327:. Such a projection can be pictured as a finite number of (oriented) arcs in the plane, separated by the crossings of the projection. The fundamental group is generated by loops winding around each arc. Each crossing gives rise to a certain relation among the generators corresponding to the arcs meeting at the crossing.
508:
380:
Conditions (3) and (4) are essentially the
Wirtinger presentation condition, restated. Kervaire proved in dimensions 5 and larger that the above conditions are necessary and sufficient. Characterizing knot groups in dimension four is an open problem.
177:
306:
399:
89:
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347:
proved that an abstract group is the fundamental group of a knot exterior (in a perhaps high-dimensional sphere) if and only if all the following conditions are satisfied:
109:
571:
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223:, which does not make a difference for the purposes of the Wirtinger presentation.) The open subspace which is the complement of the knot,
597:
114:
262:
503:{\displaystyle \pi _{1}(\mathbb {R} ^{3}\backslash {\text{trefoil}})=\langle x,y\mid (xy)^{-1}yxy=x\rangle .}
366:
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183:
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24:
20:
563:
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316:. It is therefore interesting to understand this group in an accessible way.
219:. (Alternatively, the ambient space can also be taken to be the three-sphere
539:, Mathematics Lecture Series, vol. 7, Houston, TX: Publish or Perish,
589:
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584:, Series on Knots and Everything, vol. 52, World Scientific,
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331:Wirtinger presentations of high-dimensional knots
393:, a Wirtinger presentation can be shown to be
308:is an invariant of the knot in the sense that
172:{\displaystyle \{g_{1},g_{2},\ldots ,g_{k}\}.}
8:
494:
442:
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343:are known to have Wirtinger presentations.
323:is derived from a regular projection of an
301:{\displaystyle \pi _{1}(S^{3}\setminus K)}
610:, The Mathematical Association of America
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16:Group presentations useful in knot theory
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35:where the relations are of the form
211:is an embedding of the one-sphere
84:{\displaystyle wg_{i}w^{-1}=g_{j}}
14:
335:More generally, co-dimension two
194:with presentations of this form.
249:{\displaystyle S^{3}\setminus K}
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1:
582:Algebraic invariants of links
355:of the group is the integers.
111:is a word in the generators,
606:Livingston, Charles (1993),
256:is the knot complement. Its
198:Preliminaries and definition
215:in three-dimensional space
641:
580:Hillman, Jonathan (2012),
564:10.1007/978-3-0348-9227-8
362:of the group is trivial.
556:A survey of knot theory
554:Kawauchi, Akio (1996),
504:
376:of a single generator.
365:The group is finitely
321:Wirtinger presentation
302:
250:
173:
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29:Wirtinger presentation
590:10.1142/9789814407397
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184:complements of knots
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192:fundamental groups
182:observed that the
169:
101:
81:
573:978-3-0348-9953-6
546:978-0-914098-16-4
434:
372:The group is the
258:fundamental group
180:Wilhelm Wirtinger
104:{\displaystyle w}
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312:have isomorphic
310:equivalent knots
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537:Knots and links
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526:Further reading
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345:Michel Kervaire
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558:, Birkhäuser,
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374:normal closure
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353:abelianization
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533:Rolfsen, Dale
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325:oriented knot
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22:
607:
581:
555:
550:, section 3D
536:
391:trefoil knot
388:
379:
334:
320:
318:
220:
216:
212:
208:
203:
201:
33:presentation
31:is a finite
28:
25:group theory
18:
625:Knot theory
608:Knot Theory
314:knot groups
21:mathematics
520:Knot group
495:⟩
472:−
455:∣
443:⟨
429:∖
405:π
367:presented
290:∖
268:π
241:∖
148:…
61:−
619:Category
535:(1990),
514:See also
389:For the
385:Examples
360:homology
358:The 2nd
433:trefoil
341:spheres
188:3-space
596:
570:
543:
91:where
337:knots
190:have
594:ISBN
568:ISBN
541:ISBN
351:The
205:knot
27:, a
586:doi
560:doi
339:in
186:in
19:In
621::
592:,
566:,
319:A
202:A
588::
562::
498:.
492:x
489:=
486:y
483:x
480:y
475:1
468:)
464:y
461:x
458:(
452:y
449:,
446:x
440:=
437:)
424:3
419:R
414:(
409:1
369:.
296:)
293:K
285:3
281:S
277:(
272:1
244:K
236:3
232:S
221:S
217:R
213:S
209:K
167:.
164:}
159:k
155:g
151:,
145:,
140:2
136:g
132:,
127:1
123:g
119:{
99:w
77:j
73:g
69:=
64:1
57:w
51:i
47:g
43:w
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