25:
1663:
90:
1675:
174:."). Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics. Knot invariants are thus used in knot classification, both in "enumeration" and "duplication removal".
195:
quantity defined on knot diagrams. Thus if two knot diagrams differ with respect to some knot invariant, they must represent different knots. However, as is generally the case with topological invariants, if two knot diagrams share the same values with respect to a knot invariant, then we still
224:, which are currently among the most useful invariants for distinguishing knots from one another, though currently it is not known whether there exists a knot polynomial which distinguishes all knots from each other. However, there are invariants which distinguish the
370:
There is also growing interest from both knot theorists and scientists in understanding "physical" or geometric properties of knots and relating it to topological invariants and knot type. An old result in this direction is the
343:
of the knot. It has been proven effective in deducing new results about the classical invariants. Along a different line of study, there is a combinatorially defined cohomology theory of knots called
250:
Historically, many of the early knot invariants are not defined by first selecting a diagram but defined intrinsically, which can make computing some of these invariants a challenge. For example,
239:
Other invariants can be defined by considering some integer-valued function of knot diagrams and taking its minimum value over all possible diagrams of a given knot. This category includes the
547:
464:
412:
54:
809:
312:
is an invariant for these knots and links. Volume, and other hyperbolic invariants, have proven very effective, utilized in some of the extensive efforts at
846:
363:
and Lev
Rozansky have since defined several other related cohomology theories whose Euler characteristics recover other classical invariants.
128:. Some invariants are indeed numbers (algebraic), but invariants can range from the simple, such as a yes/no answer, to those as complex as a
1040:
1608:
1527:
969:
950:
931:
743:
706:
681:
653:
629:
606:
76:
1074:
688:
is a mathematical property or quantity associated with a knot that does not change as we perform triangular moves on the knot.
1517:
1522:
1393:
240:
1094:
1156:
502:
182:
is a quantity defined on the set of all knots, which takes the same value for any two equivalent knots. For example, a
37:
420:
798:
660:
is a function from the set of knots to some other set whose value depends only on the equivalence class of the knot."
47:
41:
33:
372:
278:
1162:
270:
1226:
1221:
1033:
293:
is also a complete invariant in this sense but it is difficult to determine if two quandles are isomorphic. The
58:
1701:
762:
367:
gave a representation theoretic interpretation of
Khovanov homology by categorifying quantum group invariants.
309:
1354:
301:
1679:
1568:
1537:
255:
1398:
571:
568: – Numerical invariant that describes the linking of two closed curves in three-dimensional space
388:
1667:
1438:
1026:
835:
340:
336:
117:
1475:
1458:
294:
1496:
1443:
1057:
1053:
900:
757:
364:
113:
1593:
1542:
1492:
1448:
1408:
1403:
1321:
965:
946:
927:
892:
779:
739:
702:
677:
649:
625:
602:
344:
286:
266:
229:
205:
1628:
1453:
1349:
1084:
884:
771:
481:
360:
348:
324:
221:
1588:
1552:
1487:
1433:
1388:
1381:
1271:
1183:
1066:
376:
332:
320:
305:
274:
262:
217:
209:
129:
121:
254:
is particularly tricky to compute, but can be effective (for instance, in distinguishing
574: – Type of invariant in Knot theory (or Vassiliev or Vassiliev–Goussarov invariant)
1648:
1547:
1509:
1428:
1216:
1208:
1168:
1011:
565:
328:
233:
556:, which is the length of unit-diameter rope needed to realize a particular knot type.
1695:
1583:
1371:
1364:
1359:
244:
192:
125:
1598:
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356:
290:
201:
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1120:
1112:
1104:
756:
Horner, Kate; Miller, Mark; Steedb, Jonathan; Sutcliffe, Paul (August 20, 2016).
1613:
1376:
1150:
1130:
1049:
1018:
352:
243:, which is the minimum number of crossings for any diagram of the knot, and the
105:
101:
216:-colorability) is a particularly simple and common example. Other examples are
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1413:
1243:
1140:
553:
313:
282:
251:
183:
93:
943:
The Knot Book: an
Elementary Introduction to the Mathematical Theory of Knots
896:
273:
in the sense that it distinguishes the given knot from all other knots up to
200:
From the modern perspective, it is natural to define a knot invariant from a
1638:
1306:
783:
580: – Smallest number of edges of an equivalent polygonal path for a knot
1623:
1233:
204:. Of course, it must be unchanged (that is to say, invariant) under the
904:
872:
775:
89:
247:, which is the minimum number of bridges for any diagram of the knot.
1643:
1291:
1251:
987:
225:
888:
351:. This has recently been shown to be useful in obtaining bounds on
1532:
281:. Some invariants associated with the knot complement include the
88:
16:
Function of a knot that takes the same value for equivalent knots
1603:
1006:
1022:
18:
622:
269:) is known to be a "complete invariant" of the knot by the
873:"On Irreducible 3-Manifolds Which are Sufficiently Large"
964:(2nd rev. and extended ed.). New York: De Gruyter.
699:
505:
423:
391:
1561:
1505:
1340:
1242:
1207:
1065:
945:(Repr., with corr ed.). Providence, RI: AMS.
541:
458:
406:
304:, the hyperbolic structure on the complement of a
112:is a quantity (in a broad sense) defined for each
542:{\displaystyle \oint _{K}\kappa \,ds>4\pi .\,}
319:In recent years, there has been much interest in
736:Understanding Topology: A Practical Introduction
709:. "Likewise," with knot invariants, "a quantity
459:{\displaystyle \oint _{K}\kappa \,ds\leq 4\pi ,}
46:but its sources remain unclear because it lacks
672:Messer, Robert and Straffin, Philip D. (2018).
189:
176:
96:are organized by the crossing number invariant.
1034:
770:(23). Royal Society of Chemistry: 6409–6658.
496:is an unknot. Therefore, for knotted curves,
8:
196:cannot conclude that the knots are the same.
1041:
1027:
1019:
960:Burde, Gerhard; Zieschang, Heiner (2002).
738:, p.245. Johns Hopkins University Press.
668:
666:
640:
638:
538:
519:
510:
504:
437:
428:
422:
398:
394:
393:
390:
120:knots. The equivalence is often given by
77:Learn how and when to remove this message
601:, p.113. American Mathematical Society.
552:An example of a "physical" invariant is
843:www.dk-compmath.jku.at/people/mhodorog/
676:, p.50. American Mathematical Society.
616:
614:
590:
297:can also work as a complete invariant.
852:from the original on November 19, 2022
834:Hodorog, Mădălina (February 2, 2010).
815:from the original on November 19, 2022
648:, p.7. American Mathematical Society.
986:Cha, Jae Choon; Livingston, Charles.
7:
1674:
988:"KnotInfo: Table of Knot Invariants"
136:is a rule that assigns to any knot
347:whose Euler characteristic is the
14:
758:"Knot theory in modern chemistry"
1673:
1662:
1661:
799:"An Introduction to Knot Theory"
797:Skerritt, Matt (June 27, 2003).
407:{\displaystyle \mathbb {R} ^{3}}
191:Typically a knot invariant is a
23:
1528:Dowker–Thistlethwaite notation
871:Waldhausen, Friedhelm (1968).
624:, p.67. Springer Netherlands.
355:whose earlier proofs required
228:from all other knots, such as
1:
713:for any two equivalent links
620:Ricca, Renzo L.; ed. (2012).
941:Adams, Colin Conrad (2004).
697:Morishita, Masanori (2011).
597:Schultens, Jennifer (2014).
599:Introduction to 3-manifolds
308:is unique, which means the
1718:
323:invariants of knots which
1657:
1518:Alexander–Briggs notation
701:, p.16. Springer London.
644:Purcell, Jessica (2020).
763:Chemical Society Reviews
32:This article includes a
1609:List of knots and links
1157:Kinoshita–Terasaka knot
926:. Providence, RI: AMS.
734:Ault, Shaun V. (2018).
329:Heegaard Floer homology
327:well-known invariants.
289:of the complement. The
61:more precise citations.
922:Rolfsen, Dale (2003).
646:Hyperbolic Knot Theory
543:
460:
408:
302:Mostow–Prasad rigidity
208:("triangular moves").
198:
188:
116:which is the same for
97:
1399:Finite type invariant
877:Annals of Mathematics
572:Finite type invariant
544:
461:
409:
271:Gordon–Luecke theorem
92:
503:
421:
389:
341:Alexander polynomial
337:Euler characteristic
263:complement of a knot
186:is a knot invariant.
162:are equivalent then
124:but can be given by
1569:Alexander's theorem
836:"Basic Knot Theory"
375:states that if the
373:Fáry–Milnor theorem
295:peripheral subgroup
234:knot Floer homology
776:10.1039/c6cs00448b
539:
456:
404:
365:Catharina Stroppel
285:which is just the
206:Reidemeister moves
98:
34:list of references
1689:
1688:
1543:Reidemeister move
1409:Khovanov homology
1404:Hyperbolic volume
345:Khovanov homology
310:hyperbolic volume
287:fundamental group
267:topological space
230:Khovanov homology
132:(for example, "a
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1629:Tait conjectures
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1163:(−2,3,7) pretzel
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711:inv(L) = inv(L')
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361:Mikhail Khovanov
349:Jones polynomial
222:Jones polynomial
218:knot polynomials
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57:this article by
48:inline citations
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924:Knots and Links
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889:10.2307/1970594
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377:total curvature
333:homology theory
314:knot tabulation
306:hyperbolic link
275:ambient isotopy
241:crossing number
210:Tricolorability
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130:homology theory
122:ambient isotopy
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38:related reading
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845:. p. 47.
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808:. p. 22.
806:carmamaths.org
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686:knot invariant
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1394:Crossing no.
1389:Crosscap no.
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1075:Figure-eight
1010:
995:. Retrieved
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883:(1): 56–88.
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856:November 19,
854:. Retrieved
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817:. Retrieved
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165:
152:
145:
138:
133:
109:
102:mathematical
99:
73:
64:
53:Please help
45:
1429:Linking no.
1350:Alternating
1151:Conway knot
1131:Carrick mat
1085:Three-twist
1050:Knot theory
992:Indiana.edu
353:slice genus
321:homological
142:a quantity
106:knot theory
94:Prime knots
59:introducing
1589:Complement
1553:Tabulation
1510:operations
1434:Polynomial
1424:Link group
1419:Knot group
1382:Invertible
1360:Bridge no.
1342:Invariants
1272:Cinquefoil
1141:Perko pair
1067:Hyperbolic
1007:Invariants
554:ropelength
414:satisfies
379:of a knot
325:categorify
283:knot group
252:knot genus
184:knot group
118:equivalent
1483:Stick no.
1439:Alexander
1377:Chirality
1322:Solomon's
1282:Septafoil
1209:Satellite
1169:Whitehead
1095:Stevedore
997:17 August
897:0003-486X
533:π
517:κ
508:∮
482:curvature
451:π
445:≤
435:κ
426:∮
104:field of
1696:Category
1668:Category
1538:Mutation
1506:Notation
1459:Kauffman
1372:Brunnian
1365:2-bridge
1234:Knot sum
1165:(12n242)
847:Archived
810:Archived
784:27868114
67:May 2019
1680:Commons
1599:Fibered
1497:problem
1466:Pretzel
1444:Bracket
1262:Trefoil
1199:L10a140
1159:(11n42)
1153:(11n34)
1121:Endless
905:1970594
585:Sources
490:, then
480:is the
339:is the
256:mutants
100:In the
55:improve
1644:Writhe
1614:Ribbon
1449:HOMFLY
1292:Unlink
1252:Unknot
1227:Square
1222:Granny
968:
949:
930:
903:
895:
782:
742:
705:
680:
652:
628:
605:
469:where
335:whose
226:unknot
168:) = φ(
1634:Twist
1619:Slice
1574:Berge
1562:Other
1533:Flype
1471:Prime
1454:Jones
1414:Genus
1244:Torus
1058:links
1054:knots
962:Knots
901:JSTOR
850:(PDF)
839:(PDF)
813:(PDF)
802:(PDF)
331:is a
212:(and
40:, or
1639:Wild
1604:Knot
1508:and
1495:and
1476:list
1307:Hopf
1056:and
999:2021
966:ISBN
947:ISBN
928:ISBN
893:ISSN
858:2022
821:2022
780:PMID
740:ISBN
719:and
703:ISBN
678:ISBN
650:ISBN
626:ISBN
603:ISBN
527:>
277:and
261:The
232:and
156:and
114:knot
108:, a
1624:Sum
1145:161
1143:(10
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885:doi
772:doi
684:"A
656:"A
484:at
385:in
300:By
258:).
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875:.
841:.
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778:.
768:45
766:.
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725:."
722:L'
665:^
637:^
613:^
359:.
316:.
236:.
178:A
170:K'
164:φ(
159:K'
144:φ(
44:,
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1276:1
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1256:1
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1125:4
1115:3
1113:6
1107:2
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1099:1
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1089:2
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1042:e
1035:t
1028:v
1015:.
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955:.
936:.
907:.
887::
860:.
823:.
786:.
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746:.
716:L
632:.
536:.
530:4
524:s
521:d
512:K
493:K
487:p
478:)
476:p
474:(
472:κ
454:,
448:4
442:s
439:d
430:K
400:3
395:R
382:K
214:n
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166:K
153:K
148:)
146:K
139:K
80:)
74:(
69:)
65:(
51:.
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