:
2731:
772:
659:
604:
3023:
1181:
300:'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 the
3177:, 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,
3049:
2613:
1906:
2991:
1839:
1812:
1717:, 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.
1026:
916:
889:
1868:
3430:
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
2312:). But the AlexanderâConway 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
1355:
3437:
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.
1926:
1291:
496:
3304:
1498:
3352:'s original and subsequent knot tables, and differences in approach to correcting this error in knot tables and other publications created after this point.
2270:
Since the
AlexanderâConway polynomial is a knot invariant, this shows that the trefoil is not equivalent to the unknot. So the trefoil really is "knotted".
4826:
2421:
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.
1376:). Nonetheless, these algorithms can be extremely time-consuming, and a major issue in the theory is to understand how hard this problem really is (
4171:
2536:
4799:
3265:
3211:). While exponential upper and lower bounds for this sequence are known, it has not been proven that this sequence is strictly increasing (
2656:). Thus the codimension of a smooth knot can be arbitrarily large when not fixing the dimension of the knotted sphere; however, any smooth
925:
2540:
137:); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.
3208:
2343:. The hyperbolic structure depends only on the knot so any quantity computed from the hyperbolic structure is then a knot invariant (
3165:. Knot tables generally include only prime knots, and only one entry for a knot and its mirror image (even if they are different) (
769:
What this definition of knot equivalence means is that two knots are equivalent when there is a continuous family of homeomorphisms
1724:
one in which every component of the link has a preferred direction indicated by an arrow. For a given crossing of the diagram, let
4778:
There are a number of introductions to knot theory. A classical introduction for graduate students or advanced undergraduates is (
2391:
Geometry lets us visualize what the inside of a knot or link complement looks like by imagining light rays as traveling along the
1660:
and hyperbolic invariants were discovered. These aforementioned invariants are only the tip of the iceberg of modern knot theory.
1648:, which can be computed from the Alexander invariant, a module constructed from the infinite cyclic cover of the knot complement (
670:
3111:
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
2399:. The inhabitant of this link complement is viewing the space from near the red component. The balls in the picture are views of
2030:
The following is an example of a typical computation using a skein relation. It computes the
AlexanderâConway polynomial of the
261:
5149:
4965:
4938:
4909:
4889:
4866:
4842:
4817:
4338:
4312:
4250:
4216:
4152:
4120:
4060:
3869:
3812:
3552:
3169:). The number of nontrivial knots of a given crossing number increases rapidly, making tabulation computationally difficult (
2175:
176:
To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other
5717:
864:
of space onto itself, such that the last one of them carries the first knot onto the second knot. (In detail: Two knots
5636:
4597:
3375:
3361:
614:
to be equivalent if the knot can be pushed about smoothly, without intersecting itself, to coincide with another knot.
470:
line segment, wrapping it around itself arbitrarily, and then fusing its two free ends together to form a closed loop (
2738:
273:
5815:
3503:
1357:(final) stage of the ambient isotopy must be an orientation-preserving homeomorphism carrying one knot to the other.
304:. This record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of
289:
154:, a "quantity" which is the same when computed from different descriptions of a knot. Important invariants include
3708:. Leibniz Int. Proc. Inform. Vol. 164. Schloss DagstuhlâLeibniz-Zentrum fĂŒr Informatik. pp. 25:1â25:17.
3218:
The first knot tables by Tait, Little, and
Kirkman used knot diagrams, although Tait also used a precursor to the
1937:
512:
5183:
3241:
3229:
The early tables attempted to list all knots of at most 10 crossings, and all alternating knots of 11 crossings (
1695:
1436:
with the additional data of which strand is over and which is under at each crossing. (These diagrams are called
422:
1066:
4794:) is suitable for undergraduates who know point-set topology; knowledge of algebraic topology is not required.
2580:), although this is no longer a requirement for smoothly knotted spheres. In fact, there are smoothly knotted
2444:
Since a knot can be considered topologically a 1-dimensional sphere, the next generalization is to consider a
5631:
5626:
5502:
3855:
3609:
3544:
3393:
3387:
3162:
222:
Archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as
5788:
5203:
3963:
1186:
1031:
331:. This would be the main approach to knot theory until a series of breakthroughs transformed the subject.
2276:
5265:
3536:
3268:. Less famous is the duplicate in his 10 crossing link table: 2.-2.-20.20 is the mirror of 8*-20:-20. .
3073:
Two knots can be added by cutting both knots and joining the pairs of ends. The operation is called the
2339:(i.e., the set of points of 3-space not on the knot) admits a geometric structure, in particular that of
169:, which are knots of several components entangled with each other. More than six billion knots and links
3465:
Gauss code is limited in its ability to identify knots. This problem is partially addressed with by the
2786:
2428:
Four-dimensional space occurs in classical knot theory, however, and an important topic is the study of
2288:
4579:
3861:
3478:
3300:
1727:
1494:
316:
4754:
2927:
2898:
2861:
2832:
2753:
2663:
2622:
2509:
2480:
2451:
1304:
1221:
1116:
107:
104:. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of
78:
5335:
5330:
5271:
5142:
3922:
2563:
2408:
methods of obtaining these invariants make calculating these invariants, in practice, a simple task (
857:{\displaystyle \{h_{t}:\mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}\ \mathrm {for} \ 0\leq t\leq 1\}}
382:. A plethora of knot invariants have been invented since then, utilizing sophisticated tools such as
343:
293:
5014:
Silliman, Robert H. (December 1963), "William
Thomson: Smoke Rings and Nineteenth-Century Atomism",
720:
4588:
3157:
A table of prime knots up to seven crossings. The knots are labeled with
AlexanderâBriggs notation
2692:
2166:(unlink of two components) = 0, since the first two polynomials are of the unknot and thus equal.
5463:
4761:. Accessed February 2016. Richard Elwes points out a common mistake in describing the Perko pair.
3271:
In the late 1990s Hoste, Thistlethwaite, and Weeks tabulated all the knots through 16 crossings (
624:
246:
have made repeated appearances in different cultures, often representing strength in unity. The
141:
fundamental problem in knot theory is determining when two descriptions represent the same knot.
3173:, p. 20). Tabulation efforts have succeeded in enumerating over 6 billion knots and links (
5677:
5646:
4980:
4583:
3821:
Doll, Helmut; Hoste, Jim (1991), "A tabulation of oriented links. With microfiche supplement",
1777:
be the oriented link diagrams resulting from changing the diagram as indicated in the figure:
662:
565:
324:
206:
145:
19:
4953:
2996:
1145:
421:). Knot theory may be crucial in the construction of quantum computers, through the model of
413:, strings with both ends fixed in place, have been effectively used in studying the action of
5507:
4926:
4514:
Marc Lackenby announces a new unknot recognition algorithm that runs in quasi-polynomial time
4075:
4010:
3349:
2535:
onto itself taking the embedded 2-sphere to the standard "round" embedding of the 2-sphere.
402:
375:
3411:
The notation describes how to construct a particular link diagram of the link. Start with a
3028:
2583:
2299:
Actually, there are two trefoil knots, called the right and left-handed trefoils, which are
1876:
5810:
5776:
5547:
5135:
5107:
4710:
4669:
4628:
4518:
4420:
4226:
4208:
3982:
3951:
3830:
3734:
3593:
3408:). The advantage of this notation is that it reflects some properties of the knot or link.
3401:
2964:
1817:
1790:
1687:
1645:
1433:
993:
894:
867:
410:
394:
379:
328:
277:
165:
The original motivation for the founders of knot theory was to create a table of knots and
4574:
Levine, J.; Orr, K (2000), "A survey of applications of surgery to knot and link theory",
3222:. Different notations have been invented for knots which allow more efficient tabulation (
1844:
1841:, depending on the chosen crossing's configuration. Then the AlexanderâConway polynomial,
393:
In the last several decades of the 20th century, scientists became interested in studying
8:
5584:
5567:
5111:
4998:
4015:
3725:
3466:
3312:
3144:
2340:
1691:
1657:
1334:
507:
339:
247:
5049:
4731:
4714:
4698:
4673:
4632:
4424:
3986:
3834:
3738:
374:, and others, revealed deep connections between knot theory and mathematical methods in
5605:
5552:
5166:
5162:
5039:
5031:
4659:
4618:
4467:
4436:
4360:
4344:
4190:
4158:
4130:
4092:
4052:
4032:
3998:
3972:
3906:
3804:
3792:
3781:
3701:
3678:
3662:
3631:
3597:
3571:
3397:
3237:
2780:
2734:
1911:
1714:
1449:
1441:
1421:
1413:
1385:
1276:
607:
481:
297:
266:
182:
166:
52:
4603:â An introductory article to high dimensional knots and links for the advanced readers
4381:
3994:
3878:
3843:
3746:
3626:
5702:
5651:
5601:
5557:
5517:
5512:
5430:
5043:
4961:
4934:
4905:
4885:
4875:
4862:
4852:
4838:
4813:
4736:
4593:
4356:
4348:
4334:
4308:
4295:
4246:
4212:
4162:
4148:
4116:
4056:
3939:
3865:
3808:
3795:(1970), "An enumeration of knots and links, and some of their algebraic properties",
3785:
3750:
3692:
3666:
3601:
3548:
2316:
polynomial can in fact distinguish between the left- and right-handed trefoil knots (
1637:
1502:
1488:
4440:
4036:
2733:
is unknotted. The notion of a knot has further generalisations in mathematics, see:
5737:
5562:
5458:
5193:
5023:
4803:
4726:
4718:
4677:
4459:
4428:
4389:
4326:
4290:
4269:
4238:
4180:
4140:
4108:
4084:
4048:
4024:
4002:
3990:
3931:
3898:
3838:
3800:
3773:
3742:
3709:
3687:
3654:
3621:
3581:
3508:
3493:
3488:
3424:
3276:
3131:
2328:
1683:
1476:
1453:
371:
355:
335:
301:
281:
239:
231:
155:
5121:
2053:
5697:
5661:
5596:
5542:
5497:
5490:
5380:
5292:
5175:
5120:â software for low-dimensional topology with native support for knots and links.
5095:
4899:
4879:
4856:
4807:
4408:
4222:
4202:
4134:
3947:
3589:
3518:
3431:
3219:
3148:
2396:
2381:
2362:
2336:
2332:
2141:
2097:
2066:
2043:
1669:
1641:
1626:
1397:
1294:
347:
243:
170:
134:
72:
5127:
4112:
3714:
2395:
of the geometry. An example is provided by the picture of the complement of the
2154:
2131:
2110:
2087:
5757:
5656:
5618:
5537:
5450:
5325:
5317:
5277:
5059:
4722:
4492:
3917:
3673:
3642:
3483:
2958:
2384:
complement from the perspective of an inhabitant living near the red component.
2024:
1675:
1608:
1601:
1594:
1408:
1369:
387:
351:
150:
5117:
4242:
4144:
3585:
2546:
The mathematical technique called "general position" implies that for a given
5804:
5692:
5480:
5473:
5468:
4404:
4070:
3943:
3541:
The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots
3498:
3257:
2308:, before the invention of knot polynomials, using group theoretical methods (
1393:
665:
414:
383:
367:
359:
251:
216:
4043:
Hoste, Jim (2005). "The Enumeration and Classification of Knots and Links".
1656:). In the late 20th century, invariants such as "quantum" knot polynomials,
5707:
5687:
5591:
5574:
5370:
5307:
5083:
4740:
4636:â An introductory article to high dimensional knots and links for beginners
4447:
3851:
3754:
3308:
3116:
2031:
611:
235:
28:
24:
4330:
3366:
401:
and other polymers. Knot theory can be used to determine if a molecule is
144:
A complete algorithmic solution to this problem exists, which has unknown
5722:
5485:
5390:
5259:
5239:
5229:
5221:
5213:
4994:
3441:
For an algebraic knot such as in the first example, 1* is often omitted.
3434:. One inserts this tangle at the vertex of the basic polyhedron 1*.
3101:
3097:
2436:. A notorious open problem asks whether every slice knot is also ribbon.
2433:
2300:
255:
4682:
4647:
4512:
3857:
When topology meets chemistry: A topological look at molecular chirality
2371:
2122:
of two components) and an unknot. The unlink takes a bit of sneakiness:
1613:
A knot invariant is a "quantity" that is the same for equivalent knots (
211:
5742:
5727:
5682:
5579:
5532:
5527:
5522:
5352:
5249:
5055:
4471:
4432:
4273:
4194:
4096:
4028:
3935:
3910:
3777:
3658:
3635:
3513:
3459:
3454:
3345:
3320:
3316:
3284:
3253:
3105:
2429:
2365:
are a link with the property that removing one ring unlinks the others.
1679:
1633:
320:
227:
177:
159:
5035:
4281:
Simon, Jonathan (1986), "Topological chirality of certain molecules",
1563:
1546:
1534:
5747:
5415:
5072:
4830:
4812:, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter,
3977:
3958:
2075:
1365:
983:{\displaystyle H:\mathbb {R} ^{3}\times \rightarrow \mathbb {R} ^{3}}
503:
467:
64:
4463:
4185:
4088:
3920:(1962), "Ăber das Homöomorphieproblem der 3-Mannigfaltigkeiten. I",
3902:
2118:
gives a link deformable to one with 0 crossings (it is actually the
5732:
5342:
5027:
4664:
4555:
3761:
3236:
The first major verification of this work was done in the 1960s by
3060:
2745:
2619:-dimensional space; e.g., there is a smoothly knotted 3-sphere in
2566:
2445:
2400:
2392:
2305:
1475:), or in which all of the reducible crossings have been removed. A
1428:, where the "shadow" of the knot crosses itself once transversely (
312:
305:
187:
44:
4623:
3706:
36th International Symposium on Computational Geometry (SoCG 2020)
3576:
4699:"A tile model of circuit topology for self-entangled biopolymers"
4139:, Graduate Texts in Mathematics, vol. 175, Springer-Verlag,
2377:
2355:
1780:
1703:
451:
39:
A knot diagram of the trefoil knot, the simplest non-trivial knot
35:
3299:
This is the most traditional notation, due to the 1927 paper of
3065:
2506:). Such an embedding is knotted if there is no homeomorphism of
1713:
The AlexanderâConway polynomial is actually defined in terms of
5752:
5400:
5360:
5101:
4648:"Circuit Topology for Bottom-Up Engineering of Molecular Knots"
2119:
1929:
1381:
68:
60:
3723:
Collins, Graham (April 2006), "Computing with Quantum Knots",
1452:.) Analogously, knotted surfaces in 4-space can be related to
272:
A mathematical theory of knots was first developed in 1771 by
5641:
4697:
Flapan, Erica; Mashaghi, Alireza; Wong, Helen (1 June 2023).
3416:
3153:
2034:. The yellow patches indicate where the relation is applied.
1539:
606:. Topologists consider knots and other entanglements such as
499:
223:
3961:(1998), "Algorithms for recognizing knots and 3-manifolds",
3370:
A knot diagram with crossings labelled for a Dowker sequence
1598:
A 3D print depicting the complement of the figure eight knot
710:{\displaystyle h\colon \mathbb {R} ^{3}\to \mathbb {R} ^{3}}
5712:
3203:
56:
4576:
Surveys on Surgery Theory: Papers Dedicated to C.T.C. Wall
2078:. Applying the relation to the Hopf link where indicated,
1396:
announced a new unknot recognition algorithm that runs in
442:
4169:
Perko, Kenneth (1974), "On the classification of knots",
3115:. There is a prime decomposition for knots, analogous to
398:
350:, enabling the use of geometry in defining new, powerful
319:, and othersâstudied knots from the point of view of the
173:
since the beginnings of knot theory in the 19th century.
4646:
Golovnev, Anatoly; Mashaghi, Alireza (7 December 2021).
4561:
4235:
Die eindeutige Zerlegbarkeit eines Knotens in Primknoten
4008:
3560:
3275:). In 2003 Rankin, Flint, and Schermann, tabulated the
3272:
3230:
3166:
311:
These topologists in the early part of the 20th centuryâ
5092:â software to investigate geometric properties of knots
5089:
3085:
of two knots. This can be formally defined as follows (
2439:
1493:
In 1927, working with this diagrammatic form of knots,
5052:
of a modern recreation of Tait's smoke ring experiment
4103:
Kontsevich, M. (1993). "Vassiliev's knot invariants".
4013:; Weeks, Jeffrey (1998), "The First 1,701,935 Knots",
3161:
Traditionally, knots have been catalogued in terms of
2260:{\displaystyle C(\mathrm {trefoil} )=1+z(0+z)=1+z^{2}}
4924:
4543:
3240:, who not only developed a new notation but also the
3031:
2999:
2967:
2930:
2901:
2864:
2835:
2789:
2756:
2695:
2666:
2625:
2586:
2512:
2483:
2454:
2178:
1940:
1914:
1879:
1847:
1820:
1793:
1730:
1368:
exist to solve this problem, with the first given by
1337:
1307:
1279:
1224:
1189:
1148:
1119:
1069:
1034:
996:
928:
897:
870:
775:
723:
673:
627:
568:
515:
484:
366:, pp. 71â89), and subsequent contributions from
148:. In practice, knots are often distinguished using a
110:
81:
4531:
4260:
Silver, Daniel (2006). "Knot Theory's Odd Origins".
4073:(1965), "A classification of differentiable knots",
3645:; King, Henry C. (1981), "All knots are algebraic",
3612:(1991), "Hyperbolic invariants of knots and links",
3607:
2409:
5086:â detailed info on individual knots in knot tables
5078:Table of Knot Invariants and Knot Theory Resources
4696:
3504:Contact geometry#Legendrian submanifolds and knots
3043:
3017:
2985:
2945:
2916:
2879:
2850:
2810:
2771:
2725:
2681:
2640:
2607:
2527:
2498:
2469:
2259:
2023:The second rule is what is often referred to as a
2012:
1920:
1900:
1862:
1833:
1806:
1769:
1520:Move a strand completely over or under a crossing.
1349:
1322:
1285:
1265:
1210:
1175:
1134:
1105:
1055:
1020:
982:
910:
883:
856:
758:
709:
653:
598:
554:
490:
125:
96:
5157:
3614:Transactions of the American Mathematical Society
3355:
2543:are two typical families of such 2-sphere knots.
1870:, is recursively defined according to the rules:
5802:
4645:
4172:Proceedings of the American Mathematical Society
1364:, is determining the equivalence of two knots.
180:and objects other than circles can be used; see
4409:"Quantum field theory and the Jones polynomial"
2562:), the sphere should be unknotted. In general,
215:Intricate Celtic knotwork in the 1200-year-old
186:. For example, a higher-dimensional knot is an
5104:â online database and image generator of knots
4798:
4578:, Annals of mathematics studies, vol. 1,
5143:
5003:Proceedings of the Royal Society of Edinburgh
4782:). Other good texts from the references are (
2477:) embedded in 4-dimensional Euclidean space (
1694:. A variant of the Alexander polynomial, the
397:in order to understand knotting phenomena in
3676:(1995), "On the Vassiliev knot invariants",
3564:Journal of Knot Theory and Its Ramifications
3415:, a 4-valent connected planar graph with no
3294:
2013:{\displaystyle C(L_{+})=C(L_{-})+zC(L_{0}).}
851:
776:
555:{\displaystyle K\colon \to \mathbb {R} ^{3}}
5066:
4323:Quantum Invariants of Knots and 3-Manifolds
4207:, Mathematics Lecture Series, vol. 7,
3641:
3344:are ambiguous, due to the discovery of the
2819:
1600:by François Guéritaud, Saul Schleimer, and
63:"). In mathematical language, a knot is an
5150:
5136:
4960:, Simon & Schuster, pp. 203â218,
4825:
4549:
4450:(1963), "Unknotting combinatorial balls",
4379:
4107:. ADVSOV. Vol. 16. pp. 137â150.
4102:
3797:Computational Problems in Abstract Algebra
1106:{\displaystyle H(x,t)\in \mathbb {R} ^{3}}
466:A knot is created by beginning with a one-
294:theory that atoms were knots in the aether
23:Examples of different knots including the
4787:
4773:
4730:
4681:
4663:
4622:
4587:
4573:
4537:
4355:
4302:
4294:
4184:
4129:
3976:
3877:
3842:
3820:
3764:(1914), "Die beiden Kleeblattschlingen",
3713:
3691:
3672:
3625:
3575:
3336:. AlexanderâBriggs names in the range 10
3256:, which would only be noticed in 1974 by
3249:
2933:
2904:
2867:
2838:
2792:
2759:
2669:
2649:
2628:
2515:
2486:
2457:
2317:
1707:
1649:
1618:
1583:
1579:
1310:
1198:
1122:
1093:
1043:
970:
937:
809:
794:
697:
682:
542:
475:
363:
113:
84:
5013:
4897:
4874:
4851:
4791:
4232:
3365:
3152:
3130:Knots can also be constructed using the
3120:
3064:
2323:
1632:"Classical" knot invariants include the
1593:
1517:Move one strand completely over another.
1463:is a knot diagram in which there are no
1407:
562:, with the only "non-injectivity" being
260:
210:
34:
18:
4993:
4779:
4200:
3722:
3311:in his knot table (see image above and
1653:
1622:
1429:
1380:). The special case of recognizing the
426:
5803:
5124:of prime knots with up to 19 crossings
4615:Introduction to high dimensional knots
4491:As first sketched using the theory of
4446:
4403:
4320:
4259:
4069:
3850:
3791:
3699:
3405:
3288:
3273:Hoste, Thistlethwaite & Weeks 1998
3245:
3231:Hoste, Thistlethwaite & Weeks 1998
3167:Hoste, Thistlethwaite & Weeks 1998
3025:cases are well studied, and so is the
2653:
2577:
1514:Twist and untwist in either direction.
1360:The basic problem of knot theory, the
418:
285:
5131:
4951:
4783:
4612:
4496:
4280:
4168:
4042:
3916:
3535:
3280:
3261:
3223:
3212:
3174:
3170:
3086:
2344:
2169:Putting all this together will show:
1787:The original diagram might be either
1614:
1482:
1389:
1211:{\displaystyle x\in \mathbb {R} ^{3}}
1056:{\displaystyle x\in \mathbb {R} ^{3}}
922:if there exists a continuous mapping
471:
406:
254:lavished entire pages with intricate
5783:
5098:â software to create images of knots
4954:"Ch. 8: Unreasonable Effectiveness?"
4500:
3957:
3760:
2739:isotopy classification of embeddings
2440:Knotting spheres of higher dimension
2415:
2309:
1424:except at the double points, called
1377:
1373:
3138:
2576: + 2)-dimensional space (
1720:Consider an oriented link diagram,
1663:
1412:Tenfold Knottiness, plate IX, from
432:
13:
4884:(4th ed.), World Scientific,
4768:
3885: − 1)-spheres in 6
3805:10.1016/B978-0-08-012975-4.50034-5
3608:Adams, Colin; Hildebrand, Martin;
3381:
3096:The knot sum of oriented knots is
2811:{\displaystyle \mathbb {R} ^{n+1}}
2410:Adams, Hildebrand & Weeks 1991
2204:
2201:
2198:
2195:
2192:
2189:
2186:
1779:
1698:, is a polynomial in the variable
1682:. Well-known examples include the
1589:
829:
826:
823:
14:
5827:
4974:
3844:10.1090/S0025-5718-1991-1094946-4
3747:10.1038/scientificamerican0406-56
3627:10.1090/s0002-9947-1991-0994161-2
3396:for knots and links, named after
3127:knots in codimension at least 3.
2554:-dimensional Euclidean space, if
1770:{\displaystyle L_{+},L_{-},L_{0}}
342:into the study of knots with the
197:+2)-dimensional Euclidean space.
5782:
5771:
5770:
5114:function for investigating knots
4499:. For a more recent survey, see
4053:10.1016/B978-044451452-3/50006-X
3283:). In 2020 Burton tabulated all
2946:{\displaystyle \mathbb {R} ^{m}}
2917:{\displaystyle \mathbb {S} ^{n}}
2880:{\displaystyle \mathbb {R} ^{m}}
2851:{\displaystyle \mathbb {S} ^{n}}
2772:{\displaystyle \mathbb {S} ^{n}}
2682:{\displaystyle \mathbb {R} ^{n}}
2641:{\displaystyle \mathbb {R} ^{6}}
2528:{\displaystyle \mathbb {R} ^{4}}
2499:{\displaystyle \mathbb {R} ^{4}}
2470:{\displaystyle \mathbb {S} ^{2}}
2370:
2354:
2287:
2275:
2153:
2140:
2130:
2109:
2096:
2086:
2065:
2052:
2042:
1562:
1545:
1538:
1533:
1403:
1323:{\displaystyle \mathbb {R} ^{3}}
1266:{\displaystyle H(K_{1},1)=K_{2}}
1135:{\displaystyle \mathbb {R} ^{3}}
498:is a "simple closed curve" (see
450:
441:
126:{\displaystyle \mathbb {R} ^{3}}
97:{\displaystyle \mathbb {E} ^{3}}
4747:
4305:Knots, mathematics with a twist
3054:
1524:
423:topological quantum computation
274:Alexandre-Théophile Vandermonde
5637:DowkerâThistlethwaite notation
4904:, Cambridge University Press,
4861:, Princeton University Press,
4690:
4639:
4606:
4567:
4505:
4485:
4136:An Introduction to Knot Theory
3799:, Pergamon, pp. 329â358,
3376:DowkerâThistlethwaite notation
3362:DowkerâThistlethwaite notation
3356:DowkerâThistlethwaite notation
2602:
2587:
2558:is large enough (depending on
2294:The right-handed trefoil knot.
2235:
2223:
2208:
2182:
2004:
1991:
1979:
1966:
1957:
1944:
1889:
1883:
1857:
1851:
1247:
1228:
1164:
1152:
1085:
1073:
1015:
1003:
965:
962:
950:
804:
759:{\displaystyle h(K_{1})=K_{2}}
740:
727:
692:
661:are equivalent if there is an
593:
587:
578:
572:
537:
534:
522:
346:. Many knots were shown to be
1:
4755:The Revenge of the Perko Pair
3995:10.1016/S0960-0779(97)00109-4
3545:American Mathematical Society
3524:
3448:
3388:Conway notation (knot theory)
2783:with isolated singularity in
2282:The left-handed trefoil knot.
1388:, is of particular interest (
478:). Simply, we can say a knot
405:(has a "handedness") or not (
162:, and hyperbolic invariants.
4987:
4478:
4380:Weisstein, Eric W. (2013a).
4321:Turaev, Vladimir G. (2016).
4307:, Harvard University Press,
4296:10.1016/0040-9383(86)90041-8
3964:Chaos, Solitons and Fractals
3702:"The Next 350 Million Knots"
3700:Burton, Benjamin A. (2020).
3693:10.1016/0040-9383(95)93237-2
3400:, is based on the theory of
2726:{\displaystyle 2n-3k-3>0}
7:
4898:Cromwell, Peter R. (2004),
4835:Introduction to Knot Theory
3715:10.4230/LIPIcs.SoCG.2020.25
3472:
3242:AlexanderâConway polynomial
1696:AlexanderâConway polynomial
1416:'s article "On Knots", 1884
654:{\displaystyle K_{1},K_{2}}
16:Study of mathematical knots
10:
5832:
4918:
4723:10.1038/s41598-023-35771-8
4580:Princeton University Press
4517:, Mathematical Institute,
4303:Sossinsky, Alexei (2002),
3862:Cambridge University Press
3529:
3479:List of knot theory topics
3452:
3385:
3359:
3142:
3058:
1667:
1606:
1486:
265:The first knot tabulator,
204:
200:
5766:
5670:
5627:AlexanderâBriggs notation
5614:
5449:
5351:
5316:
5174:
4243:10.1007/978-3-642-45813-2
4145:10.1007/978-1-4612-0691-0
3923:Mathematische Zeitschrift
3586:10.1142/S021821651550011X
3295:AlexanderâBriggs notation
3287:with up to 19 crossings (
2074:gives the unknot and the
1570:
1561:
599:{\displaystyle K(0)=K(1)}
133:upon itself (known as an
5816:Low-dimensional topology
5067:Knot tables and software
4233:Schubert, Horst (1949).
4131:Lickorish, W. B. Raymond
3423:Each vertex then has an
3018:{\displaystyle m>n+2}
1176:{\displaystyle H(x,0)=x}
502:) â that is: a "nearly"
178:three-dimensional spaces
5718:List of knots and links
5266:KinoshitaâTerasaka knot
4981:"Mathematics and Knots"
4958:Is God a Mathematician?
4931:Handbook of Knot Theory
4113:10.1090/advsov/016.2/04
4045:Handbook of Knot Theory
3119:and composite numbers (
2820:Akbulut & King 1981
1674:A knot polynomial is a
354:. The discovery of the
344:hyperbolization theorem
5056:History of knot theory
4927:Thistlethwaite, Morwen
4774:Introductory textbooks
4361:"Reduced Knot Diagram"
4201:Rolfsen, Dale (1976),
4011:Thistlethwaite, Morwen
3371:
3327:and the Hopf link is 2
3307:and later extended by
3279:through 22 crossings (
3158:
3070:
3045:
3044:{\displaystyle n>1}
3019:
2987:
2947:
2918:
2881:
2852:
2812:
2773:
2727:
2683:
2642:
2609:
2608:{\displaystyle (4k-1)}
2529:
2500:
2471:
2446:two-dimensional sphere
2331:proved many knots are
2261:
2014:
1928:is any diagram of the
1922:
1902:
1901:{\displaystyle C(O)=1}
1864:
1835:
1808:
1784:
1771:
1604:
1448:when they represent a
1440:when they represent a
1417:
1351:
1324:
1287:
1267:
1212:
1177:
1136:
1113:is a homeomorphism of
1107:
1057:
1022:
990:such that a) for each
984:
912:
885:
858:
760:
711:
663:orientation-preserving
655:
600:
556:
492:
269:
250:monks who created the
219:
207:History of knot theory
127:
98:
40:
32:
5508:Finite type invariant
5058:(on the home page of
4952:Livio, Mario (2009),
4925:Menasco, William W.;
4452:Annals of Mathematics
4331:10.1515/9783110435221
4211:: Publish or Perish,
4105:I. M. Gelfand Seminar
4076:Annals of Mathematics
3891:Annals of Mathematics
3766:Mathematische Annalen
3369:
3350:Charles Newton Little
3250:Doll & Hoste 1991
3156:
3068:
3046:
3020:
2988:
2986:{\displaystyle m=n+2}
2948:
2919:
2882:
2853:
2813:
2774:
2728:
2684:
2643:
2610:
2530:
2501:
2472:
2324:Hyperbolic invariants
2262:
2015:
1923:
1903:
1865:
1836:
1834:{\displaystyle L_{-}}
1809:
1807:{\displaystyle L_{+}}
1783:
1772:
1597:
1411:
1398:quasi-polynomial time
1352:
1325:
1288:
1268:
1213:
1178:
1137:
1108:
1058:
1023:
1021:{\displaystyle t\in }
985:
913:
911:{\displaystyle K_{2}}
886:
884:{\displaystyle K_{1}}
859:
761:
712:
656:
601:
557:
493:
376:statistical mechanics
264:
224:recording information
214:
128:
99:
38:
22:
4995:Thomson, Sir William
4613:Ogasa, Eiji (2013),
4519:University of Oxford
4382:"Reducible Crossing"
4209:Berkeley, California
4047:. pp. 209â232.
3647:Comment. Math. Helv.
3514:Necktie § Knots
3029:
2997:
2965:
2928:
2899:
2862:
2833:
2787:
2754:
2693:
2664:
2660:-sphere embedded in
2623:
2584:
2572:form knots only in (
2510:
2481:
2452:
2176:
1938:
1912:
1877:
1863:{\displaystyle C(z)}
1845:
1818:
1791:
1728:
1688:Alexander polynomial
1658:Vassiliev invariants
1646:Alexander polynomial
1501:, and independently
1499:Garland Baird Briggs
1434:immersed plane curve
1392:). In February 2021
1335:
1305:
1277:
1222:
1187:
1146:
1117:
1067:
1032:
994:
926:
895:
868:
773:
721:
671:
625:
566:
513:
482:
380:quantum field theory
329:Alexander polynomial
323:and invariants from
278:Carl Friedrich Gauss
108:
79:
55:. While inspired by
5678:Alexander's theorem
5112:Wolfram Mathematica
4827:Crowell, Richard H.
4715:2023NatSR..13.8889F
4683:10.3390/sym13122353
4674:2021Symm...13.2353G
4633:2013arXiv1304.6053O
4425:1989CMaPh.121..351W
4016:Math. Intelligencer
3987:1998CSF.....9..569H
3881:(1962), "Knotted (4
3835:1991MaCom..57..747D
3739:2006SciAm.294d..56C
3726:Scientific American
3467:extended Gauss code
3313:List of prime knots
3145:List of prime knots
3077:, or sometimes the
2341:hyperbolic geometry
2335:, meaning that the
2162:which implies that
1692:Kauffman polynomial
1529:
1473:removable crossings
1465:reducible crossings
1372:in the late 1960s (
1362:recognition problem
1350:{\displaystyle t=1}
1028:the mapping taking
508:continuous function
340:hyperbolic geometry
334:In the late 1970s,
327:theory such as the
191:-dimensional sphere
171:have been tabulated
27:(top left) and the
4876:Kauffman, Louis H.
4853:Kauffman, Louis H.
4759:RichardElwes.co.uk
4703:Scientific Reports
4433:10.1007/BF01217730
4357:Weisstein, Eric W.
4274:10.1511/2006.2.158
4262:American Scientist
4029:10.1007/BF03025227
3936:10.1007/BF01162369
3778:10.1007/BF01563732
3659:10.1007/BF02566217
3570:(3): 1550011, 30,
3398:John Horton Conway
3372:
3301:James W. Alexander
3238:John Horton Conway
3159:
3071:
3041:
3015:
2983:
2943:
2914:
2891:-link consists of
2877:
2848:
2829:-knot is a single
2808:
2781:real-algebraic set
2769:
2744:Every knot in the
2735:Knot (mathematics)
2723:
2679:
2638:
2605:
2525:
2496:
2467:
2380:'s cusp view: the
2257:
2010:
1918:
1898:
1860:
1831:
1804:
1785:
1767:
1605:
1527:Reidemeister moves
1525:
1507:Reidemeister moves
1483:Reidemeister moves
1418:
1414:Peter Guthrie Tait
1386:unknotting problem
1347:
1320:
1283:
1273:. Such a function
1263:
1208:
1173:
1132:
1103:
1053:
1018:
980:
908:
881:
854:
756:
707:
651:
596:
552:
488:
298:Peter Guthrie Tait
288:). In the 1860s,
280:, who defined the
270:
267:Peter Guthrie Tait
220:
183:knot (mathematics)
123:
94:
53:mathematical knots
41:
33:
5798:
5797:
5652:Reidemeister move
5518:Khovanov homology
5513:Hyperbolic volume
4999:"On Vortex Atoms"
4967:978-0-7432-9405-8
4940:978-0-444-51452-3
4911:978-0-521-54831-1
4891:978-981-4383-00-4
4881:Knots and Physics
4868:978-0-691-08435-0
4844:978-0-387-90272-2
4819:978-3-11-008675-1
4804:Zieschang, Heiner
4562:Adams et al. 2015
4454:, Second Series,
4413:Comm. Math. Phys.
4340:978-3-11-043522-1
4314:978-0-674-00944-8
4252:978-3-540-01419-5
4218:978-0-914098-16-4
4154:978-0-387-98254-0
4122:978-0-8218-4117-4
4079:, Second Series,
4062:978-0-444-51452-3
3893:, Second Series,
3871:978-0-521-66254-3
3814:978-0-08-012975-4
3554:978-0-8218-3678-1
3305:Garland B. Briggs
3277:alternating knots
2779:is the link of a
2416:Higher dimensions
1921:{\displaystyle O}
1638:fundamental group
1575:
1574:
1503:Kurt Reidemeister
1489:Reidemeister move
1454:immersed surfaces
1286:{\displaystyle H}
835:
821:
491:{\displaystyle K}
71:in 3-dimensional
5823:
5786:
5785:
5774:
5773:
5738:Tait conjectures
5441:
5440:
5426:
5425:
5411:
5410:
5303:
5302:
5288:
5287:
5272:(â2,3,7) pretzel
5152:
5145:
5138:
5129:
5128:
5046:
5010:
4970:
4943:
4914:
4894:
4871:
4848:
4822:
4762:
4751:
4745:
4744:
4734:
4694:
4688:
4687:
4685:
4667:
4643:
4637:
4635:
4626:
4610:
4604:
4602:
4591:
4571:
4565:
4559:
4553:
4547:
4541:
4535:
4529:
4528:
4527:
4526:
4509:
4503:
4489:
4474:
4443:
4400:
4398:
4396:
4376:
4374:
4372:
4352:
4317:
4299:
4298:
4277:
4256:
4229:
4197:
4188:
4165:
4126:
4099:
4066:
4039:
4005:
3980:
3971:(4â5): 569â581,
3954:
3913:
3879:Haefliger, André
3874:
3847:
3846:
3829:(196): 747â761,
3817:
3788:
3757:
3719:
3717:
3696:
3695:
3669:
3638:
3629:
3604:
3579:
3557:
3509:Knots and graphs
3494:Quantum topology
3489:Circuit topology
3425:algebraic tangle
3413:basic polyhedron
3335:
3334:
3319:comes after the
3206:
3200:
3199:
3196:
3190:
3189:
3183:
3182:
3139:Tabulating knots
3132:circuit topology
3069:Adding two knots
3050:
3048:
3047:
3042:
3024:
3022:
3021:
3016:
2992:
2990:
2989:
2984:
2952:
2950:
2949:
2944:
2942:
2941:
2936:
2923:
2921:
2920:
2915:
2913:
2912:
2907:
2886:
2884:
2883:
2878:
2876:
2875:
2870:
2857:
2855:
2854:
2849:
2847:
2846:
2841:
2817:
2815:
2814:
2809:
2807:
2806:
2795:
2778:
2776:
2775:
2770:
2768:
2767:
2762:
2732:
2730:
2729:
2724:
2688:
2686:
2685:
2680:
2678:
2677:
2672:
2647:
2645:
2644:
2639:
2637:
2636:
2631:
2614:
2612:
2611:
2606:
2564:piecewise-linear
2534:
2532:
2531:
2526:
2524:
2523:
2518:
2505:
2503:
2502:
2497:
2495:
2494:
2489:
2476:
2474:
2473:
2468:
2466:
2465:
2460:
2374:
2358:
2333:hyperbolic knots
2329:William Thurston
2291:
2279:
2266:
2264:
2263:
2258:
2256:
2255:
2207:
2157:
2144:
2134:
2113:
2100:
2090:
2069:
2056:
2046:
2019:
2017:
2016:
2011:
2003:
2002:
1978:
1977:
1956:
1955:
1927:
1925:
1924:
1919:
1907:
1905:
1904:
1899:
1869:
1867:
1866:
1861:
1840:
1838:
1837:
1832:
1830:
1829:
1813:
1811:
1810:
1805:
1803:
1802:
1776:
1774:
1773:
1768:
1766:
1765:
1753:
1752:
1740:
1739:
1684:Jones polynomial
1664:Knot polynomials
1566:
1549:
1542:
1537:
1530:
1477:petal projection
1356:
1354:
1353:
1348:
1329:
1327:
1326:
1321:
1319:
1318:
1313:
1292:
1290:
1289:
1284:
1272:
1270:
1269:
1264:
1262:
1261:
1240:
1239:
1217:
1215:
1214:
1209:
1207:
1206:
1201:
1182:
1180:
1179:
1174:
1142:onto itself; b)
1141:
1139:
1138:
1133:
1131:
1130:
1125:
1112:
1110:
1109:
1104:
1102:
1101:
1096:
1062:
1060:
1059:
1054:
1052:
1051:
1046:
1027:
1025:
1024:
1019:
989:
987:
986:
981:
979:
978:
973:
946:
945:
940:
917:
915:
914:
909:
907:
906:
890:
888:
887:
882:
880:
879:
863:
861:
860:
855:
833:
832:
819:
818:
817:
812:
803:
802:
797:
788:
787:
765:
763:
762:
757:
755:
754:
739:
738:
716:
714:
713:
708:
706:
705:
700:
691:
690:
685:
660:
658:
657:
652:
650:
649:
637:
636:
619:knot equivalence
605:
603:
602:
597:
561:
559:
558:
553:
551:
550:
545:
497:
495:
494:
489:
454:
445:
433:Knot equivalence
372:Maxim Kontsevich
356:Jones polynomial
348:hyperbolic knots
336:William Thurston
302:Tait conjectures
282:linking integral
240:Tibetan Buddhism
232:Chinese knotting
156:knot polynomials
132:
130:
129:
124:
122:
121:
116:
103:
101:
100:
95:
93:
92:
87:
51:is the study of
5831:
5830:
5826:
5825:
5824:
5822:
5821:
5820:
5801:
5800:
5799:
5794:
5762:
5666:
5632:Conway notation
5616:
5610:
5597:Tricolorability
5445:
5439:
5436:
5435:
5434:
5424:
5421:
5420:
5419:
5409:
5406:
5405:
5404:
5396:
5386:
5376:
5366:
5347:
5326:Composite knots
5312:
5301:
5298:
5297:
5296:
5293:Borromean rings
5286:
5283:
5282:
5281:
5255:
5245:
5235:
5225:
5217:
5209:
5199:
5189:
5170:
5156:
5069:
4990:
4977:
4968:
4941:
4929:, eds. (2005),
4921:
4912:
4901:Knots and Links
4892:
4869:
4845:
4820:
4776:
4771:
4769:Further reading
4766:
4765:
4752:
4748:
4695:
4691:
4644:
4640:
4611:
4607:
4600:
4572:
4568:
4560:
4556:
4550:Weisstein 2013a
4548:
4544:
4536:
4532:
4524:
4522:
4511:
4510:
4506:
4493:Haken manifolds
4490:
4486:
4481:
4464:10.2307/1970538
4448:Zeeman, Erik C.
4394:
4392:
4370:
4368:
4341:
4315:
4253:
4219:
4204:Knots and Links
4186:10.2307/2040074
4155:
4123:
4089:10.2307/1970561
4063:
3918:Haken, Wolfgang
3903:10.2307/1970208
3872:
3815:
3793:Conway, John H.
3674:Bar-Natan, Dror
3643:Akbulut, Selman
3555:
3532:
3527:
3519:Lamp cord trick
3475:
3457:
3451:
3432:rational tangle
3394:Conway notation
3390:
3384:
3382:Conway notation
3364:
3358:
3343:
3339:
3333:
3330:
3329:
3328:
3326:
3297:
3220:Dowker notation
3202:
3197:
3194:
3192:
3187:
3185:
3180:
3178:
3163:crossing number
3151:
3149:Knot tabulation
3141:
3063:
3057:
3030:
3027:
3026:
2998:
2995:
2994:
2966:
2963:
2962:
2937:
2932:
2931:
2929:
2926:
2925:
2908:
2903:
2902:
2900:
2897:
2896:
2871:
2866:
2865:
2863:
2860:
2859:
2842:
2837:
2836:
2834:
2831:
2830:
2796:
2791:
2790:
2788:
2785:
2784:
2763:
2758:
2757:
2755:
2752:
2751:
2694:
2691:
2690:
2673:
2668:
2667:
2665:
2662:
2661:
2632:
2627:
2626:
2624:
2621:
2620:
2585:
2582:
2581:
2537:Suspended knots
2519:
2514:
2513:
2511:
2508:
2507:
2490:
2485:
2484:
2482:
2479:
2478:
2461:
2456:
2455:
2453:
2450:
2449:
2442:
2418:
2397:Borromean rings
2389:
2388:
2387:
2386:
2385:
2382:Borromean rings
2375:
2367:
2366:
2363:Borromean rings
2359:
2337:knot complement
2326:
2295:
2292:
2283:
2280:
2251:
2247:
2185:
2177:
2174:
2173:
1998:
1994:
1973:
1969:
1951:
1947:
1939:
1936:
1935:
1913:
1910:
1909:
1878:
1875:
1874:
1846:
1843:
1842:
1825:
1821:
1819:
1816:
1815:
1798:
1794:
1792:
1789:
1788:
1761:
1757:
1748:
1744:
1735:
1731:
1729:
1726:
1725:
1672:
1670:Knot polynomial
1666:
1642:knot complement
1636:, which is the
1627:tricolorability
1611:
1599:
1592:
1590:Knot invariants
1523:
1495:J. W. Alexander
1491:
1485:
1461:reduced diagram
1406:
1336:
1333:
1332:
1314:
1309:
1308:
1306:
1303:
1302:
1295:ambient isotopy
1293:is known as an
1278:
1275:
1274:
1257:
1253:
1235:
1231:
1223:
1220:
1219:
1202:
1197:
1196:
1188:
1185:
1184:
1147:
1144:
1143:
1126:
1121:
1120:
1118:
1115:
1114:
1097:
1092:
1091:
1068:
1065:
1064:
1047:
1042:
1041:
1033:
1030:
1029:
995:
992:
991:
974:
969:
968:
941:
936:
935:
927:
924:
923:
902:
898:
896:
893:
892:
875:
871:
869:
866:
865:
822:
813:
808:
807:
798:
793:
792:
783:
779:
774:
771:
770:
750:
746:
734:
730:
722:
719:
718:
701:
696:
695:
686:
681:
680:
672:
669:
668:
645:
641:
632:
628:
626:
623:
622:
567:
564:
563:
546:
541:
540:
514:
511:
510:
483:
480:
479:
464:
463:
462:
461:
457:
456:
455:
447:
446:
435:
352:knot invariants
317:J. W. Alexander
256:Celtic knotwork
244:Borromean rings
209:
203:
135:ambient isotopy
117:
112:
111:
109:
106:
105:
88:
83:
82:
80:
77:
76:
73:Euclidean space
17:
12:
11:
5:
5829:
5819:
5818:
5813:
5796:
5795:
5793:
5792:
5780:
5767:
5764:
5763:
5761:
5760:
5758:Surgery theory
5755:
5750:
5745:
5740:
5735:
5730:
5725:
5720:
5715:
5710:
5705:
5700:
5695:
5690:
5685:
5680:
5674:
5672:
5668:
5667:
5665:
5664:
5659:
5657:Skein relation
5654:
5649:
5644:
5639:
5634:
5629:
5623:
5621:
5612:
5611:
5609:
5608:
5602:Unknotting no.
5599:
5594:
5589:
5588:
5587:
5577:
5572:
5571:
5570:
5565:
5560:
5555:
5550:
5540:
5535:
5530:
5525:
5520:
5515:
5510:
5505:
5500:
5495:
5494:
5493:
5483:
5478:
5477:
5476:
5466:
5461:
5455:
5453:
5447:
5446:
5444:
5443:
5437:
5428:
5422:
5413:
5407:
5398:
5394:
5388:
5384:
5378:
5374:
5368:
5364:
5357:
5355:
5349:
5348:
5346:
5345:
5340:
5339:
5338:
5333:
5322:
5320:
5314:
5313:
5311:
5310:
5305:
5299:
5290:
5284:
5275:
5269:
5263:
5257:
5253:
5247:
5243:
5237:
5233:
5227:
5223:
5219:
5215:
5211:
5207:
5201:
5197:
5191:
5187:
5180:
5178:
5172:
5171:
5155:
5154:
5147:
5140:
5132:
5126:
5125:
5115:
5105:
5099:
5093:
5087:
5084:The Knot Atlas
5081:
5068:
5065:
5064:
5063:
5060:Andrew Ranicki
5053:
5047:
5028:10.1086/349764
5022:(4): 461â474,
5011:
4989:
4986:
4985:
4984:
4976:
4975:External links
4973:
4972:
4971:
4966:
4949:
4948:
4947:
4939:
4920:
4917:
4916:
4915:
4910:
4895:
4890:
4872:
4867:
4849:
4843:
4823:
4818:
4800:Burde, Gerhard
4788:Lickorish 1997
4775:
4772:
4770:
4767:
4764:
4763:
4746:
4689:
4638:
4605:
4599:978-0691049380
4598:
4589:10.1.1.64.4359
4566:
4554:
4542:
4538:Weisstein 2013
4530:
4504:
4483:
4482:
4480:
4477:
4476:
4475:
4458:(3): 501â526,
4444:
4419:(3): 351â399,
4405:Witten, Edward
4401:
4377:
4353:
4339:
4318:
4313:
4300:
4289:(2): 229â235,
4278:
4257:
4251:
4230:
4217:
4198:
4166:
4153:
4127:
4121:
4100:
4071:Levine, Jerome
4067:
4061:
4040:
4006:
3955:
3914:
3897:(3): 452â466,
3875:
3870:
3848:
3818:
3813:
3789:
3772:(3): 402â413,
3758:
3720:
3697:
3686:(2): 423â472,
3670:
3653:(3): 339â351,
3639:
3610:Weeks, Jeffrey
3605:
3558:
3553:
3531:
3528:
3526:
3523:
3522:
3521:
3516:
3511:
3506:
3501:
3496:
3491:
3486:
3484:Molecular knot
3481:
3474:
3471:
3453:Main article:
3450:
3447:
3386:Main article:
3383:
3380:
3360:Main article:
3357:
3354:
3341:
3337:
3331:
3324:
3296:
3293:
3201:... (sequence
3140:
3137:
3059:Main article:
3056:
3053:
3040:
3037:
3034:
3014:
3011:
3008:
3005:
3002:
2982:
2979:
2976:
2973:
2970:
2959:natural number
2940:
2935:
2911:
2906:
2874:
2869:
2845:
2840:
2805:
2802:
2799:
2794:
2766:
2761:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2676:
2671:
2650:Haefliger 1962
2635:
2630:
2604:
2601:
2598:
2595:
2592:
2589:
2522:
2517:
2493:
2488:
2464:
2459:
2441:
2438:
2417:
2414:
2376:
2369:
2368:
2360:
2353:
2352:
2351:
2350:
2349:
2325:
2322:
2318:Lickorish 1997
2297:
2296:
2293:
2286:
2284:
2281:
2274:
2268:
2267:
2254:
2250:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2184:
2181:
2160:
2159:
2116:
2115:
2072:
2071:
2057:) +
2047:) =
2025:skein relation
2021:
2020:
2009:
2006:
2001:
1997:
1993:
1990:
1987:
1984:
1981:
1976:
1972:
1968:
1965:
1962:
1959:
1954:
1950:
1946:
1943:
1933:
1917:
1897:
1894:
1891:
1888:
1885:
1882:
1859:
1856:
1853:
1850:
1828:
1824:
1801:
1797:
1764:
1760:
1756:
1751:
1747:
1743:
1738:
1734:
1708:Lickorish 1997
1706:coefficients (
1676:knot invariant
1668:Main article:
1665:
1662:
1650:Lickorish 1997
1619:Lickorish 1997
1609:Knot invariant
1607:Main article:
1602:Henry Segerman
1591:
1588:
1584:Lickorish 1997
1580:Sossinsky 2002
1573:
1572:
1568:
1567:
1559:
1558:
1555:
1551:
1550:
1543:
1522:
1521:
1518:
1515:
1511:
1487:Main article:
1484:
1481:
1405:
1402:
1370:Wolfgang Haken
1346:
1343:
1340:
1317:
1312:
1282:
1260:
1256:
1252:
1249:
1246:
1243:
1238:
1234:
1230:
1227:
1205:
1200:
1195:
1192:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1129:
1124:
1100:
1095:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1050:
1045:
1040:
1037:
1017:
1014:
1011:
1008:
1005:
1002:
999:
977:
972:
967:
964:
961:
958:
955:
952:
949:
944:
939:
934:
931:
905:
901:
878:
874:
853:
850:
847:
844:
841:
838:
831:
828:
825:
816:
811:
806:
801:
796:
791:
786:
782:
778:
753:
749:
745:
742:
737:
733:
729:
726:
704:
699:
694:
689:
684:
679:
676:
648:
644:
640:
635:
631:
595:
592:
589:
586:
583:
580:
577:
574:
571:
549:
544:
539:
536:
533:
530:
527:
524:
521:
518:
487:
476:Sossinsky 2002
459:
458:
449:
448:
440:
439:
438:
437:
436:
434:
431:
395:physical knots
388:Floer homology
384:quantum groups
364:Sossinsky 2002
205:Main article:
202:
199:
151:knot invariant
120:
115:
91:
86:
15:
9:
6:
4:
3:
2:
5828:
5817:
5814:
5812:
5809:
5808:
5806:
5791:
5790:
5781:
5779:
5778:
5769:
5768:
5765:
5759:
5756:
5754:
5751:
5749:
5746:
5744:
5741:
5739:
5736:
5734:
5731:
5729:
5726:
5724:
5721:
5719:
5716:
5714:
5711:
5709:
5706:
5704:
5701:
5699:
5696:
5694:
5693:Conway sphere
5691:
5689:
5686:
5684:
5681:
5679:
5676:
5675:
5673:
5669:
5663:
5660:
5658:
5655:
5653:
5650:
5648:
5645:
5643:
5640:
5638:
5635:
5633:
5630:
5628:
5625:
5624:
5622:
5620:
5613:
5607:
5603:
5600:
5598:
5595:
5593:
5590:
5586:
5583:
5582:
5581:
5578:
5576:
5573:
5569:
5566:
5564:
5561:
5559:
5556:
5554:
5551:
5549:
5546:
5545:
5544:
5541:
5539:
5536:
5534:
5531:
5529:
5526:
5524:
5521:
5519:
5516:
5514:
5511:
5509:
5506:
5504:
5501:
5499:
5496:
5492:
5489:
5488:
5487:
5484:
5482:
5479:
5475:
5472:
5471:
5470:
5467:
5465:
5464:Arf invariant
5462:
5460:
5457:
5456:
5454:
5452:
5448:
5432:
5429:
5417:
5414:
5402:
5399:
5392:
5389:
5382:
5379:
5372:
5369:
5362:
5359:
5358:
5356:
5354:
5350:
5344:
5341:
5337:
5334:
5332:
5329:
5328:
5327:
5324:
5323:
5321:
5319:
5315:
5309:
5306:
5294:
5291:
5279:
5276:
5273:
5270:
5267:
5264:
5261:
5258:
5251:
5248:
5241:
5238:
5231:
5228:
5226:
5220:
5218:
5212:
5205:
5202:
5195:
5192:
5185:
5182:
5181:
5179:
5177:
5173:
5168:
5164:
5160:
5153:
5148:
5146:
5141:
5139:
5134:
5133:
5130:
5123:
5119:
5116:
5113:
5109:
5108:KnotData.html
5106:
5103:
5100:
5097:
5094:
5091:
5088:
5085:
5082:
5080:
5079:
5075:
5071:
5070:
5061:
5057:
5054:
5051:
5048:
5045:
5041:
5037:
5033:
5029:
5025:
5021:
5017:
5012:
5008:
5004:
5000:
4996:
4992:
4991:
4982:
4979:
4978:
4969:
4963:
4959:
4955:
4950:
4945:
4944:
4942:
4936:
4932:
4928:
4923:
4922:
4913:
4907:
4903:
4902:
4896:
4893:
4887:
4883:
4882:
4877:
4873:
4870:
4864:
4860:
4859:
4854:
4850:
4846:
4840:
4836:
4832:
4828:
4824:
4821:
4815:
4811:
4810:
4805:
4801:
4797:
4796:
4795:
4793:
4792:Cromwell 2004
4789:
4785:
4781:
4760:
4756:
4750:
4742:
4738:
4733:
4728:
4724:
4720:
4716:
4712:
4708:
4704:
4700:
4693:
4684:
4679:
4675:
4671:
4666:
4661:
4657:
4653:
4649:
4642:
4634:
4630:
4625:
4620:
4616:
4609:
4601:
4595:
4590:
4585:
4581:
4577:
4570:
4563:
4558:
4551:
4546:
4539:
4534:
4520:
4516:
4515:
4508:
4502:
4498:
4494:
4488:
4484:
4473:
4469:
4465:
4461:
4457:
4453:
4449:
4445:
4442:
4438:
4434:
4430:
4426:
4422:
4418:
4414:
4410:
4406:
4402:
4391:
4387:
4383:
4378:
4366:
4362:
4358:
4354:
4350:
4346:
4342:
4336:
4332:
4328:
4324:
4319:
4316:
4310:
4306:
4301:
4297:
4292:
4288:
4284:
4279:
4275:
4271:
4267:
4263:
4258:
4254:
4248:
4244:
4240:
4236:
4231:
4228:
4224:
4220:
4214:
4210:
4206:
4205:
4199:
4196:
4192:
4187:
4182:
4178:
4174:
4173:
4167:
4164:
4160:
4156:
4150:
4146:
4142:
4138:
4137:
4132:
4128:
4124:
4118:
4114:
4110:
4106:
4101:
4098:
4094:
4090:
4086:
4082:
4078:
4077:
4072:
4068:
4064:
4058:
4054:
4050:
4046:
4041:
4038:
4034:
4030:
4026:
4022:
4018:
4017:
4012:
4007:
4004:
4000:
3996:
3992:
3988:
3984:
3979:
3974:
3970:
3966:
3965:
3960:
3956:
3953:
3949:
3945:
3941:
3937:
3933:
3929:
3925:
3924:
3919:
3915:
3912:
3908:
3904:
3900:
3896:
3892:
3888:
3884:
3880:
3876:
3873:
3867:
3863:
3859:
3858:
3853:
3852:Flapan, Erica
3849:
3845:
3840:
3836:
3832:
3828:
3824:
3819:
3816:
3810:
3806:
3802:
3798:
3794:
3790:
3787:
3783:
3779:
3775:
3771:
3767:
3763:
3759:
3756:
3752:
3748:
3744:
3740:
3736:
3732:
3728:
3727:
3721:
3716:
3711:
3707:
3703:
3698:
3694:
3689:
3685:
3681:
3680:
3675:
3671:
3668:
3664:
3660:
3656:
3652:
3648:
3644:
3640:
3637:
3633:
3628:
3623:
3619:
3615:
3611:
3606:
3603:
3599:
3595:
3591:
3587:
3583:
3578:
3573:
3569:
3565:
3559:
3556:
3550:
3546:
3542:
3538:
3534:
3533:
3520:
3517:
3515:
3512:
3510:
3507:
3505:
3502:
3500:
3499:Ribbon theory
3497:
3495:
3492:
3490:
3487:
3485:
3482:
3480:
3477:
3476:
3470:
3468:
3463:
3461:
3456:
3446:
3442:
3438:
3435:
3433:
3428:
3426:
3421:
3418:
3414:
3409:
3407:
3403:
3399:
3395:
3389:
3379:
3377:
3368:
3363:
3353:
3351:
3347:
3322:
3318:
3314:
3310:
3306:
3302:
3292:
3290:
3286:
3282:
3278:
3274:
3269:
3267:
3266:Alain Caudron
3263:
3259:
3258:Kenneth Perko
3255:
3251:
3247:
3243:
3239:
3234:
3232:
3227:
3225:
3221:
3216:
3214:
3210:
3205:
3176:
3172:
3168:
3164:
3155:
3150:
3146:
3136:
3133:
3128:
3126:
3122:
3121:Schubert 1949
3118:
3114:
3110:
3109:
3103:
3099:
3094:
3092:
3088:
3084:
3080:
3079:connected sum
3076:
3067:
3062:
3052:
3038:
3035:
3032:
3012:
3009:
3006:
3003:
3000:
2980:
2977:
2974:
2971:
2968:
2960:
2956:
2938:
2909:
2894:
2890:
2872:
2858:embedded in
2843:
2828:
2823:
2821:
2803:
2800:
2797:
2782:
2764:
2750:
2748:
2742:
2740:
2736:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2674:
2659:
2655:
2651:
2633:
2618:
2615:-spheres in 6
2599:
2596:
2593:
2590:
2579:
2575:
2571:
2569:
2565:
2561:
2557:
2553:
2549:
2544:
2542:
2538:
2520:
2491:
2462:
2447:
2437:
2435:
2431:
2426:
2422:
2413:
2411:
2405:
2402:
2398:
2394:
2383:
2379:
2373:
2364:
2357:
2348:
2346:
2342:
2338:
2334:
2330:
2321:
2319:
2315:
2311:
2307:
2302:
2301:mirror images
2290:
2285:
2278:
2273:
2272:
2271:
2252:
2248:
2244:
2241:
2238:
2232:
2229:
2226:
2220:
2217:
2214:
2211:
2179:
2172:
2171:
2170:
2167:
2165:
2156:
2151:
2148:
2143:
2138:
2133:
2128:
2125:
2124:
2123:
2121:
2112:
2107:
2104:
2099:
2094:
2089:
2084:
2081:
2080:
2079:
2077:
2068:
2063:
2060:
2055:
2050:
2045:
2040:
2037:
2036:
2035:
2033:
2028:
2026:
2007:
1999:
1995:
1988:
1985:
1982:
1974:
1970:
1963:
1960:
1952:
1948:
1941:
1934:
1931:
1915:
1895:
1892:
1886:
1880:
1873:
1872:
1871:
1854:
1848:
1826:
1822:
1799:
1795:
1782:
1778:
1762:
1758:
1754:
1749:
1745:
1741:
1736:
1732:
1723:
1718:
1716:
1711:
1709:
1705:
1701:
1697:
1693:
1689:
1685:
1681:
1677:
1671:
1661:
1659:
1655:
1651:
1647:
1643:
1639:
1635:
1630:
1628:
1624:
1620:
1616:
1610:
1603:
1596:
1587:
1585:
1581:
1569:
1565:
1560:
1556:
1553:
1552:
1548:
1544:
1541:
1536:
1532:
1531:
1528:
1519:
1516:
1513:
1512:
1510:
1508:
1504:
1500:
1496:
1490:
1480:
1478:
1474:
1470:
1466:
1462:
1457:
1455:
1451:
1447:
1446:link diagrams
1443:
1439:
1438:knot diagrams
1435:
1431:
1427:
1423:
1415:
1410:
1404:Knot diagrams
1401:
1399:
1395:
1394:Marc Lackenby
1391:
1387:
1384:, called the
1383:
1379:
1375:
1371:
1367:
1363:
1358:
1344:
1341:
1338:
1315:
1298:
1296:
1280:
1258:
1254:
1250:
1244:
1241:
1236:
1232:
1225:
1203:
1193:
1190:
1170:
1167:
1161:
1158:
1155:
1149:
1127:
1098:
1088:
1082:
1079:
1076:
1070:
1048:
1038:
1035:
1012:
1009:
1006:
1000:
997:
975:
959:
956:
953:
947:
942:
932:
929:
921:
903:
899:
876:
872:
848:
845:
842:
839:
836:
814:
799:
789:
784:
780:
767:
751:
747:
743:
735:
731:
724:
702:
687:
677:
674:
667:
666:homeomorphism
664:
646:
642:
638:
633:
629:
620:
615:
613:
609:
590:
584:
581:
575:
569:
547:
531:
528:
525:
519:
516:
509:
505:
501:
485:
477:
473:
469:
453:
444:
430:
428:
424:
420:
416:
415:topoisomerase
412:
408:
404:
400:
396:
391:
389:
385:
381:
377:
373:
369:
368:Edward Witten
365:
361:
360:Vaughan Jones
357:
353:
349:
345:
341:
337:
332:
330:
326:
322:
318:
314:
309:
307:
303:
299:
295:
291:
287:
283:
279:
275:
268:
263:
259:
257:
253:
252:Book of Kells
249:
245:
241:
237:
233:
229:
225:
218:
217:Book of Kells
213:
208:
198:
196:
193:embedded in (
192:
190:
185:
184:
179:
174:
172:
168:
163:
161:
157:
153:
152:
147:
142:
138:
136:
118:
89:
74:
70:
66:
62:
58:
54:
50:
46:
37:
30:
26:
21:
5787:
5775:
5703:Double torus
5688:Braid theory
5503:Crossing no.
5498:Crosscap no.
5184:Figure-eight
5158:
5077:
5073:
5019:
5015:
5006:
5002:
4957:
4933:, Elsevier,
4930:
4900:
4880:
4857:
4837:. Springer.
4834:
4808:
4780:Rolfsen 1976
4777:
4758:
4749:
4706:
4702:
4692:
4658:(12): 2353.
4655:
4651:
4641:
4614:
4608:
4575:
4569:
4557:
4545:
4533:
4523:, retrieved
4521:, 2021-02-03
4513:
4507:
4497:Haken (1962)
4487:
4455:
4451:
4416:
4412:
4393:. Retrieved
4385:
4369:. Retrieved
4364:
4322:
4304:
4286:
4282:
4265:
4261:
4234:
4203:
4179:(2): 262â6,
4176:
4170:
4135:
4104:
4083:(1): 15â50,
4080:
4074:
4044:
4023:(4): 33â48,
4020:
4014:
4009:Hoste, Jim;
3978:math/9712269
3968:
3962:
3927:
3921:
3894:
3890:
3886:
3882:
3856:
3826:
3822:
3796:
3769:
3765:
3733:(4): 56â63,
3730:
3724:
3705:
3683:
3677:
3650:
3646:
3617:
3613:
3567:
3563:
3540:
3537:Adams, Colin
3464:
3458:
3443:
3439:
3436:
3429:
3422:
3412:
3410:
3391:
3373:
3309:Dale Rolfsen
3298:
3270:
3235:
3228:
3217:
3160:
3129:
3124:
3112:
3107:
3095:
3090:
3082:
3078:
3074:
3072:
3055:Adding knots
2954:
2924:embedded in
2892:
2888:
2826:
2824:
2746:
2743:
2657:
2616:
2573:
2567:
2559:
2555:
2551:
2547:
2545:
2443:
2434:ribbon knots
2427:
2423:
2419:
2406:
2390:
2327:
2313:
2298:
2269:
2168:
2163:
2161:
2149:
2146:
2136:
2126:
2117:
2105:
2102:
2092:
2082:
2073:
2061:
2058:
2048:
2038:
2032:trefoil knot
2029:
2022:
1786:
1721:
1719:
1712:
1699:
1673:
1654:Rolfsen 1976
1631:
1623:Rolfsen 1976
1612:
1576:
1526:
1506:
1492:
1472:
1468:
1464:
1460:
1458:
1456:in 3-space.
1445:
1437:
1430:Rolfsen 1976
1425:
1419:
1361:
1359:
1299:
919:
768:
618:
617:The idea of
616:
465:
427:Collins 2006
392:
333:
310:
271:
242:, while the
236:endless knot
221:
194:
188:
181:
175:
164:
149:
143:
139:
48:
42:
29:trefoil knot
25:trivial knot
5811:Knot theory
5538:Linking no.
5459:Alternating
5260:Conway knot
5240:Carrick mat
5194:Three-twist
5159:Knot theory
4709:(1): 8889.
4501:Hass (1998)
3860:, Outlook,
3823:Math. Comp.
3620:(1): 1â56,
3406:Conway 1970
3289:Burton 2020
3285:prime knots
3246:Conway 1970
3102:associative
3098:commutative
3083:composition
2961:. Both the
2895:-copies of
2654:Levine 1965
2578:Zeeman 1963
2550:-sphere in
2430:slice knots
468:dimensional
419:Flapan 2000
338:introduced
290:Lord Kelvin
286:Silver 2006
238:appears in
160:knot groups
49:knot theory
5805:Categories
5698:Complement
5662:Tabulation
5619:operations
5543:Polynomial
5533:Link group
5528:Knot group
5491:Invertible
5469:Bridge no.
5451:Invariants
5381:Cinquefoil
5250:Perko pair
5176:Hyperbolic
4831:Fox, Ralph
4784:Adams 2004
4665:2106.03925
4525:2021-02-03
4268:(2): 158.
3959:Hass, Joel
3930:: 89â120,
3525:References
3460:Gauss code
3455:Gauss code
3449:Gauss code
3346:Perko pair
3321:torus knot
3317:twist knot
3281:Hoste 2005
3262:Perko 1974
3254:Perko pair
3224:Hoste 2005
3213:Adams 2004
3175:Hoste 2005
3171:Hoste 2005
3143:See also:
3087:Adams 2004
2541:spun knots
2345:Adams 2004
1690:, and the
1680:polynomial
1678:that is a
1644:, and the
1634:knot group
1615:Adams 2004
1586:, ch. 1).
1582:, ch. 3) (
1422:one-to-one
1390:Hoste 2005
1366:Algorithms
920:equivalent
472:Adams 2004
407:Simon 1986
321:knot group
146:complexity
31:(below it)
5592:Stick no.
5548:Alexander
5486:Chirality
5431:Solomon's
5391:Septafoil
5318:Satellite
5278:Whitehead
5204:Stevedore
5102:Knoutilus
5096:Knotscape
5044:144988108
4624:1304.6053
4584:CiteSeerX
4479:Footnotes
4386:MathWorld
4367:. Wolfram
4365:MathWorld
4349:118682559
4163:122824389
3944:0025-5874
3889:-space",
3786:120452571
3762:Dehn, Max
3667:120218312
3602:119320887
3577:1208.5742
3113:composite
2712:−
2703:−
2597:−
2393:geodesics
2310:Dehn 1914
2076:Hopf link
1975:−
1827:−
1750:−
1571:Type III
1426:crossings
1378:Hass 1998
1374:Hass 1998
1218:; and c)
1194:∈
1089:∈
1039:∈
1001:∈
966:→
948:×
846:≤
840:≤
805:→
693:→
678::
538:→
520::
504:injective
362:in 1984 (
65:embedding
5777:Category
5647:Mutation
5615:Notation
5568:Kauffman
5481:Brunnian
5474:2-bridge
5343:Knot sum
5274:(12n242)
5090:KnotPlot
5074:KnotInfo
5009:: 94â105
4997:(1867),
4878:(2013),
4858:On Knots
4855:(1987),
4833:(1977).
4806:(1985),
4741:37264056
4732:10235088
4652:Symmetry
4441:14951363
4407:(1989),
4359:(2013).
4283:Topology
4133:(1997),
4037:18027155
3854:(2000),
3755:16596880
3679:Topology
3539:(2004),
3473:See also
3106:knot is
3091:oriented
3075:knot sum
3061:Knot sum
2993:and the
2953:, where
2570:-spheres
2401:horoball
2306:Max Dehn
1557:Type II
1469:nugatory
1183:for all
417:on DNA (
325:homology
313:Max Dehn
306:topology
45:topology
5789:Commons
5708:Fibered
5606:problem
5575:Pretzel
5553:Bracket
5371:Trefoil
5308:L10a140
5268:(11n42)
5262:(11n34)
5230:Endless
4988:History
4919:Surveys
4786:) and (
4711:Bibcode
4670:Bibcode
4629:Bibcode
4472:1970538
4421:Bibcode
4390:Wolfram
4227:0515288
4195:2040074
4097:1970561
4003:7381505
3983:Bibcode
3952:0160196
3911:1970208
3831:Bibcode
3735:Bibcode
3636:2001854
3594:3342136
3530:Sources
3402:tangles
3207:in the
3204:A002863
2749:-sphere
2378:SnapPea
1908:(where
1704:integer
1640:of the
1509:, are:
411:Tangles
296:led to
234:). The
201:History
5753:Writhe
5723:Ribbon
5558:HOMFLY
5401:Unlink
5361:Unknot
5336:Square
5331:Granny
5122:Tables
5118:Regina
5042:
5036:228151
5034:
4964:
4937:
4908:
4888:
4865:
4841:
4816:
4739:
4729:
4596:
4586:
4470:
4439:
4347:
4337:
4311:
4249:
4225:
4215:
4193:
4161:
4151:
4119:
4095:
4059:
4035:
4001:
3950:
3942:
3909:
3868:
3811:
3784:
3753:
3665:
3634:
3600:
3592:
3551:
3125:smooth
3051:case.
2120:unlink
1930:unknot
1686:, the
1554:Type I
1467:(also
1382:unknot
834:
820:
612:braids
403:chiral
248:Celtic
69:circle
61:unknot
5743:Twist
5728:Slice
5683:Berge
5671:Other
5642:Flype
5580:Prime
5563:Jones
5523:Genus
5353:Torus
5167:links
5163:knots
5050:Movie
5040:S2CID
5032:JSTOR
4809:Knots
4660:arXiv
4619:arXiv
4468:JSTOR
4437:S2CID
4395:8 May
4371:8 May
4345:S2CID
4191:JSTOR
4159:S2CID
4093:JSTOR
4033:S2CID
3999:S2CID
3973:arXiv
3907:JSTOR
3782:S2CID
3663:S2CID
3632:JSTOR
3598:S2CID
3572:arXiv
3417:digon
3340:to 10
3117:prime
3108:prime
2957:is a
2887:. An
2689:with
2314:Jones
1715:links
1702:with
717:with
608:links
500:Curve
228:tying
167:links
67:of a
57:knots
5748:Wild
5713:Knot
5617:and
5604:and
5585:list
5416:Hopf
5165:and
5016:Isis
4962:ISBN
4935:ISBN
4906:ISBN
4886:ISBN
4863:ISBN
4839:ISBN
4814:ISBN
4737:PMID
4594:ISBN
4397:2013
4373:2013
4335:ISBN
4309:ISBN
4247:ISBN
4213:ISBN
4149:ISBN
4117:ISBN
4081:1982
4057:ISBN
3940:ISSN
3866:ISBN
3809:ISBN
3751:PMID
3549:ISBN
3392:The
3374:The
3303:and
3209:OEIS
3147:and
3104:. A
3100:and
3036:>
3004:>
2718:>
2539:and
2432:and
2361:The
2145:) +
2135:) =
2101:) +
2091:) =
1722:i.e.
1497:and
1450:link
1444:and
1442:knot
918:are
891:and
610:and
506:and
386:and
378:and
226:and
5733:Sum
5254:161
5252:(10
5024:doi
4757:",
4727:PMC
4719:doi
4678:doi
4495:by
4460:doi
4429:doi
4417:121
4327:doi
4291:doi
4270:doi
4239:doi
4181:doi
4141:doi
4109:doi
4085:doi
4049:doi
4025:doi
3991:doi
3932:doi
3899:doi
3839:doi
3801:doi
3774:doi
3743:doi
3731:294
3710:doi
3688:doi
3655:doi
3622:doi
3618:326
3582:doi
3348:in
3342:166
3338:162
3291:).
3248:) (
3226:).
3215:).
3198:705
3195:388
3188:293
3186:253
3181:972
3081:or
2825:An
2822:).
2652:) (
2412:).
2347:).
2320:).
1814:or
1710:).
1621:) (
1617:) (
1471:or
1297:.)
1063:to
474:) (
429:).
409:).
399:DNA
358:by
292:'s
43:In
5807::
5433:(4
5418:(2
5403:(0
5393:(7
5383:(5
5373:(3
5363:(0
5295:(6
5280:(5
5244:18
5242:(8
5232:(7
5206:(6
5196:(5
5186:(4
5110:â
5076::
5038:,
5030:,
5020:54
5018:,
5007:VI
5005:,
5001:,
4956:,
4829:;
4802:;
4735:.
4725:.
4717:.
4707:13
4705:.
4701:.
4676:.
4668:.
4656:13
4654:.
4650:.
4627:,
4617:,
4592:,
4582:,
4466:,
4456:78
4435:,
4427:,
4415:,
4411:,
4388:.
4384:.
4363:.
4343:.
4333:.
4325:.
4287:25
4285:,
4266:94
4264:.
4245:.
4237:.
4223:MR
4221:,
4189:,
4177:45
4175:,
4157:,
4147:,
4115:.
4091:,
4055:.
4031:,
4021:20
4019:,
3997:,
3989:,
3981:,
3967:,
3948:MR
3946:,
3938:,
3928:80
3926:,
3905:,
3895:75
3864:,
3837:,
3827:57
3825:,
3807:,
3780:,
3770:75
3768:,
3749:,
3741:,
3729:,
3704:.
3684:34
3682:,
3661:,
3651:56
3649:,
3630:,
3616:,
3596:,
3590:MR
3588:,
3580:,
3568:24
3566:,
3547:,
3543:,
3469:.
3191:,
3184:,
3179:46
2741:.
2737:,
1652:)(
1629:.
1459:A
1400:.
766:.
390:.
370:,
315:,
308:.
258:.
158:,
75:,
47:,
5442:)
5438:1
5427:)
5423:1
5412:)
5408:1
5397:)
5395:1
5387:)
5385:1
5377:)
5375:1
5367:)
5365:1
5304:)
5300:2
5289:)
5285:1
5256:)
5246:)
5236:)
5234:4
5224:3
5222:6
5216:2
5214:6
5210:)
5208:1
5200:)
5198:2
5190:)
5188:1
5169:)
5161:(
5151:e
5144:t
5137:v
5062:)
5026::
4847:.
4753:"
4743:.
4721::
4713::
4686:.
4680::
4672::
4662::
4631::
4621::
4564:.
4552:.
4540:.
4462::
4431::
4423::
4399:.
4375:.
4351:.
4329::
4293::
4276:.
4272::
4255:.
4241::
4183::
4143::
4125:.
4111::
4087::
4065:.
4051::
4027::
3993::
3985::
3975::
3969:9
3934::
3901::
3887:k
3883:k
3841::
3833::
3803::
3776::
3745::
3737::
3718:.
3712::
3690::
3657::
3624::
3584::
3574::
3404:(
3332:1
3325:1
3260:(
3244:(
3193:1
3039:1
3033:n
3013:2
3010:+
3007:n
3001:m
2981:2
2978:+
2975:n
2972:=
2969:m
2955:k
2939:m
2934:R
2910:n
2905:S
2893:k
2889:n
2873:m
2868:R
2844:n
2839:S
2827:n
2818:(
2804:1
2801:+
2798:n
2793:R
2765:n
2760:S
2747:n
2721:0
2715:3
2709:k
2706:3
2700:n
2697:2
2675:n
2670:R
2658:k
2648:(
2634:6
2629:R
2617:k
2603:)
2600:1
2594:k
2591:4
2588:(
2574:n
2568:n
2560:n
2556:m
2552:m
2548:n
2521:4
2516:R
2492:4
2487:R
2463:2
2458:S
2448:(
2253:2
2249:z
2245:+
2242:1
2239:=
2236:)
2233:z
2230:+
2227:0
2224:(
2221:z
2218:+
2215:1
2212:=
2209:)
2205:l
2202:i
2199:o
2196:f
2193:e
2190:r
2187:t
2183:(
2180:C
2164:C
2158:)
2152:(
2150:C
2147:z
2139:(
2137:C
2129:(
2127:C
2114:)
2108:(
2106:C
2103:z
2095:(
2093:C
2085:(
2083:C
2070:)
2064:(
2062:C
2059:z
2051:(
2049:C
2041:(
2039:C
2008:.
2005:)
2000:0
1996:L
1992:(
1989:C
1986:z
1983:+
1980:)
1971:L
1967:(
1964:C
1961:=
1958:)
1953:+
1949:L
1945:(
1942:C
1932:)
1916:O
1896:1
1893:=
1890:)
1887:O
1884:(
1881:C
1858:)
1855:z
1852:(
1849:C
1823:L
1800:+
1796:L
1763:0
1759:L
1755:,
1746:L
1742:,
1737:+
1733:L
1700:z
1345:1
1342:=
1339:t
1316:3
1311:R
1281:H
1259:2
1255:K
1251:=
1248:)
1245:1
1242:,
1237:1
1233:K
1229:(
1226:H
1204:3
1199:R
1191:x
1171:x
1168:=
1165:)
1162:0
1159:,
1156:x
1153:(
1150:H
1128:3
1123:R
1099:3
1094:R
1086:)
1083:t
1080:,
1077:x
1074:(
1071:H
1049:3
1044:R
1036:x
1016:]
1013:1
1010:,
1007:0
1004:[
998:t
976:3
971:R
963:]
960:1
957:,
954:0
951:[
943:3
938:R
933::
930:H
904:2
900:K
877:1
873:K
852:}
849:1
843:t
837:0
830:r
827:o
824:f
815:3
810:R
800:3
795:R
790::
785:t
781:h
777:{
752:2
748:K
744:=
741:)
736:1
732:K
728:(
725:h
703:3
698:R
688:3
683:R
675:h
647:2
643:K
639:,
634:1
630:K
594:)
591:1
588:(
585:K
582:=
579:)
576:0
573:(
570:K
548:3
543:R
535:]
532:1
529:,
526:0
523:[
517:K
486:K
425:(
284:(
195:n
189:n
119:3
114:R
90:3
85:E
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
