Knowledge (XXG)

Ambient isotopy

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the same if one can distort one knot into the other without breaking it. Such a distortion is an example of an ambient isotopy. More precisely, let
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since one cannot be deformed into the other through a continuous path of homeomorphisms of the ambient space. They are ambient-isotopic in
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must be preserved by ambient isotopies. For example, two knots that are
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An Illustrated Introduction to Topology and Homotopy
556:, CRC Press, 2010, Chapter 10: Isotopy and Homotopy 494: 455: 431: 400: 373: 353: 330: 274: 254: 230: 210: 190: 170: 104: 63: 510:of each other are, in general, not equivalent. 604: 8: 341:is defined to be an ambient isotopy taking 611: 597: 474: 468: 448: 423: 417: 392: 386: 366: 346: 293: 267: 247: 223: 203: 183: 163: 96: 92: 91: 88: 55: 51: 50: 47: 331:{\displaystyle F:M\times \rightarrow M} 150:to another submanifold. For example in 7: 565: 563: 583:. You can help Knowledge (XXG) by 14: 567: 105:{\displaystyle \mathbb {R} ^{4}} 64:{\displaystyle \mathbb {R} ^{3}} 32: 23: 495:{\displaystyle F_{1}\circ g=h} 322: 319: 307: 1: 656: 562: 502:. This implies that the 496: 457: 433: 402: 375: 355: 332: 276: 256: 232: 212: 192: 172: 106: 65: 552:Sasho Kalajdzievski, 497: 458: 434: 432:{\displaystyle F_{t}} 403: 401:{\displaystyle F_{0}} 376: 356: 333: 277: 257: 233: 213: 193: 173: 107: 66: 467: 447: 416: 385: 365: 345: 292: 266: 246: 222: 202: 182: 162: 154:, one considers two 87: 46: 492: 453: 429: 398: 371: 351: 328: 272: 252: 228: 208: 188: 168: 102: 61: 16:Concept in toplogy 635:Maps of manifolds 592: 591: 541:M. A. Armstrong, 456:{\displaystyle M} 374:{\displaystyle h} 354:{\displaystyle g} 275:{\displaystyle M} 255:{\displaystyle N} 231:{\displaystyle h} 211:{\displaystyle g} 198:be manifolds and 191:{\displaystyle M} 171:{\displaystyle N} 138:distortion of an 130:, also called an 647: 613: 606: 599: 577:topology-related 571: 564: 525:Regular homotopy 501: 499: 498: 493: 479: 478: 462: 460: 459: 454: 438: 436: 435: 430: 428: 427: 407: 405: 404: 399: 397: 396: 380: 378: 377: 372: 360: 358: 357: 352: 337: 335: 334: 329: 281: 279: 278: 273: 261: 259: 258: 253: 237: 235: 234: 229: 217: 215: 214: 209: 197: 195: 194: 189: 177: 175: 174: 169: 142:, for example a 111: 109: 108: 103: 101: 100: 95: 77:ambient-isotopic 70: 68: 67: 62: 60: 59: 54: 36: 27: 655: 654: 650: 649: 648: 646: 645: 644: 620: 619: 618: 617: 560: 547:Springer-Verlag 538: 530:Regular isotopy 516: 470: 465: 464: 463:to itself, and 445: 444: 419: 414: 413: 388: 383: 382: 363: 362: 343: 342: 290: 289: 264: 263: 244: 243: 220: 219: 200: 199: 180: 179: 160: 159: 134:, is a kind of 128:ambient isotopy 116: 115: 114: 113: 90: 85: 84: 49: 44: 43: 39: 38: 37: 29: 28: 17: 12: 11: 5: 653: 651: 643: 642: 640:Topology stubs 637: 632: 622: 621: 616: 615: 608: 601: 593: 590: 589: 572: 558: 557: 550: 543:Basic Topology 537: 534: 533: 532: 527: 522: 515: 512: 491: 488: 485: 482: 477: 473: 452: 426: 422: 395: 391: 370: 350: 339: 338: 327: 324: 321: 318: 315: 312: 309: 306: 303: 300: 297: 284:continuous map 271: 251: 227: 207: 187: 167: 99: 94: 58: 53: 41: 40: 31: 30: 22: 21: 20: 19: 18: 15: 13: 10: 9: 6: 4: 3: 2: 652: 641: 638: 636: 633: 631: 628: 627: 625: 614: 609: 607: 602: 600: 595: 594: 588: 586: 582: 579:article is a 578: 573: 570: 566: 561: 555: 551: 548: 544: 540: 539: 535: 531: 528: 526: 523: 521: 518: 517: 513: 511: 509: 508:mirror images 505: 489: 486: 483: 480: 475: 471: 450: 442: 441:homeomorphism 424: 420: 411: 393: 389: 368: 348: 325: 316: 313: 310: 304: 301: 298: 295: 288: 287: 286: 285: 269: 249: 241: 225: 205: 185: 165: 157: 153: 149: 145: 141: 140:ambient space 137: 133: 129: 125: 121: 97: 82: 78: 74: 56: 35: 26: 585:expanding it 574: 559: 553: 542: 410:identity map 340: 131: 127: 120:mathematical 117: 81:trefoil knot 76: 504:orientation 412:, each map 152:knot theory 148:submanifold 146:, taking a 122:subject of 624:Categories 536:References 240:embeddings 136:continuous 481:∘ 323:→ 305:× 132:h-isotopy 630:Topology 514:See also 144:manifold 124:topology 520:Isotopy 408:is the 118:In the 79:to the 75:is not 549:, 1983 73:unknot 71:, the 575:This 443:from 439:is a 282:. A 156:knots 126:, an 581:stub 218:and 178:and 381:if 361:to 262:in 242:of 238:be 42:In 626:: 545:, 612:e 605:t 598:v 587:. 490:h 487:= 484:g 476:1 472:F 451:M 425:t 421:F 394:0 390:F 369:h 349:g 326:M 320:] 317:1 314:, 311:0 308:[ 302:M 299:: 296:F 270:M 250:N 226:h 206:g 186:M 166:N 112:. 98:4 93:R 57:3 52:R

Index



unknot
trefoil knot
mathematical
topology
continuous
ambient space
manifold
submanifold
knot theory
knots
embeddings
continuous map
identity map
homeomorphism
orientation
mirror images
Isotopy
Regular homotopy
Regular isotopy
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