585:
sufficiently small real 3-sphere centred at the singularity. If the singular point is isolated, then the real link is a one-dimensional manifold in a 3-dimensional space. Thus the link is of interest to knot theorists. And as you increase the dimensions of things you get these more general links. Some mention of braid monodromy would be nice too. I don't want to touch the article: it's very beautiful and my
Knowledge editing skills are next to nothing. If you need advice on the singularity theory then I can help you with that :o)
253:
243:
222:
191:
529:
collection of manifolds. This agrees with the historical usage of the word link. Is that what you're getting at? The knot theory articles on wikipedia all have that slant. This "link" definition is still a stub and biased towards plain-jane traditional 3-dimensional knot/link theory. Not much effort has been put into it yet. You're welcome to improve it.
605:
p.s. The milnor fibre is the wedged product of μ spheres where μ is the Milnor number of the singular function germ, i.e. the local multiplicity, i.e. the dimension of the local algebra. To compute this number you take the ring of function germs and quotient out by the
Jacobian ideal, i.e. the ideal
528:
Your example fits the basic definition of this article -- in fact the usage of "link" in singularity theory is derived from the manifold-theoretic usage of the word. I suppose "link" in the singularity-theoretic sense does not have to be a knotted collection of spheres but in general just a knotted
584:
It's getting better. I guess I'm just biased: I would love to see some mention of links in applications, e.g. singularity theory. For example a complex curve in the plane is a real two-dimensional body in four-dimensional space. The real link of a singularity on the curve is the intersection of a
512:
These are called the Milnor fibres. Then for small ε we intersect the milnor fibre with a sufficiently small hypersphere. The resulting intersection is called the link of the singularity. The topology of this link is of great importance. Moreover, varying ε along a closed path which does not pass
925:
IMO most of what you're talking about would be more appropriately placed in a topic like "fibred link". Links in general are not fibred so there's no monodromy to speak of -- there is a vague analogue in that you could talk about the action of the fundamental group of the link on the commutator
870:
507:
606:
generated by all first order partial derivatives. Since the function is assumed to have an isolated singularity it follows that this is a finite dimensional vector space: the local algebra. For an example, consider
153:
547:
You're quite right. Although I am feeling quite puzzled. I don't remember seeing the "more generally" section the last time I looked. There must have been a problem with my web browser, or my eyes. Sorry
407:
733:
744:
309:
665:
926:
subgroup or equivalently on the homotopy-fibre of the "abelianization map" from the link complement to the appropriate product of circles. But that's kind of a different topic.
147:
908:
418:
981:
299:
79:
44:
331:
First I'd like to say that it is an attractive article with some nice pictures. However, it should be mentioned that there are other uses of the word
976:
275:
85:
953:
266:
227:
946:
While not as scientifically "sexy" a subject as singularities, has there been any attempt to apply knot theory to chainmaille?
345:
168:
99:
30:
135:
104:
20:
74:
865:{\displaystyle {\mathcal {A}}_{f}={\frac {{\mathcal {O}}(x,y)}{\langle x^{2},y^{2}\rangle }}=\langle 1,x,y,xy\rangle \ .}
674:
202:
513:
through the origin gives a mapping from the homology of the link to itself, and induces what is called the monodromy.
65:
129:
915:
590:
553:
518:
125:
957:
109:
190:
911:
586:
549:
514:
208:
175:
609:
252:
949:
161:
55:
274:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
258:
70:
24:
242:
221:
878:
141:
51:
412:
Then we get a locally trivial fibration over the complex plane minus the origin. We consider
502:{\displaystyle F_{\epsilon }:=\{z\in \mathbb {C} ^{n}:f(z)=\epsilon ,\ \epsilon \neq 0\}\ .}
931:
571:
534:
970:
271:
248:
961:
935:
927:
919:
594:
575:
567:
557:
538:
530:
522:
910:, and so the link is homeomorphic to the wedged product of four spheres.
566:
I've touched it up a little. What do you think? Still needs some work.
332:
184:
15:
770:
751:
402:{\displaystyle f:(\mathbb {C} ^{n},0)\to (\mathbb {C} ,0)}
160:
881:
747:
677:
612:
421:
348:
728:{\displaystyle J_{f}=\langle x^{2},y^{2}\rangle \ .}
270:, a collaborative effort to improve the coverage of
902:
864:
727:
659:
501:
401:
33:for general discussion of the article's subject.
335:. For example, consider a smooth function germ
174:
8:
854:
827:
818:
792:
717:
691:
491:
435:
339:with an isolated singularity at the origin:
216:
188:
880:
812:
799:
769:
768:
765:
756:
750:
749:
746:
711:
698:
682:
676:
651:
638:
611:
450:
446:
445:
426:
420:
386:
385:
364:
360:
359:
347:
218:
7:
264:This article is within the scope of
738:The local aglebra is then given by
207:It is of interest to the following
23:for discussing improvements to the
660:{\displaystyle f(x,y)=x^{3}+y^{3}}
14:
982:Low-priority mathematics articles
667:. Then the Jacobian ideal is just
284:Knowledge:WikiProject Mathematics
287:Template:WikiProject Mathematics
251:
241:
220:
189:
45:Click here to start a new topic.
977:Stub-Class mathematics articles
304:This article has been rated as
891:
885:
787:
775:
628:
616:
468:
462:
396:
382:
379:
376:
355:
1:
278:and see a list of open tasks.
42:Put new text under old text.
962:17:23, 3 January 2012 (UTC)
936:01:09, 14 August 2008 (UTC)
920:22:31, 13 August 2008 (UTC)
595:22:14, 13 August 2008 (UTC)
576:21:59, 13 August 2008 (UTC)
558:21:51, 13 August 2008 (UTC)
539:21:37, 13 August 2008 (UTC)
523:20:16, 13 August 2008 (UTC)
50:New to Knowledge? Welcome!
998:
903:{\displaystyle \mu (f)=4}
333:link (and disambiguation)
303:
236:
215:
80:Be welcoming to newcomers
310:project's priority scale
267:WikiProject Mathematics
904:
866:
729:
661:
503:
403:
197:This article is rated
75:avoid personal attacks
905:
875:It follows then that
867:
730:
662:
504:
404:
100:Neutral point of view
879:
745:
675:
610:
419:
346:
290:mathematics articles
105:No original research
900:
862:
725:
657:
499:
399:
259:Mathematics portal
203:content assessment
86:dispute resolution
47:
25:Link (knot theory)
952:comment added by
858:
822:
721:
495:
481:
324:
323:
320:
319:
316:
315:
183:
182:
66:Assume good faith
43:
989:
964:
909:
907:
906:
901:
871:
869:
868:
863:
857:
823:
821:
817:
816:
804:
803:
790:
774:
773:
766:
761:
760:
755:
754:
734:
732:
731:
726:
720:
716:
715:
703:
702:
687:
686:
666:
664:
663:
658:
656:
655:
643:
642:
508:
506:
505:
500:
494:
480:
455:
454:
449:
431:
430:
408:
406:
405:
400:
389:
369:
368:
363:
292:
291:
288:
285:
282:
261:
256:
255:
245:
238:
237:
232:
224:
217:
200:
194:
193:
185:
179:
178:
164:
95:Article policies
16:
997:
996:
992:
991:
990:
988:
987:
986:
967:
966:
947:
944:
877:
876:
808:
795:
791:
767:
748:
743:
742:
707:
694:
678:
673:
672:
647:
634:
608:
607:
444:
422:
417:
416:
358:
344:
343:
329:
289:
286:
283:
280:
279:
257:
250:
230:
201:on Knowledge's
198:
121:
116:
115:
114:
91:
61:
12:
11:
5:
995:
993:
985:
984:
979:
969:
968:
943:
940:
939:
938:
899:
896:
893:
890:
887:
884:
873:
872:
861:
856:
853:
850:
847:
844:
841:
838:
835:
832:
829:
826:
820:
815:
811:
807:
802:
798:
794:
789:
786:
783:
780:
777:
772:
764:
759:
753:
736:
735:
724:
719:
714:
710:
706:
701:
697:
693:
690:
685:
681:
670:
669:
668:
654:
650:
646:
641:
637:
633:
630:
627:
624:
621:
618:
615:
600:
599:
598:
597:
579:
578:
563:
562:
561:
560:
542:
541:
510:
509:
498:
493:
490:
487:
484:
479:
476:
473:
470:
467:
464:
461:
458:
453:
448:
443:
440:
437:
434:
429:
425:
410:
409:
398:
395:
392:
388:
384:
381:
378:
375:
372:
367:
362:
357:
354:
351:
328:
325:
322:
321:
318:
317:
314:
313:
302:
296:
295:
293:
276:the discussion
263:
262:
246:
234:
233:
225:
213:
212:
206:
195:
181:
180:
118:
117:
113:
112:
107:
102:
93:
92:
90:
89:
82:
77:
68:
62:
60:
59:
48:
39:
38:
35:
34:
28:
13:
10:
9:
6:
4:
3:
2:
994:
983:
980:
978:
975:
974:
972:
965:
963:
959:
955:
954:67.142.178.20
951:
941:
937:
933:
929:
924:
923:
922:
921:
917:
913:
912:Dharma6662000
897:
894:
888:
882:
859:
851:
848:
845:
842:
839:
836:
833:
830:
824:
813:
809:
805:
800:
796:
784:
781:
778:
762:
757:
741:
740:
739:
722:
712:
708:
704:
699:
695:
688:
683:
679:
671:
652:
648:
644:
639:
635:
631:
625:
622:
619:
613:
604:
603:
602:
601:
596:
592:
588:
587:Dharma6662000
583:
582:
581:
580:
577:
573:
569:
565:
564:
559:
555:
551:
550:Dharma6662000
546:
545:
544:
543:
540:
536:
532:
527:
526:
525:
524:
520:
516:
515:Dharma6662000
496:
488:
485:
482:
477:
474:
471:
465:
459:
456:
451:
441:
438:
432:
427:
423:
415:
414:
413:
393:
390:
373:
370:
365:
352:
349:
342:
341:
340:
338:
334:
326:
311:
307:
301:
298:
297:
294:
277:
273:
269:
268:
260:
254:
249:
247:
244:
240:
239:
235:
229:
226:
223:
219:
214:
210:
204:
196:
192:
187:
186:
177:
173:
170:
167:
163:
159:
155:
152:
149:
146:
143:
140:
137:
134:
131:
127:
124:
123:Find sources:
120:
119:
111:
110:Verifiability
108:
106:
103:
101:
98:
97:
96:
87:
83:
81:
78:
76:
72:
69:
67:
64:
63:
57:
53:
52:Learn to edit
49:
46:
41:
40:
37:
36:
32:
26:
22:
18:
17:
948:— Preceding
945:
874:
737:
511:
411:
336:
330:
327:Too Specific
306:Low-priority
305:
265:
231:Low‑priority
209:WikiProjects
171:
165:
157:
150:
144:
138:
132:
122:
94:
19:This is the
942:Chainmaille
281:Mathematics
272:mathematics
228:Mathematics
148:free images
31:not a forum
971:Categories
199:Stub-class
88:if needed
71:Be polite
21:talk page
950:unsigned
56:get help
29:This is
27:article.
308:on the
154:WP refs
142:scholar
205:scale.
126:Google
169:JSTOR
130:books
84:Seek
958:talk
932:talk
928:Rybu
916:talk
591:talk
572:talk
568:Rybu
554:talk
535:talk
531:Rybu
519:talk
162:FENS
136:news
73:and
300:Low
176:TWL
973::
960:)
934:)
918:)
883:μ
855:⟩
828:⟨
819:⟩
793:⟨
718:⟩
692:⟨
593:)
574:)
556:)
537:)
521:)
486:≠
483:ϵ
475:ϵ
442:∈
433::=
428:ϵ
380:→
156:)
54:;
956:(
930:(
914:(
898:4
895:=
892:)
889:f
886:(
860:.
852:y
849:x
846:,
843:y
840:,
837:x
834:,
831:1
825:=
814:2
810:y
806:,
801:2
797:x
788:)
785:y
782:,
779:x
776:(
771:O
763:=
758:f
752:A
723:.
713:2
709:y
705:,
700:2
696:x
689:=
684:f
680:J
653:3
649:y
645:+
640:3
636:x
632:=
629:)
626:y
623:,
620:x
617:(
614:f
589:(
570:(
552:(
533:(
517:(
497:.
492:}
489:0
478:,
472:=
469:)
466:z
463:(
460:f
457::
452:n
447:C
439:z
436:{
424:F
397:)
394:0
391:,
387:C
383:(
377:)
374:0
371:,
366:n
361:C
356:(
353::
350:f
337:f
312:.
211::
172:·
166:·
158:·
151:·
145:·
139:·
133:·
128:(
58:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.