3389:, 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.
273:
31:
3100:): 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
1420:
1589:
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 (
1606:
3378:
2288:
2383:
2300:
223:
5783:
1792:
1575:
3473:, 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
2367:
47:
463:
3077:
3275:). 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
1558:
1546:
3165:
5795:
454:
2431:
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
3451:
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.
2414:
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
1588:
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
1636:). 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
2418:
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
151:
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
1311:
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
3134:). 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
3326:). 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
2415:
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.
3145:
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.
3455:
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.
3430:
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.
873:
3244:). 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.
1341:
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
3334:). 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
3263:). 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
2435:
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.
2314:
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
1443:). 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
2038:. 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.
999:
3572:
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",
726:
1431:
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
287:
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
2276:
4957:
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.
2029:
571:
4801:). 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. (
1122:
4994:
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.
241:
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
70:
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 "
3104:, 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.
1516:, 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
1227:
1072:
632:
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
1490:
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".
2827:
1786:
2962:
2933:
2896:
2867:
2788:
2698:
2657:
2544:
2515:
2486:
1339:
1282:
1151:
142:
113:
471:
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.
3438:
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.
775:
2742:
783:
670:
615:
4182:
3034:
1192:
311:'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
3188:, 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,
3060:
2624:
1917:
3002:
1850:
1823:
1728:, 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.
1037:
927:
900:
1879:
3441:
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
2323:). 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
1366:
3448:
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.
1937:
1302:
507:
272:
3363:'s original and subsequent knot tables, and differences in approach to correcting this error in knot tables and other publications created after this point.
2281:
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".
2432:
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.
3219:
2065:
2153:
2109:
2078:
2055:
2166:
2143:
2122:
2099:
1387:). Nonetheless, these algorithms can be extremely time-consuming, and a major issue in the theory is to understand how hard this problem really is (
5160:
4976:
4949:
4920:
4900:
4877:
4853:
4828:
4349:
4323:
4261:
4227:
4163:
4131:
4071:
3880:
3823:
3563:
5728:
3222:). While exponential upper and lower bounds for this sequence are known, it has not been proven that this sequence is strictly increasing (
2667:). Thus the codimension of a smooth knot can be arbitrarily large when not fixing the dimension of the knotted sphere; however, any smooth
936:
5647:
3386:
3372:
4608:
284:
148:); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.
2354:. The hyperbolic structure depends only on the knot so any quantity computed from the hyperbolic structure is then a knot invariant (
5826:
3176:. Knot tables generally include only prime knots, and only one entry for a knot and its mirror image (even if they are different) (
780:
What this definition of knot equivalence means is that two knots are equivalent when there is a continuous family of homeomorphisms
300:
1735:
one in which every component of the link has a preferred direction indicated by an arrow. For a given crossing of the diagram, let
4789:
There are a number of introductions to knot theory. A classical introduction for graduate students or advanced undergraduates is (
2402:
Geometry lets us visualize what the inside of a knot or link complement looks like by imagining light rays as traveling along the
1671:
and hyperbolic invariants were discovered. These aforementioned invariants are only the tip of the iceberg of modern knot theory.
1659:, which can be computed from the Alexander invariant, a module constructed from the infinite cyclic cover of the knot complement (
681:
3122:
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
2410:. The inhabitant of this link complement is viewing the space from near the red component. The balls in the picture are views of
1551:
3252:
2041:
The following is an example of a typical computation using a skein relation. It computes the
AlexanderâConway polynomial of the
1706:
5194:
433:
3180:). The number of nontrivial knots of a given crossing number increases rapidly, making tabulation computationally difficult (
5637:
2186:
187:
To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other
5642:
5513:
3866:
3620:
3555:
3404:
3398:
3173:
875:
of space onto itself, such that the last one of them carries the first knot onto the second knot. (In detail: Two knots
5214:
3974:
5276:
625:
to be equivalent if the knot can be pushed about smoothly, without intersecting itself, to coincide with another knot.
481:
line segment, wrapping it around itself arbitrarily, and then fusing its two free ends together to form a closed loop (
3547:
2287:
1368:(final) stage of the ambient isotopy must be an orientation-preserving homeomorphism carrying one knot to the other.
315:. This record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of
30:
4590:
3872:
3489:
3311:
2299:
1505:
327:
165:, a "quantity" which is the same when computed from different descriptions of a knot. Important invariants include
5282:
3719:. Leibniz Int. Proc. Inform. Vol. 164. Schloss DagstuhlâLeibniz-Zentrum fĂŒr Informatik. pp. 25:1â25:17.
3229:
The first knot tables by Tait, Little, and
Kirkman used knot diagrams, although Tait also used a precursor to the
1948:
523:
5346:
5341:
5153:
3933:
3240:
The early tables attempted to list all knots of at most 10 crossings, and all alternating knots of 11 crossings (
2749:
2574:
1447:
with the additional data of which strand is over and which is under at each crossing. (These diagrams are called
354:
304:
1077:
4805:) is suitable for undergraduates who know point-set topology; knowledge of algebraic topology is not required.
2591:), although this is no longer a requirement for smoothly knotted spheres. In fact, there are smoothly knotted
2455:
Since a knot can be considered topologically a 1-dimensional sphere, the next generalization is to consider a
5474:
233:
Archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as
5799:
1197:
1042:
342:. This would be the main approach to knot theory until a series of breakthroughs transformed the subject.
5688:
5657:
4594:
3315:
1509:
673:
335:
217:
156:
3279:. Less famous is the duplicate in his 10 crossing link table: 2.-2.-20.20 is the mirror of 8*-20:-20. .
3084:
Two knots can be added by cutting both knots and joining the pairs of ends. The operation is called the
2350:(i.e., the set of points of 3-space not on the knot) admits a geometric structure, in particular that of
180:, which are knots of several components entangled with each other. More than six billion knots and links
5518:
4937:
4837:
4086:
4021:
3476:
Gauss code is limited in its ability to identify knots. This problem is partially addressed with by the
3360:
2797:
2439:
Four-dimensional space occurs in classical knot theory, however, and an important topic is the study of
413:
386:
1738:
4765:
2938:
2909:
2872:
2843:
2764:
2674:
2633:
2520:
2491:
2462:
1315:
1232:
1127:
118:
115:. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of
89:
5821:
5787:
5558:
5146:
4721:
4680:
4639:
4529:
4431:
4219:
3993:
3841:
3745:
3412:
2419:
methods of obtaining these invariants make calculating these invariants, in practice, a simple task (
1698:
1656:
1444:
868:{\displaystyle \{h_{t}:\mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}\ \mathrm {for} \ 0\leq t\leq 1\}}
421:
405:
393:. A plethora of knot invariants have been invented since then, utilizing sophisticated tools such as
390:
339:
288:
5025:
Silliman, Robert H. (December 1963), "William
Thomson: Smoke Rings and Nineteenth-Century Atomism",
2547:
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3323:
3276:
3155:
2351:
2064:
1702:
1668:
518:
350:
258:
3168:
A table of prime knots up to seven crossings. The knots are labeled with
AlexanderâBriggs notation
2703:
2177:(unlink of two components) = 0, since the first two polynomials are of the unknot and thus equal.
2152:
2108:
2077:
2054:
5616:
5563:
5177:
5173:
5050:
5042:
4772:. Accessed February 2016. Richard Elwes points out a common mistake in describing the Perko pair.
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4009:
3983:
3917:
3803:
3792:
3689:
3673:
3642:
3608:
3582:
3408:
3282:
In the late 1990s Hoste, Thistlethwaite, and Weeks tabulated all the knots through 16 crossings (
3248:
2791:
2745:
1725:
1460:
1452:
1432:
1424:
1396:
635:
618:
308:
277:
257:
have made repeated appearances in different cultures, often representing strength in unity. The
193:
177:
152:
fundamental problem in knot theory is determining when two descriptions represent the same knot.
63:
3889:
3184:, p. 20). Tabulation efforts have succeeded in enumerating over 6 billion knots and links (
2165:
2142:
2121:
2098:
4991:
3832:
Doll, Helmut; Hoste, Jim (1991), "A tabulation of oriented links. With microfiche supplement",
2551:
1788:
be the oriented link diagrams resulting from changing the diagram as indicated in the figure:
1605:
1419:
576:
5713:
5662:
5612:
5568:
5528:
5523:
5441:
4972:
4964:
4945:
4916:
4896:
4886:
4873:
4863:
4849:
4824:
4747:
4604:
4367:
4345:
4319:
4257:
4223:
4159:
4127:
4067:
3950:
3876:
3819:
3761:
3559:
3007:
1648:
1513:
1499:
1156:
432:). Knot theory may be crucial in the construction of quantum computers, through the model of
424:, strings with both ends fixed in place, have been effectively used in studying the action of
4525:
Marc Lackenby announces a new unknot recognition algorithm that runs in quasi-polynomial time
2546:
onto itself taking the embedded 2-sphere to the standard "round" embedding of the 2-sphere.
5748:
5573:
5469:
5204:
5034:
4814:
4737:
4729:
4688:
4470:
4439:
4400:
4337:
4301:
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4249:
4191:
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4001:
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3909:
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3811:
3784:
3753:
3720:
3698:
3665:
3632:
3592:
3519:
3514:
3504:
3499:
3435:
3422:
The notation describes how to construct a particular link diagram of the link. Start with a
3287:
3142:
3039:
2594:
2339:
2310:
Actually, there are two trefoil knots, called the right and left-handed trefoils, which are
1887:
1694:
1487:
1464:
382:
366:
346:
312:
292:
250:
242:
166:
5118:
4237:
3962:
3604:
3419:). The advantage of this notation is that it reflects some properties of the knot or link.
2975:
1828:
1801:
1004:
905:
878:
176:
The original motivation for the founders of knot theory was to create a table of knots and
5708:
5672:
5607:
5553:
5508:
5501:
5391:
5303:
5186:
4585:
Levine, J.; Orr, K (2000), "A survey of applications of surgery to knot and link theory",
4233:
3958:
3600:
3529:
3442:
3233:. Different notations have been invented for knots which allow more efficient tabulation (
3230:
3159:
2407:
2392:
2373:
2347:
2343:
1855:
1852:, depending on the chosen crossing's configuration. Then the AlexanderâConway polynomial,
1680:
1652:
1637:
1408:
1305:
404:
In the last several decades of the 20th century, scientists became interested in studying
358:
254:
181:
145:
83:
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1345:
17:
5060:
4742:
4725:
4709:
4684:
4643:
4435:
3997:
3845:
3749:
3377:
385:, and others, revealed deep connections between knot theory and mathematical methods in
5768:
5667:
5629:
5548:
5461:
5336:
5328:
5288:
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4371:
4063:
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2969:
2035:
1922:
1686:
1619:
1612:
1380:
1287:
492:
398:
362:
161:
4614:â An introductory article to high dimensional knots and links for the advanced readers
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4005:
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222:
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5703:
5491:
5484:
5479:
5054:
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4173:
4081:
3806:(1970), "An enumeration of knots and links, and some of their algebraic properties",
3796:
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2327:
polynomial can in fact distinguish between the left- and right-handed trefoil knots (
1404:
676:
425:
394:
378:
370:
262:
227:
4451:
4047:
2744:
is unknotted. The notion of a knot has further generalisations in mathematics, see:
5718:
5698:
5602:
5585:
5381:
5318:
4458:
4013:
3862:
3319:
3127:
2042:
1574:
1557:
1545:
622:
246:
39:
35:
5401:
5240:
5232:
5224:
5132:
5131:â software for low-dimensional topology with native support for knots and links.
5106:
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4890:
4867:
4818:
4419:
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5270:
5250:
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3725:
3112:
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2444:
2406:
of the geometry. An example is provided by the picture of the complement of the
2311:
266:
4733:
2395:
complement from the perspective of an inhabitant living near the red component.
5753:
5738:
5693:
5590:
5543:
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5533:
5363:
5260:
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4253:
4155:
3596:
3470:
3465:
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3295:
3264:
3116:
2557:
The mathematical technique called "general position" implies that for a given
2440:
1690:
1644:
331:
238:
188:
170:
3954:
3552:
The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots
2319:, before the invention of knot polynomials, using group theoretical methods (
5758:
5426:
4841:
4054:
Hoste, Jim (2005). "The Enumeration and Classification of Knots and Links".
3969:
2366:
2086:
1791:
1667:). In the late 20th century, invariants such as "quantum" knot polynomials,
1376:
514:
478:
462:
75:
46:
5094:
4751:
4647:â An introductory article to high dimensional knots and links for beginners
3765:
3076:
4341:
412:
and other polymers. Knot theory can be used to determine if a molecule is
155:
A complete algorithmic solution to this problem exists, which has unknown
5743:
5353:
3772:
3452:
For an algebraic knot such as in the first example, 1* is often omitted.
3445:. One inserts this tangle at the vertex of the basic polyhedron 1*.
3071:
2756:
2577:
2456:
2447:. A notorious open problem asks whether every slice knot is also ribbon.
2411:
2403:
2316:
323:
316:
198:
55:
4693:
4658:
4523:
3868:
When topology meets chemistry: A topological look at molecular chirality
2133:
of two components) and an unknot. The unlink takes a bit of sneakiness:
1624:
A knot invariant is a "quantity" that is the same for equivalent knots (
5066:
4482:
4443:
4284:
4205:
4107:
4039:
3946:
3921:
3788:
3669:
3646:
3524:
3164:
2388:
2376:
are a link with the property that removing one ring unlinks the others.
1714:
1550:
5046:
4292:
Simon, Jonathan (1986), "Topological chirality of certain molecules",
5763:
5411:
5371:
5083:
4823:, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter,
3988:
2130:
1940:
1392:
994:{\displaystyle H:\mathbb {R} ^{3}\times \rightarrow \mathbb {R} ^{3}}
79:
71:
4474:
4196:
4099:
3931:(1962), "Ăber das Homöomorphieproblem der 3-Mannigfaltigkeiten. I",
3913:
2129:
gives a link deformable to one with 0 crossings (it is actually the
5038:
4675:
3247:
The first major verification of this work was done in the 1960s by
2630:-dimensional space; e.g., there is a smoothly knotted 3-sphere in
1486:), or in which all of the reducible crossings have been removed. A
1439:, where the "shadow" of the knot crosses itself once transversely (
453:
5652:
4634:
3717:
36th International Symposium on Computational Geometry (SoCG 2020)
3587:
3427:
3376:
3163:
3075:
1604:
1418:
510:
271:
234:
221:
45:
29:
4710:"A tile model of circuit topology for self-entangled biopolymers"
4150:, Graduate Texts in Mathematics, vol. 175, Springer-Verlag,
50:
A knot diagram of the trefoil knot, the simplest non-trivial knot
5723:
3310:
This is the most traditional notation, due to the 1927 paper of
2517:). Such an embedding is knotted if there is no homeomorphism of
1724:
The AlexanderâConway polynomial is actually defined in terms of
67:
5142:
5112:
4659:"Circuit Topology for Bottom-Up Engineering of Molecular Knots"
3734:
Collins, Graham (April 2006), "Computing with Quantum Knots",
1463:.) Analogously, knotted surfaces in 4-space can be related to
409:
283:
A mathematical theory of knots was first developed in 1771 by
4708:
Flapan, Erica; Mashaghi, Alireza; Wong, Helen (1 June 2023).
2045:. The yellow patches indicate where the relation is applied.
617:. Topologists consider knots and other entanglements such as
3972:(1998), "Algorithms for recognizing knots and 3-manifolds",
3381:
A knot diagram with crossings labelled for a Dowker sequence
1790:
1609:
A 3D print depicting the complement of the figure eight knot
721:{\displaystyle h\colon \mathbb {R} ^{3}\to \mathbb {R} ^{3}}
3214:
4587:
Surveys on Surgery Theory: Papers Dedicated to C.T.C. Wall
2089:. Applying the relation to the Hopf link where indicated,
1407:
announced a new unknot recognition algorithm that runs in
4180:
Perko, Kenneth (1974), "On the classification of knots",
3126:. There is a prime decomposition for knots, analogous to
361:, enabling the use of geometry in defining new, powerful
330:, and othersâstudied knots from the point of view of the
184:
since the beginnings of knot theory in the 19th century.
4657:
Golovnev, Anatoly; Mashaghi, Alireza (7 December 2021).
4572:
4246:
Die eindeutige Zerlegbarkeit eines Knotens in Primknoten
3286:). In 2003 Rankin, Flint, and Schermann, tabulated the
3283:
3241:
3177:
322:
These topologists in the early part of the 20th centuryâ
5103:â software to investigate geometric properties of knots
5100:
3096:
of two knots. This can be formally defined as follows (
1504:
In 1927, working with this diagrammatic form of knots,
5063:
of a modern recreation of Tait's smoke ring experiment
4114:
Kontsevich, M. (1993). "Vassiliev's knot invariants".
4024:; Weeks, Jeffrey (1998), "The First 1,701,935 Knots",
3172:
Traditionally, knots have been catalogued in terms of
2271:{\displaystyle C(\mathrm {trefoil} )=1+z(0+z)=1+z^{2}}
3251:, who not only developed a new notation but also the
3042:
3010:
2978:
2941:
2912:
2875:
2846:
2800:
2767:
2706:
2677:
2636:
2597:
2523:
2494:
2465:
2189:
1951:
1925:
1890:
1858:
1831:
1804:
1741:
1379:
exist to solve this problem, with the first given by
1348:
1318:
1290:
1235:
1200:
1159:
1130:
1080:
1045:
1007:
939:
908:
881:
786:
734:
684:
638:
579:
526:
495:
377:, pp. 71â89), and subsequent contributions from
159:. In practice, knots are often distinguished using a
121:
92:
4271:
Silver, Daniel (2006). "Knot Theory's Odd Origins".
4084:(1965), "A classification of differentiable knots",
3656:; King, Henry C. (1981), "All knots are algebraic",
3623:(1991), "Hyperbolic invariants of knots and links",
2420:
5681:
5625:
5460:
5362:
5327:
5185:
5097:â detailed info on individual knots in knot tables
5089:Table of Knot Invariants and Knot Theory Resources
3515:Contact geometry#Legendrian submanifolds and knots
3054:
3028:
2996:
2956:
2927:
2890:
2861:
2821:
2782:
2736:
2692:
2651:
2618:
2538:
2509:
2480:
2270:
2034:The second rule is what is often referred to as a
2023:
1931:
1911:
1873:
1844:
1817:
1780:
1531:Move a strand completely over or under a crossing.
1360:
1333:
1296:
1276:
1221:
1186:
1145:
1116:
1066:
1031:
993:
921:
894:
867:
769:
720:
664:
609:
565:
501:
136:
107:
3625:Transactions of the American Mathematical Society
2554:are two typical families of such 2-sphere knots.
1881:, is recursively defined according to the rules:
4183:Proceedings of the American Mathematical Society
1375:, is determining the equivalence of two knots.
191:and objects other than circles can be used; see
4420:"Quantum field theory and the Jones polynomial"
2573:), the sphere should be unknotted. In general,
226:Intricate Celtic knotwork in the 1200-year-old
197:. For example, a higher-dimensional knot is an
5115:â online database and image generator of knots
4589:, Annals of mathematics studies, vol. 1,
5154:
5014:Proceedings of the Royal Society of Edinburgh
4793:). Other good texts from the references are (
2488:) embedded in 4-dimensional Euclidean space (
1705:. A variant of the Alexander polynomial, the
408:in order to understand knotting phenomena in
8:
3687:(1995), "On the Vassiliev knot invariants",
3575:Journal of Knot Theory and Its Ramifications
3426:, a 4-valent connected planar graph with no
2024:{\displaystyle C(L_{+})=C(L_{-})+zC(L_{0}).}
862:
787:
566:{\displaystyle K\colon \to \mathbb {R} ^{3}}
4334:Quantum Invariants of Knots and 3-Manifolds
4218:, Mathematics Lecture Series, vol. 7,
3355:are ambiguous, due to the discovery of the
2830:
1611:by François Guéritaud, Saul Schleimer, and
74:"). In mathematical language, a knot is an
5161:
5147:
5139:
4971:, Simon & Schuster, pp. 203â218,
4560:
4461:(1963), "Unknotting combinatorial balls",
4118:. ADVSOV. Vol. 16. pp. 137â150.
3808:Computational Problems in Abstract Algebra
1117:{\displaystyle H(x,t)\in \mathbb {R} ^{3}}
477:A knot is created by beginning with a one-
305:theory that atoms were knots in the aether
34:Examples of different knots including the
4798:
4741:
4692:
4674:
4633:
4598:
4548:
4305:
4195:
3987:
3853:
3775:(1914), "Die beiden Kleeblattschlingen",
3724:
3702:
3636:
3586:
3347:. AlexanderâBriggs names in the range 10
3267:, which would only be noticed in 1974 by
3260:
3041:
3009:
2977:
2948:
2944:
2943:
2940:
2919:
2915:
2914:
2911:
2882:
2878:
2877:
2874:
2853:
2849:
2848:
2845:
2807:
2803:
2802:
2799:
2774:
2770:
2769:
2766:
2705:
2684:
2680:
2679:
2676:
2660:
2643:
2639:
2638:
2635:
2596:
2530:
2526:
2525:
2522:
2501:
2497:
2496:
2493:
2472:
2468:
2467:
2464:
2328:
2262:
2196:
2188:
2009:
1984:
1962:
1950:
1924:
1889:
1857:
1836:
1830:
1809:
1803:
1772:
1759:
1746:
1740:
1718:
1660:
1629:
1594:
1590:
1347:
1325:
1321:
1320:
1317:
1289:
1268:
1246:
1234:
1213:
1209:
1208:
1199:
1158:
1137:
1133:
1132:
1129:
1108:
1104:
1103:
1079:
1058:
1054:
1053:
1044:
1006:
985:
981:
980:
952:
948:
947:
938:
913:
907:
886:
880:
833:
824:
820:
819:
809:
805:
804:
794:
785:
761:
745:
733:
712:
708:
707:
697:
693:
692:
683:
656:
643:
637:
578:
557:
553:
552:
525:
494:
486:
374:
128:
124:
123:
120:
99:
95:
94:
91:
4802:
3141:Knots can also be constructed using the
3131:
1643:"Classical" knot invariants include the
1528:Move one strand completely over another.
1474:is a knot diagram in which there are no
573:, with the only "non-injectivity" being
4790:
4495:
3322:in his knot table (see image above and
2283:
1664:
1633:
1440:
1391:). The special case of recognizing the
437:
5135:of prime knots with up to 19 crossings
4626:Introduction to high dimensional knots
4502:As first sketched using the theory of
3416:
3299:
3284:Hoste, Thistlethwaite & Weeks 1998
3256:
3242:Hoste, Thistlethwaite & Weeks 1998
3178:Hoste, Thistlethwaite & Weeks 1998
3036:cases are well studied, and so is the
2664:
2588:
1525:Twist and untwist in either direction.
1371:The basic problem of knot theory, the
429:
296:
4794:
4507:
3291:
3272:
3234:
3223:
3185:
3181:
3097:
2355:
2180:Putting all this together will show:
1798:The original diagram might be either
1625:
1400:
1222:{\displaystyle x\in \mathbb {R} ^{3}}
1067:{\displaystyle x\in \mathbb {R} ^{3}}
933:if there exists a continuous mapping
482:
417:
265:lavished entire pages with intricate
7:
5794:
5109:â software to create images of knots
4965:"Ch. 8: Unreasonable Effectiveness?"
4511:
2750:isotopy classification of embeddings
2451:Knotting spheres of higher dimension
2320:
1435:except at the double points, called
1388:
1384:
2587: + 2)-dimensional space (
1731:Consider an oriented link diagram,
1423:Tenfold Knottiness, plate IX, from
4895:(4th ed.), World Scientific,
3896: − 1)-spheres in 6
3816:10.1016/B978-0-08-012975-4.50034-5
3619:Adams, Colin; Hildebrand, Martin;
3107:The knot sum of oriented knots is
2822:{\displaystyle \mathbb {R} ^{n+1}}
2421:Adams, Hildebrand & Weeks 1991
2215:
2212:
2209:
2206:
2203:
2200:
2197:
1709:, is a polynomial in the variable
1693:. Well-known examples include the
840:
837:
834:
25:
3855:10.1090/S0025-5718-1991-1094946-4
3758:10.1038/scientificamerican0406-56
3638:10.1090/s0002-9947-1991-0994161-2
3407:for knots and links, named after
3138:knots in codimension at least 3.
2565:-dimensional Euclidean space, if
1781:{\displaystyle L_{+},L_{-},L_{0}}
353:into the study of knots with the
208:+2)-dimensional Euclidean space.
5793:
5782:
5781:
5125:function for investigating knots
4510:. For a more recent survey, see
4064:10.1016/B978-044451452-3/50006-X
3294:). In 2020 Burton tabulated all
2957:{\displaystyle \mathbb {R} ^{m}}
2928:{\displaystyle \mathbb {S} ^{n}}
2891:{\displaystyle \mathbb {R} ^{m}}
2862:{\displaystyle \mathbb {S} ^{n}}
2783:{\displaystyle \mathbb {S} ^{n}}
2693:{\displaystyle \mathbb {R} ^{n}}
2652:{\displaystyle \mathbb {R} ^{6}}
2539:{\displaystyle \mathbb {R} ^{4}}
2510:{\displaystyle \mathbb {R} ^{4}}
2481:{\displaystyle \mathbb {S} ^{2}}
2381:
2365:
2298:
2286:
2164:
2151:
2141:
2120:
2107:
2097:
2076:
2063:
2053:
1573:
1556:
1549:
1544:
1334:{\displaystyle \mathbb {R} ^{3}}
1277:{\displaystyle H(K_{1},1)=K_{2}}
1146:{\displaystyle \mathbb {R} ^{3}}
509:is a "simple closed curve" (see
461:
452:
137:{\displaystyle \mathbb {R} ^{3}}
108:{\displaystyle \mathbb {E} ^{3}}
4316:Knots, mathematics with a twist
1535:
434:topological quantum computation
285:Alexandre-Théophile Vandermonde
5648:DowkerâThistlethwaite notation
4915:, Cambridge University Press,
4872:, Princeton University Press,
4147:An Introduction to Knot Theory
3810:, Pergamon, pp. 329â358,
3387:DowkerâThistlethwaite notation
3373:DowkerâThistlethwaite notation
3367:DowkerâThistlethwaite notation
2613:
2598:
2569:is large enough (depending on
2305:The right-handed trefoil knot.
2246:
2234:
2219:
2193:
2015:
2002:
1990:
1977:
1968:
1955:
1900:
1894:
1868:
1862:
1258:
1239:
1175:
1163:
1096:
1084:
1026:
1014:
976:
973:
961:
815:
770:{\displaystyle h(K_{1})=K_{2}}
751:
738:
703:
672:are equivalent if there is an
604:
598:
589:
583:
548:
545:
533:
357:. Many knots were shown to be
1:
4766:The Revenge of the Perko Pair
4006:10.1016/S0960-0779(97)00109-4
3556:American Mathematical Society
3399:Conway notation (knot theory)
2794:with isolated singularity in
2293:The left-handed trefoil knot.
1399:, is of particular interest (
489:). Simply, we can say a knot
416:(has a "handedness") or not (
173:, and hyperbolic invariants.
4391:Weisstein, Eric W. (2013a).
4332:Turaev, Vladimir G. (2016).
4318:, Harvard University Press,
4307:10.1016/0040-9383(86)90041-8
3975:Chaos, Solitons and Fractals
3713:"The Next 350 Million Knots"
3711:Burton, Benjamin A. (2020).
3704:10.1016/0040-9383(95)93237-2
3411:, is based on the theory of
2737:{\displaystyle 2n-3k-3>0}
4909:Cromwell, Peter R. (2004),
4846:Introduction to Knot Theory
3726:10.4230/LIPIcs.SoCG.2020.25
3253:AlexanderâConway polynomial
1707:AlexanderâConway polynomial
1427:'s article "On Knots", 1884
665:{\displaystyle K_{1},K_{2}}
27:Study of mathematical knots
5843:
4734:10.1038/s41598-023-35771-8
4591:Princeton University Press
4528:, Mathematical Institute,
4314:Sossinsky, Alexei (2002),
3873:Cambridge University Press
3490:List of knot theory topics
3463:
3396:
3370:
3153:
3069:
1678:
1617:
1497:
276:The first knot tabulator,
215:
5777:
5638:AlexanderâBriggs notation
4254:10.1007/978-3-642-45813-2
4156:10.1007/978-1-4612-0691-0
3934:Mathematische Zeitschrift
3597:10.1142/S021821651550011X
3306:AlexanderâBriggs notation
3298:with up to 19 crossings (
2085:gives the unknot and the
1581:
1572:
610:{\displaystyle K(0)=K(1)}
144:upon itself (known as an
18:AlexanderâBriggs notation
5827:Low-dimensional topology
5078:Knot tables and software
4244:Schubert, Horst (1949).
4142:Lickorish, W. B. Raymond
3434:Each vertex then has an
3029:{\displaystyle m>n+2}
1187:{\displaystyle H(x,0)=x}
513:) â that is: a "nearly"
189:three-dimensional spaces
5729:List of knots and links
5277:KinoshitaâTerasaka knot
4992:"Mathematics and Knots"
4969:Is God a Mathematician?
4942:Handbook of Knot Theory
4124:10.1090/advsov/016.2/04
4056:Handbook of Knot Theory
3130:and composite numbers (
2831:Akbulut & King 1981
1685:A knot polynomial is a
365:. The discovery of the
355:hyperbolization theorem
5067:History of knot theory
4938:Thistlethwaite, Morwen
4785:Introductory textbooks
4372:"Reduced Knot Diagram"
4212:Rolfsen, Dale (1976),
4022:Thistlethwaite, Morwen
3382:
3338:and the Hopf link is 2
3318:and later extended by
3290:through 22 crossings (
3169:
3081:
3056:
3055:{\displaystyle n>1}
3030:
2998:
2958:
2929:
2892:
2863:
2823:
2784:
2738:
2694:
2653:
2620:
2619:{\displaystyle (4k-1)}
2540:
2511:
2482:
2457:two-dimensional sphere
2342:proved many knots are
2272:
2025:
1939:is any diagram of the
1933:
1913:
1912:{\displaystyle C(O)=1}
1875:
1846:
1819:
1795:
1782:
1615:
1459:when they represent a
1451:when they represent a
1428:
1362:
1335:
1298:
1278:
1223:
1188:
1147:
1124:is a homeomorphism of
1118:
1068:
1033:
1001:such that a) for each
995:
923:
896:
869:
771:
722:
674:orientation-preserving
666:
611:
567:
503:
280:
261:monks who created the
230:
218:History of knot theory
138:
109:
51:
43:
5519:Finite type invariant
5069:(on the home page of
4963:Livio, Mario (2009),
4936:Menasco, William W.;
4463:Annals of Mathematics
4342:10.1515/9783110435221
4222:: Publish or Perish,
4116:I. M. Gelfand Seminar
4087:Annals of Mathematics
3902:Annals of Mathematics
3777:Mathematische Annalen
3380:
3361:Charles Newton Little
3261:Doll & Hoste 1991
3167:
3079:
3057:
3031:
2999:
2997:{\displaystyle m=n+2}
2959:
2930:
2893:
2864:
2824:
2785:
2739:
2695:
2654:
2621:
2541:
2512:
2483:
2335:Hyperbolic invariants
2273:
2026:
1934:
1914:
1876:
1847:
1845:{\displaystyle L_{-}}
1820:
1818:{\displaystyle L_{+}}
1794:
1783:
1608:
1422:
1409:quasi-polynomial time
1363:
1336:
1299:
1279:
1224:
1189:
1148:
1119:
1069:
1034:
1032:{\displaystyle t\in }
996:
924:
922:{\displaystyle K_{2}}
897:
895:{\displaystyle K_{1}}
870:
772:
723:
667:
612:
568:
504:
387:statistical mechanics
275:
235:recording information
225:
139:
110:
49:
33:
5006:Thomson, Sir William
4624:Ogasa, Eiji (2013),
4530:University of Oxford
4393:"Reducible Crossing"
4220:Berkeley, California
4058:. pp. 209â232.
3658:Comment. Math. Helv.
3525:Necktie § Knots
3040:
3008:
2976:
2939:
2910:
2873:
2844:
2798:
2765:
2704:
2675:
2671:-sphere embedded in
2634:
2595:
2583:form knots only in (
2521:
2492:
2463:
2187:
1949:
1923:
1888:
1874:{\displaystyle C(z)}
1856:
1829:
1802:
1739:
1699:Alexander polynomial
1669:Vassiliev invariants
1657:Alexander polynomial
1512:, and independently
1510:Garland Baird Briggs
1445:immersed plane curve
1403:). In February 2021
1346:
1316:
1288:
1233:
1198:
1157:
1128:
1078:
1043:
1005:
937:
906:
879:
784:
732:
682:
636:
577:
524:
493:
391:quantum field theory
340:Alexander polynomial
334:and invariants from
289:Carl Friedrich Gauss
119:
90:
66:. While inspired by
5689:Alexander's theorem
5123:Wolfram Mathematica
4838:Crowell, Richard H.
4726:2023NatSR..13.8889F
4694:10.3390/sym13122353
4685:2021Symm...13.2353G
4644:2013arXiv1304.6053O
4436:1989CMaPh.121..351W
4027:Math. Intelligencer
3998:1998CSF.....9..569H
3892:(1962), "Knotted (4
3846:1991MaCom..57..747D
3750:2006SciAm.294d..56C
3737:Scientific American
3478:extended Gauss code
3324:List of prime knots
3156:List of prime knots
3088:, or sometimes the
2352:hyperbolic geometry
2346:, meaning that the
2173:which implies that
1703:Kauffman polynomial
1540:
1484:removable crossings
1476:reducible crossings
1383:in the late 1960s (
1373:recognition problem
1361:{\displaystyle t=1}
1039:the mapping taking
519:continuous function
351:hyperbolic geometry
345:In the late 1970s,
338:theory such as the
202:-dimensional sphere
182:have been tabulated
38:(top left) and the
4887:Kauffman, Louis H.
4864:Kauffman, Louis H.
4770:RichardElwes.co.uk
4714:Scientific Reports
4444:10.1007/BF01217730
4368:Weisstein, Eric W.
4285:10.1511/2006.2.158
4273:American Scientist
4040:10.1007/BF03025227
3947:10.1007/BF01162369
3789:10.1007/BF01563732
3670:10.1007/BF02566217
3581:(3): 1550011, 30,
3409:John Horton Conway
3383:
3312:James W. Alexander
3249:John Horton Conway
3170:
3082:
3052:
3026:
2994:
2954:
2925:
2902:-link consists of
2888:
2859:
2840:-knot is a single
2819:
2792:real-algebraic set
2780:
2755:Every knot in the
2746:Knot (mathematics)
2734:
2690:
2649:
2616:
2536:
2507:
2478:
2391:'s cusp view: the
2268:
2021:
1929:
1909:
1871:
1842:
1815:
1796:
1778:
1616:
1538:Reidemeister moves
1536:
1518:Reidemeister moves
1494:Reidemeister moves
1429:
1425:Peter Guthrie Tait
1397:unknotting problem
1358:
1331:
1294:
1284:. Such a function
1274:
1219:
1184:
1143:
1114:
1064:
1029:
991:
919:
892:
865:
767:
718:
662:
607:
563:
499:
309:Peter Guthrie Tait
299:). In the 1860s,
291:, who defined the
281:
278:Peter Guthrie Tait
231:
194:knot (mathematics)
134:
105:
64:mathematical knots
52:
44:
5809:
5808:
5663:Reidemeister move
5529:Khovanov homology
5524:Hyperbolic volume
5010:"On Vortex Atoms"
4978:978-0-7432-9405-8
4951:978-0-444-51452-3
4922:978-0-521-54831-1
4902:978-981-4383-00-4
4892:Knots and Physics
4879:978-0-691-08435-0
4855:978-0-387-90272-2
4830:978-3-11-008675-1
4815:Zieschang, Heiner
4573:Adams et al. 2015
4465:, Second Series,
4424:Comm. Math. Phys.
4351:978-3-11-043522-1
4325:978-0-674-00944-8
4263:978-3-540-01419-5
4229:978-0-914098-16-4
4165:978-0-387-98254-0
4133:978-0-8218-4117-4
4090:, Second Series,
4073:978-0-444-51452-3
3904:, Second Series,
3882:978-0-521-66254-3
3825:978-0-08-012975-4
3565:978-0-8218-3678-1
3316:Garland B. Briggs
3288:alternating knots
2790:is the link of a
2427:Higher dimensions
1932:{\displaystyle O}
1649:fundamental group
1586:
1585:
1514:Kurt Reidemeister
1500:Reidemeister move
1465:immersed surfaces
1297:{\displaystyle H}
846:
832:
502:{\displaystyle K}
82:in 3-dimensional
16:(Redirected from
5834:
5797:
5796:
5785:
5784:
5749:Tait conjectures
5452:
5451:
5437:
5436:
5422:
5421:
5314:
5313:
5299:
5298:
5283:(â2,3,7) pretzel
5163:
5156:
5149:
5140:
5057:
5021:
4981:
4954:
4925:
4905:
4882:
4859:
4833:
4773:
4762:
4756:
4755:
4745:
4705:
4699:
4698:
4696:
4678:
4654:
4648:
4646:
4637:
4621:
4615:
4613:
4602:
4582:
4576:
4570:
4564:
4558:
4552:
4546:
4540:
4539:
4538:
4537:
4520:
4514:
4500:
4485:
4454:
4411:
4409:
4407:
4387:
4385:
4383:
4363:
4328:
4310:
4309:
4288:
4267:
4240:
4208:
4199:
4176:
4137:
4110:
4077:
4050:
4016:
3991:
3982:(4â5): 569â581,
3965:
3924:
3890:Haefliger, André
3885:
3858:
3857:
3840:(196): 747â761,
3828:
3799:
3768:
3730:
3728:
3707:
3706:
3680:
3649:
3640:
3615:
3590:
3568:
3520:Knots and graphs
3505:Quantum topology
3500:Circuit topology
3436:algebraic tangle
3424:basic polyhedron
3346:
3345:
3330:comes after the
3217:
3211:
3210:
3207:
3201:
3200:
3194:
3193:
3150:Tabulating knots
3143:circuit topology
3080:Adding two knots
3061:
3059:
3058:
3053:
3035:
3033:
3032:
3027:
3003:
3001:
3000:
2995:
2963:
2961:
2960:
2955:
2953:
2952:
2947:
2934:
2932:
2931:
2926:
2924:
2923:
2918:
2897:
2895:
2894:
2889:
2887:
2886:
2881:
2868:
2866:
2865:
2860:
2858:
2857:
2852:
2828:
2826:
2825:
2820:
2818:
2817:
2806:
2789:
2787:
2786:
2781:
2779:
2778:
2773:
2743:
2741:
2740:
2735:
2699:
2697:
2696:
2691:
2689:
2688:
2683:
2658:
2656:
2655:
2650:
2648:
2647:
2642:
2625:
2623:
2622:
2617:
2575:piecewise-linear
2545:
2543:
2542:
2537:
2535:
2534:
2529:
2516:
2514:
2513:
2508:
2506:
2505:
2500:
2487:
2485:
2484:
2479:
2477:
2476:
2471:
2385:
2369:
2344:hyperbolic knots
2340:William Thurston
2302:
2290:
2277:
2275:
2274:
2269:
2267:
2266:
2218:
2168:
2155:
2145:
2124:
2111:
2101:
2080:
2067:
2057:
2030:
2028:
2027:
2022:
2014:
2013:
1989:
1988:
1967:
1966:
1938:
1936:
1935:
1930:
1918:
1916:
1915:
1910:
1880:
1878:
1877:
1872:
1851:
1849:
1848:
1843:
1841:
1840:
1824:
1822:
1821:
1816:
1814:
1813:
1787:
1785:
1784:
1779:
1777:
1776:
1764:
1763:
1751:
1750:
1695:Jones polynomial
1675:Knot polynomials
1577:
1560:
1553:
1548:
1541:
1488:petal projection
1367:
1365:
1364:
1359:
1340:
1338:
1337:
1332:
1330:
1329:
1324:
1303:
1301:
1300:
1295:
1283:
1281:
1280:
1275:
1273:
1272:
1251:
1250:
1228:
1226:
1225:
1220:
1218:
1217:
1212:
1193:
1191:
1190:
1185:
1153:onto itself; b)
1152:
1150:
1149:
1144:
1142:
1141:
1136:
1123:
1121:
1120:
1115:
1113:
1112:
1107:
1073:
1071:
1070:
1065:
1063:
1062:
1057:
1038:
1036:
1035:
1030:
1000:
998:
997:
992:
990:
989:
984:
957:
956:
951:
928:
926:
925:
920:
918:
917:
901:
899:
898:
893:
891:
890:
874:
872:
871:
866:
844:
843:
830:
829:
828:
823:
814:
813:
808:
799:
798:
776:
774:
773:
768:
766:
765:
750:
749:
727:
725:
724:
719:
717:
716:
711:
702:
701:
696:
671:
669:
668:
663:
661:
660:
648:
647:
630:knot equivalence
616:
614:
613:
608:
572:
570:
569:
564:
562:
561:
556:
508:
506:
505:
500:
465:
456:
444:Knot equivalence
383:Maxim Kontsevich
367:Jones polynomial
359:hyperbolic knots
347:William Thurston
313:Tait conjectures
293:linking integral
251:Tibetan Buddhism
243:Chinese knotting
167:knot polynomials
143:
141:
140:
135:
133:
132:
127:
114:
112:
111:
106:
104:
103:
98:
62:is the study of
21:
5842:
5841:
5837:
5836:
5835:
5833:
5832:
5831:
5812:
5811:
5810:
5805:
5773:
5677:
5643:Conway notation
5627:
5621:
5608:Tricolorability
5456:
5450:
5447:
5446:
5445:
5435:
5432:
5431:
5430:
5420:
5417:
5416:
5415:
5407:
5397:
5387:
5377:
5358:
5337:Composite knots
5323:
5312:
5309:
5308:
5307:
5304:Borromean rings
5297:
5294:
5293:
5292:
5266:
5256:
5246:
5236:
5228:
5220:
5210:
5200:
5181:
5167:
5080:
5024:
5004:
5001:
4988:
4979:
4962:
4952:
4940:, eds. (2005),
4935:
4932:
4923:
4912:Knots and Links
4908:
4903:
4885:
4880:
4862:
4856:
4836:
4831:
4809:
4787:
4782:
4780:Further reading
4777:
4776:
4763:
4759:
4707:
4706:
4702:
4656:
4655:
4651:
4623:
4622:
4618:
4611:
4584:
4583:
4579:
4571:
4567:
4561:Weisstein 2013a
4559:
4555:
4547:
4543:
4535:
4533:
4522:
4521:
4517:
4504:Haken manifolds
4501:
4497:
4492:
4475:10.2307/1970538
4459:Zeeman, Erik C.
4457:
4414:
4405:
4403:
4390:
4381:
4379:
4366:
4352:
4331:
4326:
4313:
4291:
4270:
4264:
4243:
4230:
4215:Knots and Links
4211:
4197:10.2307/2040074
4179:
4166:
4140:
4134:
4113:
4100:10.2307/1970561
4080:
4074:
4053:
4019:
3968:
3929:Haken, Wolfgang
3927:
3914:10.2307/1970208
3888:
3883:
3861:
3831:
3826:
3804:Conway, John H.
3802:
3771:
3733:
3710:
3685:Bar-Natan, Dror
3683:
3654:Akbulut, Selman
3652:
3618:
3571:
3566:
3546:
3543:
3538:
3530:Lamp cord trick
3486:
3468:
3462:
3443:rational tangle
3405:Conway notation
3401:
3395:
3393:Conway notation
3375:
3369:
3354:
3350:
3344:
3341:
3340:
3339:
3337:
3308:
3231:Dowker notation
3213:
3208:
3205:
3203:
3198:
3196:
3191:
3189:
3174:crossing number
3162:
3160:Knot tabulation
3152:
3074:
3068:
3038:
3037:
3006:
3005:
2974:
2973:
2942:
2937:
2936:
2913:
2908:
2907:
2876:
2871:
2870:
2847:
2842:
2841:
2801:
2796:
2795:
2768:
2763:
2762:
2702:
2701:
2678:
2673:
2672:
2637:
2632:
2631:
2593:
2592:
2548:Suspended knots
2524:
2519:
2518:
2495:
2490:
2489:
2466:
2461:
2460:
2453:
2429:
2408:Borromean rings
2400:
2399:
2398:
2397:
2396:
2393:Borromean rings
2386:
2378:
2377:
2374:Borromean rings
2370:
2348:knot complement
2337:
2306:
2303:
2294:
2291:
2258:
2185:
2184:
2005:
1980:
1958:
1947:
1946:
1921:
1920:
1886:
1885:
1854:
1853:
1832:
1827:
1826:
1805:
1800:
1799:
1768:
1755:
1742:
1737:
1736:
1683:
1681:Knot polynomial
1677:
1653:knot complement
1647:, which is the
1638:tricolorability
1622:
1610:
1603:
1601:Knot invariants
1534:
1506:J. W. Alexander
1502:
1496:
1472:reduced diagram
1417:
1344:
1343:
1319:
1314:
1313:
1306:ambient isotopy
1304:is known as an
1286:
1285:
1264:
1242:
1231:
1230:
1207:
1196:
1195:
1155:
1154:
1131:
1126:
1125:
1102:
1076:
1075:
1052:
1041:
1040:
1003:
1002:
979:
946:
935:
934:
909:
904:
903:
882:
877:
876:
818:
803:
790:
782:
781:
757:
741:
730:
729:
706:
691:
680:
679:
652:
639:
634:
633:
575:
574:
551:
522:
521:
491:
490:
475:
474:
473:
472:
468:
467:
466:
458:
457:
446:
363:knot invariants
328:J. W. Alexander
267:Celtic knotwork
255:Borromean rings
220:
214:
146:ambient isotopy
122:
117:
116:
93:
88:
87:
84:Euclidean space
28:
23:
22:
15:
12:
11:
5:
5840:
5838:
5830:
5829:
5824:
5814:
5813:
5807:
5806:
5804:
5803:
5791:
5778:
5775:
5774:
5772:
5771:
5769:Surgery theory
5766:
5761:
5756:
5751:
5746:
5741:
5736:
5731:
5726:
5721:
5716:
5711:
5706:
5701:
5696:
5691:
5685:
5683:
5679:
5678:
5676:
5675:
5670:
5668:Skein relation
5665:
5660:
5655:
5650:
5645:
5640:
5634:
5632:
5623:
5622:
5620:
5619:
5613:Unknotting no.
5610:
5605:
5600:
5599:
5598:
5588:
5583:
5582:
5581:
5576:
5571:
5566:
5561:
5551:
5546:
5541:
5536:
5531:
5526:
5521:
5516:
5511:
5506:
5505:
5504:
5494:
5489:
5488:
5487:
5477:
5472:
5466:
5464:
5458:
5457:
5455:
5454:
5448:
5439:
5433:
5424:
5418:
5409:
5405:
5399:
5395:
5389:
5385:
5379:
5375:
5368:
5366:
5360:
5359:
5357:
5356:
5351:
5350:
5349:
5344:
5333:
5331:
5325:
5324:
5322:
5321:
5316:
5310:
5301:
5295:
5286:
5280:
5274:
5268:
5264:
5258:
5254:
5248:
5244:
5238:
5234:
5230:
5226:
5222:
5218:
5212:
5208:
5202:
5198:
5191:
5189:
5183:
5182:
5168:
5166:
5165:
5158:
5151:
5143:
5137:
5136:
5126:
5116:
5110:
5104:
5098:
5095:The Knot Atlas
5092:
5079:
5076:
5075:
5074:
5071:Andrew Ranicki
5064:
5058:
5039:10.1086/349764
5033:(4): 461â474,
5022:
5000:
4997:
4996:
4995:
4987:
4986:External links
4984:
4983:
4982:
4977:
4960:
4959:
4958:
4950:
4931:
4928:
4927:
4926:
4921:
4906:
4901:
4883:
4878:
4860:
4854:
4834:
4829:
4811:Burde, Gerhard
4799:Lickorish 1997
4786:
4783:
4781:
4778:
4775:
4774:
4757:
4700:
4649:
4616:
4610:978-0691049380
4609:
4600:10.1.1.64.4359
4577:
4565:
4553:
4549:Weisstein 2013
4541:
4515:
4494:
4493:
4491:
4488:
4487:
4486:
4469:(3): 501â526,
4455:
4430:(3): 351â399,
4416:Witten, Edward
4412:
4388:
4364:
4350:
4329:
4324:
4311:
4300:(2): 229â235,
4289:
4268:
4262:
4241:
4228:
4209:
4177:
4164:
4138:
4132:
4111:
4082:Levine, Jerome
4078:
4072:
4051:
4017:
3966:
3925:
3908:(3): 452â466,
3886:
3881:
3859:
3829:
3824:
3800:
3783:(3): 402â413,
3769:
3731:
3708:
3697:(2): 423â472,
3681:
3664:(3): 339â351,
3650:
3621:Weeks, Jeffrey
3616:
3569:
3564:
3542:
3539:
3537:
3534:
3533:
3532:
3527:
3522:
3517:
3512:
3507:
3502:
3497:
3495:Molecular knot
3492:
3485:
3482:
3464:Main article:
3461:
3458:
3397:Main article:
3394:
3391:
3371:Main article:
3368:
3365:
3352:
3348:
3342:
3335:
3307:
3304:
3212:... (sequence
3151:
3148:
3070:Main article:
3067:
3064:
3051:
3048:
3045:
3025:
3022:
3019:
3016:
3013:
2993:
2990:
2987:
2984:
2981:
2970:natural number
2951:
2946:
2922:
2917:
2885:
2880:
2856:
2851:
2816:
2813:
2810:
2805:
2777:
2772:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2712:
2709:
2687:
2682:
2661:Haefliger 1962
2646:
2641:
2615:
2612:
2609:
2606:
2603:
2600:
2533:
2528:
2504:
2499:
2475:
2470:
2452:
2449:
2428:
2425:
2387:
2380:
2379:
2371:
2364:
2363:
2362:
2361:
2360:
2336:
2333:
2329:Lickorish 1997
2308:
2307:
2304:
2297:
2295:
2292:
2285:
2279:
2278:
2265:
2261:
2257:
2254:
2251:
2248:
2245:
2242:
2239:
2236:
2233:
2230:
2227:
2224:
2221:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2195:
2192:
2171:
2170:
2127:
2126:
2083:
2082:
2068:) +
2058:) =
2036:skein relation
2032:
2031:
2020:
2017:
2012:
2008:
2004:
2001:
1998:
1995:
1992:
1987:
1983:
1979:
1976:
1973:
1970:
1965:
1961:
1957:
1954:
1944:
1928:
1908:
1905:
1902:
1899:
1896:
1893:
1870:
1867:
1864:
1861:
1839:
1835:
1812:
1808:
1775:
1771:
1767:
1762:
1758:
1754:
1749:
1745:
1719:Lickorish 1997
1717:coefficients (
1687:knot invariant
1679:Main article:
1676:
1673:
1661:Lickorish 1997
1630:Lickorish 1997
1620:Knot invariant
1618:Main article:
1613:Henry Segerman
1602:
1599:
1595:Lickorish 1997
1591:Sossinsky 2002
1584:
1583:
1579:
1578:
1570:
1569:
1566:
1562:
1561:
1554:
1533:
1532:
1529:
1526:
1522:
1498:Main article:
1495:
1492:
1416:
1413:
1381:Wolfgang Haken
1357:
1354:
1351:
1328:
1323:
1293:
1271:
1267:
1263:
1260:
1257:
1254:
1249:
1245:
1241:
1238:
1216:
1211:
1206:
1203:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1140:
1135:
1111:
1106:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1061:
1056:
1051:
1048:
1028:
1025:
1022:
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1016:
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945:
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582:
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541:
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532:
529:
498:
487:Sossinsky 2002
470:
469:
460:
459:
451:
450:
449:
448:
447:
445:
442:
406:physical knots
399:Floer homology
395:quantum groups
375:Sossinsky 2002
216:Main article:
213:
210:
162:knot invariant
131:
126:
102:
97:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5839:
5828:
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5802:
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5792:
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5730:
5727:
5725:
5722:
5720:
5717:
5715:
5712:
5710:
5707:
5705:
5704:Conway sphere
5702:
5700:
5697:
5695:
5692:
5690:
5687:
5686:
5684:
5680:
5674:
5671:
5669:
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5664:
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5499:
5498:
5495:
5493:
5490:
5486:
5483:
5482:
5481:
5478:
5476:
5475:Arf invariant
5473:
5471:
5468:
5467:
5465:
5463:
5459:
5443:
5440:
5428:
5425:
5413:
5410:
5403:
5400:
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5380:
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5120:
5119:KnotData.html
5117:
5114:
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5007:
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4989:
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4907:
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4808:
4807:
4806:
4804:
4803:Cromwell 2004
4800:
4796:
4792:
4784:
4779:
4771:
4767:
4761:
4758:
4753:
4749:
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4723:
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4527:
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4509:
4505:
4499:
4496:
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4480:
4476:
4472:
4468:
4464:
4460:
4456:
4453:
4449:
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4437:
4433:
4429:
4425:
4421:
4417:
4413:
4402:
4398:
4394:
4389:
4377:
4373:
4369:
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4347:
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4339:
4335:
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4327:
4321:
4317:
4312:
4308:
4303:
4299:
4295:
4290:
4286:
4282:
4278:
4274:
4269:
4265:
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4255:
4251:
4247:
4242:
4239:
4235:
4231:
4225:
4221:
4217:
4216:
4210:
4207:
4203:
4198:
4193:
4189:
4185:
4184:
4178:
4175:
4171:
4167:
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4157:
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4149:
4148:
4143:
4139:
4135:
4129:
4125:
4121:
4117:
4112:
4109:
4105:
4101:
4097:
4093:
4089:
4088:
4083:
4079:
4075:
4069:
4065:
4061:
4057:
4052:
4049:
4045:
4041:
4037:
4033:
4029:
4028:
4023:
4018:
4015:
4011:
4007:
4003:
3999:
3995:
3990:
3985:
3981:
3977:
3976:
3971:
3967:
3964:
3960:
3956:
3952:
3948:
3944:
3940:
3936:
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3930:
3926:
3923:
3919:
3915:
3911:
3907:
3903:
3899:
3895:
3891:
3887:
3884:
3878:
3874:
3870:
3869:
3864:
3863:Flapan, Erica
3860:
3856:
3851:
3847:
3843:
3839:
3835:
3830:
3827:
3821:
3817:
3813:
3809:
3805:
3801:
3798:
3794:
3790:
3786:
3782:
3778:
3774:
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3767:
3763:
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3700:
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3639:
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3606:
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3598:
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3561:
3557:
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3544:
3540:
3535:
3531:
3528:
3526:
3523:
3521:
3518:
3516:
3513:
3511:
3510:Ribbon theory
3508:
3506:
3503:
3501:
3498:
3496:
3493:
3491:
3488:
3487:
3483:
3481:
3479:
3474:
3472:
3467:
3459:
3457:
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3392:
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3362:
3358:
3333:
3329:
3325:
3321:
3317:
3313:
3305:
3303:
3301:
3297:
3293:
3289:
3285:
3280:
3278:
3277:Alain Caudron
3274:
3270:
3269:Kenneth Perko
3266:
3262:
3258:
3254:
3250:
3245:
3243:
3238:
3236:
3232:
3227:
3225:
3221:
3216:
3187:
3183:
3179:
3175:
3166:
3161:
3157:
3149:
3147:
3144:
3139:
3137:
3133:
3132:Schubert 1949
3129:
3125:
3121:
3120:
3114:
3110:
3105:
3103:
3099:
3095:
3091:
3090:connected sum
3087:
3078:
3073:
3065:
3063:
3049:
3046:
3043:
3023:
3020:
3017:
3014:
3011:
2991:
2988:
2985:
2982:
2979:
2971:
2967:
2949:
2920:
2905:
2901:
2883:
2869:embedded in
2854:
2839:
2834:
2832:
2814:
2811:
2808:
2793:
2775:
2761:
2759:
2753:
2751:
2747:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2685:
2670:
2666:
2662:
2644:
2629:
2626:-spheres in 6
2610:
2607:
2604:
2601:
2590:
2586:
2582:
2580:
2576:
2572:
2568:
2564:
2560:
2555:
2553:
2549:
2531:
2502:
2473:
2458:
2450:
2448:
2446:
2442:
2437:
2433:
2426:
2424:
2422:
2416:
2413:
2409:
2405:
2394:
2390:
2384:
2375:
2368:
2359:
2357:
2353:
2349:
2345:
2341:
2334:
2332:
2330:
2326:
2322:
2318:
2313:
2312:mirror images
2301:
2296:
2289:
2284:
2282:
2263:
2259:
2255:
2252:
2249:
2243:
2240:
2237:
2231:
2228:
2225:
2222:
2190:
2183:
2182:
2181:
2178:
2176:
2167:
2162:
2159:
2154:
2149:
2144:
2139:
2136:
2135:
2134:
2132:
2123:
2118:
2115:
2110:
2105:
2100:
2095:
2092:
2091:
2090:
2088:
2079:
2074:
2071:
2066:
2061:
2056:
2051:
2048:
2047:
2046:
2044:
2039:
2037:
2018:
2010:
2006:
1999:
1996:
1993:
1985:
1981:
1974:
1971:
1963:
1959:
1952:
1945:
1942:
1926:
1906:
1903:
1897:
1891:
1884:
1883:
1882:
1865:
1859:
1837:
1833:
1810:
1806:
1793:
1789:
1773:
1769:
1765:
1760:
1756:
1752:
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1734:
1729:
1727:
1722:
1720:
1716:
1712:
1708:
1704:
1700:
1696:
1692:
1688:
1682:
1674:
1672:
1670:
1666:
1662:
1658:
1654:
1650:
1646:
1641:
1639:
1635:
1631:
1627:
1621:
1614:
1607:
1600:
1598:
1596:
1592:
1580:
1576:
1571:
1567:
1564:
1563:
1559:
1555:
1552:
1547:
1543:
1542:
1539:
1530:
1527:
1524:
1523:
1521:
1519:
1515:
1511:
1507:
1501:
1493:
1491:
1489:
1485:
1481:
1477:
1473:
1468:
1466:
1462:
1458:
1457:link diagrams
1454:
1450:
1449:knot diagrams
1446:
1442:
1438:
1434:
1426:
1421:
1415:Knot diagrams
1414:
1412:
1410:
1406:
1405:Marc Lackenby
1402:
1398:
1395:, called the
1394:
1390:
1386:
1382:
1378:
1374:
1369:
1355:
1352:
1349:
1326:
1309:
1307:
1291:
1269:
1265:
1261:
1255:
1252:
1247:
1243:
1236:
1214:
1204:
1201:
1181:
1178:
1172:
1169:
1166:
1160:
1138:
1109:
1099:
1093:
1090:
1087:
1081:
1059:
1049:
1046:
1023:
1020:
1017:
1011:
1008:
986:
970:
967:
964:
958:
953:
943:
940:
932:
914:
910:
887:
883:
859:
856:
853:
850:
847:
825:
810:
800:
795:
791:
778:
762:
758:
754:
746:
742:
735:
713:
698:
688:
685:
678:
677:homeomorphism
675:
657:
653:
649:
644:
640:
631:
626:
624:
620:
601:
595:
592:
586:
580:
558:
542:
539:
536:
530:
527:
520:
516:
512:
496:
488:
484:
480:
464:
455:
443:
441:
439:
435:
431:
427:
426:topoisomerase
423:
419:
415:
411:
407:
402:
400:
396:
392:
388:
384:
380:
379:Edward Witten
376:
372:
371:Vaughan Jones
368:
364:
360:
356:
352:
348:
343:
341:
337:
333:
329:
325:
320:
318:
314:
310:
306:
302:
298:
294:
290:
286:
279:
274:
270:
268:
264:
263:Book of Kells
260:
256:
252:
248:
244:
240:
236:
229:
228:Book of Kells
224:
219:
211:
209:
207:
204:embedded in (
203:
201:
196:
195:
190:
185:
183:
179:
174:
172:
168:
164:
163:
158:
153:
149:
147:
129:
100:
85:
81:
77:
73:
69:
65:
61:
57:
48:
41:
37:
32:
19:
5798:
5786:
5714:Double torus
5699:Braid theory
5514:Crossing no.
5509:Crosscap no.
5195:Figure-eight
5169:
5088:
5084:
5030:
5026:
5017:
5013:
4968:
4944:, Elsevier,
4941:
4911:
4891:
4868:
4848:. Springer.
4845:
4819:
4791:Rolfsen 1976
4788:
4769:
4760:
4717:
4713:
4703:
4669:(12): 2353.
4666:
4662:
4652:
4625:
4619:
4586:
4580:
4568:
4556:
4544:
4534:, retrieved
4532:, 2021-02-03
4524:
4518:
4508:Haken (1962)
4498:
4466:
4462:
4427:
4423:
4404:. Retrieved
4396:
4380:. Retrieved
4375:
4333:
4315:
4297:
4293:
4276:
4272:
4245:
4214:
4190:(2): 262â6,
4187:
4181:
4146:
4115:
4094:(1): 15â50,
4091:
4085:
4055:
4034:(4): 33â48,
4031:
4025:
4020:Hoste, Jim;
3989:math/9712269
3979:
3973:
3938:
3932:
3905:
3901:
3897:
3893:
3867:
3837:
3833:
3807:
3780:
3776:
3744:(4): 56â63,
3741:
3735:
3716:
3694:
3688:
3661:
3657:
3628:
3624:
3578:
3574:
3551:
3548:Adams, Colin
3475:
3469:
3454:
3450:
3447:
3440:
3433:
3423:
3421:
3402:
3384:
3320:Dale Rolfsen
3309:
3281:
3246:
3239:
3228:
3171:
3140:
3135:
3123:
3118:
3106:
3101:
3093:
3089:
3085:
3083:
3066:Adding knots
2965:
2935:embedded in
2903:
2899:
2837:
2835:
2757:
2754:
2668:
2627:
2584:
2578:
2570:
2566:
2562:
2558:
2556:
2454:
2445:ribbon knots
2438:
2434:
2430:
2417:
2401:
2338:
2324:
2309:
2280:
2179:
2174:
2172:
2160:
2157:
2147:
2137:
2128:
2116:
2113:
2103:
2093:
2084:
2072:
2069:
2059:
2049:
2043:trefoil knot
2040:
2033:
1797:
1732:
1730:
1723:
1710:
1684:
1665:Rolfsen 1976
1642:
1634:Rolfsen 1976
1623:
1587:
1537:
1517:
1503:
1483:
1479:
1475:
1471:
1469:
1467:in 3-space.
1456:
1448:
1441:Rolfsen 1976
1436:
1430:
1372:
1370:
1310:
930:
779:
629:
628:The idea of
627:
476:
438:Collins 2006
403:
344:
321:
282:
253:, while the
247:endless knot
232:
205:
199:
192:
186:
175:
160:
154:
150:
59:
53:
40:trefoil knot
36:trivial knot
5822:Knot theory
5549:Linking no.
5470:Alternating
5271:Conway knot
5251:Carrick mat
5205:Three-twist
5170:Knot theory
4720:(1): 8889.
4512:Hass (1998)
3871:, Outlook,
3834:Math. Comp.
3631:(1): 1â56,
3417:Conway 1970
3300:Burton 2020
3296:prime knots
3257:Conway 1970
3113:associative
3109:commutative
3094:composition
2972:. Both the
2906:-copies of
2665:Levine 1965
2589:Zeeman 1963
2561:-sphere in
2441:slice knots
479:dimensional
430:Flapan 2000
349:introduced
301:Lord Kelvin
297:Silver 2006
249:appears in
171:knot groups
60:knot theory
5816:Categories
5709:Complement
5673:Tabulation
5630:operations
5554:Polynomial
5544:Link group
5539:Knot group
5502:Invertible
5480:Bridge no.
5462:Invariants
5392:Cinquefoil
5261:Perko pair
5187:Hyperbolic
4842:Fox, Ralph
4795:Adams 2004
4676:2106.03925
4536:2021-02-03
4279:(2): 158.
3970:Hass, Joel
3941:: 89â120,
3536:References
3471:Gauss code
3466:Gauss code
3460:Gauss code
3357:Perko pair
3332:torus knot
3328:twist knot
3292:Hoste 2005
3273:Perko 1974
3265:Perko pair
3235:Hoste 2005
3224:Adams 2004
3186:Hoste 2005
3182:Hoste 2005
3154:See also:
3098:Adams 2004
2552:spun knots
2356:Adams 2004
1701:, and the
1691:polynomial
1689:that is a
1655:, and the
1645:knot group
1626:Adams 2004
1597:, ch. 1).
1593:, ch. 3) (
1433:one-to-one
1401:Hoste 2005
1377:Algorithms
931:equivalent
483:Adams 2004
418:Simon 1986
332:knot group
157:complexity
42:(below it)
5603:Stick no.
5559:Alexander
5497:Chirality
5442:Solomon's
5402:Septafoil
5329:Satellite
5289:Whitehead
5215:Stevedore
5113:Knoutilus
5107:Knotscape
5055:144988108
4635:1304.6053
4595:CiteSeerX
4490:Footnotes
4397:MathWorld
4378:. Wolfram
4376:MathWorld
4360:118682559
4174:122824389
3955:0025-5874
3900:-space",
3797:120452571
3773:Dehn, Max
3678:120218312
3613:119320887
3588:1208.5742
3124:composite
2723:−
2714:−
2608:−
2404:geodesics
2321:Dehn 1914
2087:Hopf link
1986:−
1838:−
1761:−
1582:Type III
1437:crossings
1389:Hass 1998
1385:Hass 1998
1229:; and c)
1205:∈
1100:∈
1050:∈
1012:∈
977:→
959:×
857:≤
851:≤
816:→
704:→
689::
549:→
531::
515:injective
373:in 1984 (
76:embedding
5788:Category
5658:Mutation
5626:Notation
5579:Kauffman
5492:Brunnian
5485:2-bridge
5354:Knot sum
5285:(12n242)
5101:KnotPlot
5085:KnotInfo
5020:: 94â105
5008:(1867),
4889:(2013),
4869:On Knots
4866:(1987),
4844:(1977).
4817:(1985),
4752:37264056
4743:10235088
4663:Symmetry
4452:14951363
4418:(1989),
4370:(2013).
4294:Topology
4144:(1997),
4048:18027155
3865:(2000),
3766:16596880
3690:Topology
3550:(2004),
3484:See also
3117:knot is
3102:oriented
3086:knot sum
3072:Knot sum
3004:and the
2964:, where
2581:-spheres
2412:horoball
2317:Max Dehn
1568:Type II
1480:nugatory
1194:for all
428:on DNA (
336:homology
324:Max Dehn
317:topology
56:topology
5800:Commons
5719:Fibered
5617:problem
5586:Pretzel
5564:Bracket
5382:Trefoil
5319:L10a140
5279:(11n42)
5273:(11n34)
5241:Endless
4999:History
4930:Surveys
4797:) and (
4722:Bibcode
4681:Bibcode
4640:Bibcode
4483:1970538
4432:Bibcode
4401:Wolfram
4238:0515288
4206:2040074
4108:1970561
4014:7381505
3994:Bibcode
3963:0160196
3922:1970208
3842:Bibcode
3746:Bibcode
3647:2001854
3605:3342136
3541:Sources
3413:tangles
3218:in the
3215:A002863
2760:-sphere
2389:SnapPea
1919:(where
1715:integer
1651:of the
1520:, are:
422:Tangles
307:led to
245:). The
212:History
5764:Writhe
5734:Ribbon
5569:HOMFLY
5412:Unlink
5372:Unknot
5347:Square
5342:Granny
5133:Tables
5129:Regina
5053:
5047:228151
5045:
4975:
4948:
4919:
4899:
4876:
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3603:
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3136:smooth
3062:case.
2131:unlink
1941:unknot
1697:, the
1565:Type I
1478:(also
1393:unknot
845:
831:
623:braids
414:chiral
259:Celtic
80:circle
72:unknot
5754:Twist
5739:Slice
5694:Berge
5682:Other
5653:Flype
5591:Prime
5574:Jones
5534:Genus
5364:Torus
5178:links
5174:knots
5061:Movie
5051:S2CID
5043:JSTOR
4820:Knots
4671:arXiv
4630:arXiv
4479:JSTOR
4448:S2CID
4406:8 May
4382:8 May
4356:S2CID
4202:JSTOR
4170:S2CID
4104:JSTOR
4044:S2CID
4010:S2CID
3984:arXiv
3918:JSTOR
3793:S2CID
3674:S2CID
3643:JSTOR
3609:S2CID
3583:arXiv
3428:digon
3351:to 10
3128:prime
3119:prime
2968:is a
2898:. An
2700:with
2325:Jones
1726:links
1713:with
728:with
619:links
511:Curve
239:tying
178:links
78:of a
68:knots
5759:Wild
5724:Knot
5628:and
5615:and
5596:list
5427:Hopf
5176:and
5027:Isis
4973:ISBN
4946:ISBN
4917:ISBN
4897:ISBN
4874:ISBN
4850:ISBN
4825:ISBN
4748:PMID
4605:ISBN
4408:2013
4384:2013
4346:ISBN
4320:ISBN
4258:ISBN
4224:ISBN
4160:ISBN
4128:ISBN
4092:1982
4068:ISBN
3951:ISSN
3877:ISBN
3820:ISBN
3762:PMID
3560:ISBN
3403:The
3385:The
3314:and
3220:OEIS
3158:and
3115:. A
3111:and
3047:>
3015:>
2729:>
2550:and
2443:and
2372:The
2156:) +
2146:) =
2112:) +
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1733:i.e.
1508:and
1461:link
1455:and
1453:knot
929:are
902:and
621:and
517:and
397:and
389:and
237:and
5744:Sum
5265:161
5263:(10
5035:doi
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4738:PMC
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4506:by
4471:doi
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