84:
74:
53:
22:
188:
The resulting connected sum knot inherits an orientation consistent with the orientations of the two original knots, and the oriented ambient isotopy class of the result is well-defined, depending only on the oriented ambient isotopy classes of the original two knots. In this manner, oriented
361:
are unoriented equivalent, simply note that they both may be constructed from the same pair of disjoint knot projections as above, the only difference being the orientations of the knots. Similarly, one sees that
140:
196:
take into account the orientations of the knots, the connected sum operation is not well defined on isotopy classes of (nonoriented) knots. To see this, consider two noninvertible knots
333:
The oriented ambient istotopy classes of these four oriented knots are all distinct. And, when one considers ambient isotopy of the knots without regard to orientation, there are
407:
130:
106:
402:
184:
Now join the two knots together by deleting these arcs from the knots and adding the arcs that form the other pair of sides of the rectangle.
97:
58:
173:
Find a rectangle in the plane where one pair of sides are arcs along each knot but is otherwise disjoint from the knots
33:
177:
so that the arcs of the knots on the sides of the rectangle are oriented around the boundary of the rectangle in the
39:
384:
83:
21:
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
89:
73:
52:
252:
with its two inequivalent orientations. There are four oriented connected sums we may form:
189:
ambient isotopy classes of oriented knots form a commutative unique factorization monoid.
200:
which are not equivalent (as unoriented knots); for example take the two pretzel knots
396:
170:
Consider a planar projection of each knot and suppose these projections are disjoint.
383:
Hi, Chuck. The article would be much improved if you were to add this stuff to it.
374:
102:
79:
158:
Connected sum of knots requires orientation of the knots to be well defined.
162:
For the connected sum of knots to be well defined, one has to consider
387:
377:
370:
may be constructed from the same pair of disjoint knot projections.
166:
in 3-space. To define the connected sum for two oriented knots:
15:
101:, a collaborative effort to improve the coverage of
234:with its two inequivalent orientations, and let
8:
47:
49:
19:
7:
95:This article is within the scope of
38:It is of interest to the following
14:
408:Mid-priority mathematics articles
115:Knowledge:WikiProject Mathematics
118:Template:WikiProject Mathematics
82:
72:
51:
20:
135:This article has been rated as
1:
109:and see a list of open tasks.
403:B-Class mathematics articles
424:
134:
67:
46:
388:18:11, 18 May 2007 (UTC)
378:14:48, 18 May 2007 (UTC)
337:equivalence classes: {
141:project's priority scale
98:WikiProject Mathematics
28:This article is rated
121:mathematics articles
90:Mathematics portal
34:content assessment
155:
154:
151:
150:
147:
146:
415:
353:}. To see that
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
423:
422:
418:
417:
416:
414:
413:
412:
393:
392:
385:Joshua R. Davis
329:
322:
310:
303:
291:
284:
272:
265:
247:
240:
229:
222:
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
421:
419:
411:
410:
405:
395:
394:
391:
390:
331:
330:
327:
320:
311:
308:
301:
292:
289:
282:
273:
270:
263:
245:
238:
227:
220:
216:(3,5,9). Let
186:
185:
182:
179:same direction
171:
164:oriented knots
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
420:
409:
406:
404:
401:
400:
398:
389:
386:
382:
381:
380:
379:
376:
371:
369:
365:
360:
356:
352:
348:
344:
340:
336:
326:
319:
315:
312:
307:
300:
296:
293:
288:
281:
277:
274:
269:
262:
258:
255:
254:
253:
251:
244:
237:
233:
226:
219:
215:
211:
207:
203:
199:
195:
190:
183:
180:
176:
172:
169:
168:
167:
165:
157:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
372:
367:
363:
358:
354:
350:
346:
342:
338:
335:two distinct
334:
332:
324:
317:
313:
305:
298:
294:
286:
279:
275:
267:
260:
256:
249:
242:
235:
231:
224:
217:
213:
209:
208:(3,5,7) and
205:
201:
197:
193:
192:If one does
191:
187:
178:
174:
163:
161:
137:Mid-priority
136:
96:
62:Mid‑priority
40:WikiProjects
112:Mathematics
103:mathematics
59:Mathematics
397:Categories
345:} and {
139:on the
30:B-class
36:scale.
375:Chuck
366:and
357:and
241:and
223:and
198:K, L
373:--
248:be
230:be
194:not
175:and
131:Mid
399::
349:~
341:~
323:#
316:=
304:#
297:=
285:#
278:=
266:#
259:=
212:=
204:=
368:D
364:C
359:B
355:A
351:D
347:C
343:B
339:A
328:+
325:L
321:-
318:K
314:D
309:-
306:L
302:+
299:K
295:C
290:-
287:L
283:-
280:K
276:B
271:+
268:L
264:+
261:K
257:A
250:L
246:-
243:L
239:+
236:L
232:K
228:-
225:K
221:+
218:K
214:P
210:L
206:P
202:K
181:.
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.