1549:
134:
3009:
1913:
108:
99:
1225:
3021:
673:
1753:
86:
A direction is assigned to the link at a point in each component and this direction is followed all the way around each component. For each crossing one comes across while traveling in this direction, if the strand underneath goes from right to left, the crossing is positive; if the lower strand goes
1544:{\displaystyle n_{1}={\frac {r_{13}\times r_{14}}{\left|r_{13}\times r_{14}\right|}},\;n_{2}={\frac {r_{14}\times r_{24}}{\left|r_{14}\times r_{24}\right|}},\;n_{3}={\frac {r_{24}\times r_{23}}{\left|r_{24}\times r_{23}\right|}},\;n_{4}={\frac {r_{23}\times r_{13}}{\left|r_{23}\times r_{13}\right|}}}
1947:
Any elastic rod, not just DNA, relieves torsional stress by coiling, an action which simultaneously untwists and bends the rod. F. Brock Fuller shows mathematically how the “elastic energy due to local twisting of the rod may be reduced if the central curve of the rod forms coils that increase its
1919:
1918:
1915:
1914:
867:
1920:
507:
1903:
1917:
1560:
2273:
714:
1908:
In addition, other methods to calculate writhe can be fully described mathematically and algorithmically, some of them outperform method above (which has quadratic computational complexity, by having linear complexity).
389:
668:{\displaystyle \operatorname {Wr} ={\frac {1}{4\pi }}\int _{C}\int _{C}d\mathbf {r} _{1}\times d\mathbf {r} _{2}\cdot {\frac {\mathbf {r} _{1}-\mathbf {r} _{2}}{\left|\mathbf {r} _{1}-\mathbf {r} _{2}\right|^{3}}}}
1787:
350:
708:
line segments. A procedure that was first derived by
Michael Levitt for the description of protein folding and later used for supercoiled DNA by Konstantin Klenin and Jörg Langowski is to compute
1916:
67:
as values. In both cases, writhe is a geometric quantity, meaning that while deforming a curve (or diagram) in such a way that does not change its topology, one may still change its writhe.
1061:
1000:
234:(in the space curve sense) is equal to the average of the integral writhe values obtained from the projections from all vantage points. Hence, writhe in this situation can take on any
1748:{\displaystyle \Omega ^{*}=\arcsin \left(n_{1}\cdot n_{2}\right)+\arcsin \left(n_{2}\cdot n_{3}\right)+\arcsin \left(n_{3}\cdot n_{4}\right)+\arcsin \left(n_{4}\cdot n_{1}\right).}
1107:
914:
320:
296:
232:
45:
479:
450:
276:
204:
153:: moves of Type II and Type III do not affect the writhe. Reidemeister move Type I, however, increases or decreases the writhe by 1. This implies that the writhe of a knot is
1940:
to describe the amount a piece of DNA is deformed as a result of this torsional stress. In general, this phenomenon of forming coils due to writhe is referred to as
1177:
1779:
1217:
1197:
1147:
1127:
954:
934:
706:
499:
417:
2038:
862:{\displaystyle \operatorname {Wr} =\sum _{i=1}^{N}\sum _{j=1}^{N}{\frac {\Omega _{ij}}{4\pi }}=2\sum _{i=2}^{N}\sum _{j<i}{\frac {\Omega _{ij}}{4\pi }}}
130:
For a knot diagram, using the right-hand rule with either orientation gives the same result, so the writhe is well-defined on unoriented knot diagrams.
161:
of the knot itself — only the diagram. By a series of Type I moves one can set the writhe of a diagram for a given knot to be any integer at all.
362:
2386:
2356:
2017:
1898:{\displaystyle {\frac {\Omega }{4\pi }}={\frac {\Omega ^{*}}{4\pi }}{\text{sign}}\left(\left(r_{34}\times r_{12}\right)\cdot r_{13}\right).}
2954:
2873:
2420:
2245:
329:
2863:
2868:
2739:
2348:
1972:
323:
2440:
2174:
2502:
2340:
1936:
will coil when twisted, just like a rubber hose or a rope will, and that is why biomathematicians use the quantity of
1005:
2135:
2091:
242:
2508:
2572:
2567:
2379:
959:
2700:
2009:
169:
Writhe is also a property of a knot represented as a curve in three-dimensional space. Strictly speaking, a
83:
diagram. The writhe is the total number of positive crossings minus the total number of negative crossings.
3025:
2914:
2883:
2183:
1069:
876:
305:
281:
217:
30:
2744:
2232:
87:
from left to right, the crossing is negative. One way of remembering this is to use a variation of the
455:
426:
252:
180:
3047:
3013:
2784:
2372:
2282:
1967:
353:
246:
2230:
Klenin, Konstantin; Langowski, Jörg (2000). "Computation of writhe in modeling of supercoiled DNA".
2172:
Levitt, Michael (1986). "Protein
Folding by Restrained Energy Minimization and Molecular Dynamics".
2821:
2804:
2188:
2003:
2842:
2789:
2403:
2399:
2073:
2047:
170:
80:
56:
48:
688:, we can approximate its value numerically by first representing our curve as a finite chain of
2939:
2888:
2838:
2794:
2754:
2749:
2667:
2352:
2318:
2249:
2201:
2013:
150:
138:
2974:
2799:
2695:
2430:
2308:
2290:
2241:
2193:
2107:
2057:
1941:
1152:
207:
2304:
2158:
2121:
2069:
1761:
2934:
2898:
2833:
2779:
2734:
2727:
2617:
2529:
2412:
2300:
2154:
2117:
2065:
685:
174:
88:
173:
is such a curve, defined mathematically as an embedding of a circle in three-dimensional
2286:
2994:
2893:
2855:
2774:
2687:
2562:
2554:
2514:
1977:
1962:
1944:
and is quite commonplace, and in fact in most organisms DNA is negatively supercoiled.
1202:
1182:
1132:
1112:
939:
919:
691:
484:
402:
299:
158:
55:
values. In another sense, it is a quantity that describes the amount of "coiling" of a
2313:
2268:
2197:
3041:
2929:
2717:
2710:
2705:
1957:
395:
In a paper from 1959, Călugăreanu also showed how to calculate the writhe Wr with an
2077:
2944:
2924:
2828:
2811:
2607:
2544:
211:
133:
107:
64:
2627:
2466:
2458:
2450:
98:
2959:
2722:
2496:
2476:
2395:
2364:
235:
76:
20:
2274:
Proceedings of the
National Academy of Sciences of the United States of America
206:. By viewing the curve from different vantage points, one can obtain different
2979:
2964:
2919:
2816:
2769:
2764:
2759:
2589:
2486:
2061:
2345:
The Knot Book: An
Elementary Introduction to the Mathematical Theory of Knots
2112:
2095:
2984:
2652:
2253:
2139:
2096:"Sur les classes d'isotopie des nœuds tridimensionnels et leurs invariants"
680:
Numerically approximating the Gauss integral for writhe of a curve in space
384:{\displaystyle \operatorname {Wr} =\operatorname {Lk} -\operatorname {Tw} }
2322:
2295:
2205:
1924:
A simulation of an elastic rod relieving torsional stress by forming coils
2969:
2579:
396:
52:
1758:
Finally, we compensate for the possible sign difference and divide by
2637:
2597:
2246:
10.1002/1097-0282(20001015)54:5<307::aid-bip20>3.0.co;2-y
2052:
2878:
1911:
916:
is the exact evaluation of the double integral over line segments
420:
60:
2225:
2223:
2221:
2219:
2217:
2215:
2949:
2368:
1149:, number the endpoints of the two segments 1, 2, 3, and 4. Let
2140:"L'intégrale de Gauss et l'analyse des nœuds tridimensionnels"
1933:
2036:
Cimasoni, David (2001). "Computing the writhe of a knot".
2031:
2029:
1997:
1995:
1993:
345:{\displaystyle \operatorname {Lk} -\operatorname {Tw} }
149:
The writhe of a knot is unaffected by two of the three
47:. In one sense, it is purely a property of an oriented
23:, there are several competing notions of the quantity
1790:
1764:
1563:
1228:
1205:
1185:
1155:
1135:
1115:
1072:
1008:
962:
942:
922:
879:
717:
694:
510:
487:
458:
429:
405:
365:
332:
308:
284:
255:
220:
183:
33:
2907:
2851:
2686:
2588:
2553:
2411:
1897:
1773:
1747:
1543:
1211:
1191:
1171:
1141:
1121:
1101:
1055:
994:
948:
928:
908:
861:
700:
684:Since writhe for a curve in space is defined as a
667:
493:
473:
444:
411:
383:
344:
314:
290:
270:
226:
198:
39:
501:. Then the writhe is equal to the Gauss integral
1056:{\displaystyle \Omega _{i,i+1}=\Omega _{ii}=0}
2380:
8:
2039:Journal of Knot Theory and Its Ramifications
2387:
2373:
2365:
1465:
1386:
1307:
79:, the writhe is a property of an oriented
2312:
2294:
2187:
2147:Revue de Mathématiques Pure et Appliquées
2111:
2051:
1881:
1863:
1850:
1831:
1815:
1809:
1791:
1789:
1763:
1731:
1718:
1689:
1676:
1647:
1634:
1605:
1592:
1568:
1562:
1528:
1515:
1499:
1486:
1479:
1470:
1449:
1436:
1420:
1407:
1400:
1391:
1370:
1357:
1341:
1328:
1321:
1312:
1291:
1278:
1262:
1249:
1242:
1233:
1227:
1204:
1184:
1160:
1154:
1134:
1114:
1091:
1086:
1077:
1071:
1038:
1013:
1007:
995:{\displaystyle \Omega _{ij}=\Omega _{ji}}
983:
967:
961:
941:
921:
898:
893:
884:
878:
840:
834:
822:
812:
801:
772:
766:
760:
749:
739:
728:
716:
693:
657:
646:
641:
631:
626:
612:
607:
597:
592:
588:
579:
574:
561:
556:
546:
536:
517:
509:
486:
465:
460:
457:
436:
431:
428:
404:
364:
331:
307:
283:
262:
258:
257:
254:
219:
190:
186:
185:
182:
63:) in three-dimensional space and assumes
32:
132:
1989:
2269:"The writhing number of a space curve"
1179:be the vector that begins at endpoint
352:depends only on the core curve of the
245:proved the following theorem: take a
7:
3020:
1102:{\displaystyle \Omega _{ij}/{4\pi }}
909:{\displaystyle \Omega _{ij}/{4\pi }}
1219:. Define the following quantities:
315:{\displaystyle \operatorname {Tw} }
291:{\displaystyle \operatorname {Lk} }
227:{\displaystyle \operatorname {Wr} }
40:{\displaystyle \operatorname {Wr} }
1812:
1793:
1565:
1074:
1035:
1010:
980:
964:
881:
837:
769:
302:of its border components, and let
14:
2100:Czechoslovak Mathematical Journal
3019:
3008:
3007:
642:
627:
608:
593:
575:
557:
474:{\displaystyle \mathbf {r} _{2}}
461:
445:{\displaystyle \mathbf {r} _{1}}
432:
271:{\displaystyle \mathbb {R} ^{3}}
199:{\displaystyle \mathbb {R} ^{3}}
106:
97:
2874:Dowker–Thistlethwaite notation
1:
2349:American Mathematical Society
2198:10.1016/s0022-2836(83)80129-6
2175:Journal of Molecular Biology
1929:Applications in DNA topology
1109:for given segments numbered
421:smooth, simple, closed curve
210:and draw the corresponding
103:
16:Invariant of a knot diagram
3064:
3003:
2864:Alexander–Briggs notation
2267:Fuller, F. Brock (1971).
2062:10.1142/S0218216501000913
2113:10.21136/CMJ.1961.100486
165:Writhe of a closed curve
2955:List of knots and links
2503:Kinoshita–Terasaka knot
2010:Oxford University Press
71:Writhe of link diagrams
2002:Bates, Andrew (2005).
1925:
1899:
1775:
1749:
1545:
1213:
1193:
1173:
1172:{\displaystyle r_{pq}}
1143:
1123:
1103:
1057:
996:
950:
930:
910:
863:
817:
765:
744:
702:
669:
495:
475:
446:
413:
385:
346:
326:. Then the difference
316:
292:
272:
241:In a paper from 1961,
228:
200:
146:
41:
2745:Finite type invariant
2296:10.1073/pnas.68.4.815
2136:Călugăreanu, Gheorghe
2092:Călugăreanu, Gheorghe
1923:
1900:
1776:
1774:{\displaystyle 4\pi }
1750:
1546:
1214:
1199:and ends at endpoint
1194:
1174:
1144:
1124:
1104:
1058:
997:
951:
931:
911:
864:
797:
745:
724:
703:
670:
496:
476:
447:
414:
386:
347:
317:
293:
273:
238:as a possible value.
229:
201:
136:
42:
1788:
1762:
1561:
1226:
1203:
1183:
1153:
1133:
1113:
1070:
1006:
960:
940:
920:
877:
715:
692:
508:
485:
456:
427:
403:
363:
330:
306:
282:
253:
243:Gheorghe Călugăreanu
218:
181:
51:diagram and assumes
31:
2915:Alexander's theorem
2287:1971PNAS...68..815B
1973:Twist (mathematics)
61:closed simple curve
2012:. pp. 36–37.
1948:writhing number”.
1926:
1895:
1771:
1745:
1554:Then we calculate
1541:
1209:
1189:
1169:
1139:
1119:
1099:
1053:
992:
946:
926:
906:
859:
833:
698:
665:
491:
471:
442:
409:
381:
342:
312:
288:
268:
224:
196:
151:Reidemeister moves
147:
37:
3035:
3034:
2889:Reidemeister move
2755:Khovanov homology
2750:Hyperbolic volume
2358:978-0-8218-3678-1
2019:978-0-19-850655-3
1921:
1834:
1829:
1804:
1539:
1460:
1381:
1302:
1212:{\displaystyle q}
1192:{\displaystyle p}
1142:{\displaystyle j}
1122:{\displaystyle i}
949:{\displaystyle j}
929:{\displaystyle i}
857:
818:
789:
701:{\displaystyle N}
663:
530:
494:{\displaystyle C}
412:{\displaystyle C}
159:isotopy invariant
139:Reidemeister move
128:
127:
57:mathematical knot
3055:
3023:
3022:
3011:
3010:
2975:Tait conjectures
2678:
2677:
2663:
2662:
2648:
2647:
2540:
2539:
2525:
2524:
2509:(−2,3,7) pretzel
2389:
2382:
2375:
2366:
2361:
2327:
2326:
2316:
2298:
2264:
2258:
2257:
2227:
2210:
2209:
2191:
2169:
2163:
2162:
2144:
2132:
2126:
2125:
2115:
2088:
2082:
2081:
2055:
2046:(387): 387–395.
2033:
2024:
2023:
1999:
1958:DNA supercoiling
1942:DNA supercoiling
1922:
1904:
1902:
1901:
1896:
1891:
1887:
1886:
1885:
1873:
1869:
1868:
1867:
1855:
1854:
1835:
1832:
1830:
1828:
1820:
1819:
1810:
1805:
1803:
1792:
1780:
1778:
1777:
1772:
1754:
1752:
1751:
1746:
1741:
1737:
1736:
1735:
1723:
1722:
1699:
1695:
1694:
1693:
1681:
1680:
1657:
1653:
1652:
1651:
1639:
1638:
1615:
1611:
1610:
1609:
1597:
1596:
1573:
1572:
1550:
1548:
1547:
1542:
1540:
1538:
1534:
1533:
1532:
1520:
1519:
1505:
1504:
1503:
1491:
1490:
1480:
1475:
1474:
1461:
1459:
1455:
1454:
1453:
1441:
1440:
1426:
1425:
1424:
1412:
1411:
1401:
1396:
1395:
1382:
1380:
1376:
1375:
1374:
1362:
1361:
1347:
1346:
1345:
1333:
1332:
1322:
1317:
1316:
1303:
1301:
1297:
1296:
1295:
1283:
1282:
1268:
1267:
1266:
1254:
1253:
1243:
1238:
1237:
1218:
1216:
1215:
1210:
1198:
1196:
1195:
1190:
1178:
1176:
1175:
1170:
1168:
1167:
1148:
1146:
1145:
1140:
1128:
1126:
1125:
1120:
1108:
1106:
1105:
1100:
1098:
1090:
1085:
1084:
1062:
1060:
1059:
1054:
1046:
1045:
1030:
1029:
1001:
999:
998:
993:
991:
990:
975:
974:
955:
953:
952:
947:
935:
933:
932:
927:
915:
913:
912:
907:
905:
897:
892:
891:
868:
866:
865:
860:
858:
856:
848:
847:
835:
832:
816:
811:
790:
788:
780:
779:
767:
764:
759:
743:
738:
707:
705:
704:
699:
674:
672:
671:
666:
664:
662:
661:
656:
652:
651:
650:
645:
636:
635:
630:
618:
617:
616:
611:
602:
601:
596:
589:
584:
583:
578:
566:
565:
560:
551:
550:
541:
540:
531:
529:
518:
500:
498:
497:
492:
480:
478:
477:
472:
470:
469:
464:
451:
449:
448:
443:
441:
440:
435:
418:
416:
415:
410:
390:
388:
387:
382:
351:
349:
348:
343:
321:
319:
318:
313:
297:
295:
294:
289:
277:
275:
274:
269:
267:
266:
261:
233:
231:
230:
225:
205:
203:
202:
197:
195:
194:
189:
110:
101:
94:
93:
46:
44:
43:
38:
3063:
3062:
3058:
3057:
3056:
3054:
3053:
3052:
3038:
3037:
3036:
3031:
2999:
2903:
2869:Conway notation
2853:
2847:
2834:Tricolorability
2682:
2676:
2673:
2672:
2671:
2661:
2658:
2657:
2656:
2646:
2643:
2642:
2641:
2633:
2623:
2613:
2603:
2584:
2563:Composite knots
2549:
2538:
2535:
2534:
2533:
2530:Borromean rings
2523:
2520:
2519:
2518:
2492:
2482:
2472:
2462:
2454:
2446:
2436:
2426:
2407:
2393:
2359:
2339:
2336:
2334:Further reading
2331:
2330:
2266:
2265:
2261:
2229:
2228:
2213:
2171:
2170:
2166:
2142:
2134:
2133:
2129:
2090:
2089:
2085:
2035:
2034:
2027:
2020:
2001:
2000:
1991:
1986:
1954:
1931:
1912:
1877:
1859:
1846:
1845:
1841:
1840:
1836:
1821:
1811:
1796:
1786:
1785:
1760:
1759:
1727:
1714:
1713:
1709:
1685:
1672:
1671:
1667:
1643:
1630:
1629:
1625:
1601:
1588:
1587:
1583:
1564:
1559:
1558:
1524:
1511:
1510:
1506:
1495:
1482:
1481:
1466:
1445:
1432:
1431:
1427:
1416:
1403:
1402:
1387:
1366:
1353:
1352:
1348:
1337:
1324:
1323:
1308:
1287:
1274:
1273:
1269:
1258:
1245:
1244:
1229:
1224:
1223:
1201:
1200:
1181:
1180:
1156:
1151:
1150:
1131:
1130:
1111:
1110:
1073:
1068:
1067:
1034:
1009:
1004:
1003:
979:
963:
958:
957:
938:
937:
918:
917:
880:
875:
874:
849:
836:
781:
768:
713:
712:
690:
689:
686:double integral
682:
640:
625:
624:
620:
619:
606:
591:
590:
573:
555:
542:
532:
522:
506:
505:
483:
482:
459:
454:
453:
430:
425:
424:
401:
400:
361:
360:
328:
327:
304:
303:
280:
279:
256:
251:
250:
216:
215:
184:
179:
178:
175:Euclidean space
167:
123:
116:
89:right-hand rule
73:
29:
28:
17:
12:
11:
5:
3061:
3059:
3051:
3050:
3040:
3039:
3033:
3032:
3030:
3029:
3017:
3004:
3001:
3000:
2998:
2997:
2995:Surgery theory
2992:
2987:
2982:
2977:
2972:
2967:
2962:
2957:
2952:
2947:
2942:
2937:
2932:
2927:
2922:
2917:
2911:
2909:
2905:
2904:
2902:
2901:
2896:
2894:Skein relation
2891:
2886:
2881:
2876:
2871:
2866:
2860:
2858:
2849:
2848:
2846:
2845:
2839:Unknotting no.
2836:
2831:
2826:
2825:
2824:
2814:
2809:
2808:
2807:
2802:
2797:
2792:
2787:
2777:
2772:
2767:
2762:
2757:
2752:
2747:
2742:
2737:
2732:
2731:
2730:
2720:
2715:
2714:
2713:
2703:
2698:
2692:
2690:
2684:
2683:
2681:
2680:
2674:
2665:
2659:
2650:
2644:
2635:
2631:
2625:
2621:
2615:
2611:
2605:
2601:
2594:
2592:
2586:
2585:
2583:
2582:
2577:
2576:
2575:
2570:
2559:
2557:
2551:
2550:
2548:
2547:
2542:
2536:
2527:
2521:
2512:
2506:
2500:
2494:
2490:
2484:
2480:
2474:
2470:
2464:
2460:
2456:
2452:
2448:
2444:
2438:
2434:
2428:
2424:
2417:
2415:
2409:
2408:
2394:
2392:
2391:
2384:
2377:
2369:
2363:
2362:
2357:
2335:
2332:
2329:
2328:
2281:(4): 815–819.
2259:
2240:(5): 307–317.
2211:
2189:10.1.1.26.3656
2182:(3): 723–764.
2164:
2127:
2106:(4): 588–625.
2083:
2025:
2018:
1988:
1987:
1985:
1982:
1981:
1980:
1978:Winding number
1975:
1970:
1965:
1963:Linking number
1960:
1953:
1950:
1930:
1927:
1906:
1905:
1894:
1890:
1884:
1880:
1876:
1872:
1866:
1862:
1858:
1853:
1849:
1844:
1839:
1827:
1824:
1818:
1814:
1808:
1802:
1799:
1795:
1770:
1767:
1756:
1755:
1744:
1740:
1734:
1730:
1726:
1721:
1717:
1712:
1708:
1705:
1702:
1698:
1692:
1688:
1684:
1679:
1675:
1670:
1666:
1663:
1660:
1656:
1650:
1646:
1642:
1637:
1633:
1628:
1624:
1621:
1618:
1614:
1608:
1604:
1600:
1595:
1591:
1586:
1582:
1579:
1576:
1571:
1567:
1552:
1551:
1537:
1531:
1527:
1523:
1518:
1514:
1509:
1502:
1498:
1494:
1489:
1485:
1478:
1473:
1469:
1464:
1458:
1452:
1448:
1444:
1439:
1435:
1430:
1423:
1419:
1415:
1410:
1406:
1399:
1394:
1390:
1385:
1379:
1373:
1369:
1365:
1360:
1356:
1351:
1344:
1340:
1336:
1331:
1327:
1320:
1315:
1311:
1306:
1300:
1294:
1290:
1286:
1281:
1277:
1272:
1265:
1261:
1257:
1252:
1248:
1241:
1236:
1232:
1208:
1188:
1166:
1163:
1159:
1138:
1118:
1097:
1094:
1089:
1083:
1080:
1076:
1052:
1049:
1044:
1041:
1037:
1033:
1028:
1025:
1022:
1019:
1016:
1012:
989:
986:
982:
978:
973:
970:
966:
945:
925:
904:
901:
896:
890:
887:
883:
871:
870:
855:
852:
846:
843:
839:
831:
828:
825:
821:
815:
810:
807:
804:
800:
796:
793:
787:
784:
778:
775:
771:
763:
758:
755:
752:
748:
742:
737:
734:
731:
727:
723:
720:
697:
681:
678:
677:
676:
660:
655:
649:
644:
639:
634:
629:
623:
615:
610:
605:
600:
595:
587:
582:
577:
572:
569:
564:
559:
554:
549:
545:
539:
535:
528:
525:
521:
516:
513:
490:
468:
463:
439:
434:
408:
393:
392:
380:
377:
374:
371:
368:
341:
338:
335:
311:
300:linking number
287:
265:
260:
223:
193:
188:
166:
163:
126:
125:
120:
118:
112:
111:
104:
102:
72:
69:
36:
15:
13:
10:
9:
6:
4:
3:
2:
3060:
3049:
3046:
3045:
3043:
3028:
3027:
3018:
3016:
3015:
3006:
3005:
3002:
2996:
2993:
2991:
2988:
2986:
2983:
2981:
2978:
2976:
2973:
2971:
2968:
2966:
2963:
2961:
2958:
2956:
2953:
2951:
2948:
2946:
2943:
2941:
2938:
2936:
2933:
2931:
2930:Conway sphere
2928:
2926:
2923:
2921:
2918:
2916:
2913:
2912:
2910:
2906:
2900:
2897:
2895:
2892:
2890:
2887:
2885:
2882:
2880:
2877:
2875:
2872:
2870:
2867:
2865:
2862:
2861:
2859:
2857:
2850:
2844:
2840:
2837:
2835:
2832:
2830:
2827:
2823:
2820:
2819:
2818:
2815:
2813:
2810:
2806:
2803:
2801:
2798:
2796:
2793:
2791:
2788:
2786:
2783:
2782:
2781:
2778:
2776:
2773:
2771:
2768:
2766:
2763:
2761:
2758:
2756:
2753:
2751:
2748:
2746:
2743:
2741:
2738:
2736:
2733:
2729:
2726:
2725:
2724:
2721:
2719:
2716:
2712:
2709:
2708:
2707:
2704:
2702:
2701:Arf invariant
2699:
2697:
2694:
2693:
2691:
2689:
2685:
2669:
2666:
2654:
2651:
2639:
2636:
2629:
2626:
2619:
2616:
2609:
2606:
2599:
2596:
2595:
2593:
2591:
2587:
2581:
2578:
2574:
2571:
2569:
2566:
2565:
2564:
2561:
2560:
2558:
2556:
2552:
2546:
2543:
2531:
2528:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2488:
2485:
2478:
2475:
2468:
2465:
2463:
2457:
2455:
2449:
2442:
2439:
2432:
2429:
2422:
2419:
2418:
2416:
2414:
2410:
2405:
2401:
2397:
2390:
2385:
2383:
2378:
2376:
2371:
2370:
2367:
2360:
2354:
2350:
2346:
2342:
2338:
2337:
2333:
2324:
2320:
2315:
2310:
2306:
2302:
2297:
2292:
2288:
2284:
2280:
2276:
2275:
2270:
2263:
2260:
2255:
2251:
2247:
2243:
2239:
2235:
2234:
2226:
2224:
2222:
2220:
2218:
2216:
2212:
2207:
2203:
2199:
2195:
2190:
2185:
2181:
2177:
2176:
2168:
2165:
2160:
2156:
2152:
2149:(in French).
2148:
2141:
2137:
2131:
2128:
2123:
2119:
2114:
2109:
2105:
2102:(in French).
2101:
2097:
2093:
2087:
2084:
2079:
2075:
2071:
2067:
2063:
2059:
2054:
2049:
2045:
2041:
2040:
2032:
2030:
2026:
2021:
2015:
2011:
2007:
2006:
1998:
1996:
1994:
1990:
1983:
1979:
1976:
1974:
1971:
1969:
1968:Ribbon theory
1966:
1964:
1961:
1959:
1956:
1955:
1951:
1949:
1945:
1943:
1939:
1935:
1928:
1910:
1892:
1888:
1882:
1878:
1874:
1870:
1864:
1860:
1856:
1851:
1847:
1842:
1837:
1825:
1822:
1816:
1806:
1800:
1797:
1784:
1783:
1782:
1768:
1765:
1742:
1738:
1732:
1728:
1724:
1719:
1715:
1710:
1706:
1703:
1700:
1696:
1690:
1686:
1682:
1677:
1673:
1668:
1664:
1661:
1658:
1654:
1648:
1644:
1640:
1635:
1631:
1626:
1622:
1619:
1616:
1612:
1606:
1602:
1598:
1593:
1589:
1584:
1580:
1577:
1574:
1569:
1557:
1556:
1555:
1535:
1529:
1525:
1521:
1516:
1512:
1507:
1500:
1496:
1492:
1487:
1483:
1476:
1471:
1467:
1462:
1456:
1450:
1446:
1442:
1437:
1433:
1428:
1421:
1417:
1413:
1408:
1404:
1397:
1392:
1388:
1383:
1377:
1371:
1367:
1363:
1358:
1354:
1349:
1342:
1338:
1334:
1329:
1325:
1318:
1313:
1309:
1304:
1298:
1292:
1288:
1284:
1279:
1275:
1270:
1263:
1259:
1255:
1250:
1246:
1239:
1234:
1230:
1222:
1221:
1220:
1206:
1186:
1164:
1161:
1157:
1136:
1116:
1095:
1092:
1087:
1081:
1078:
1064:
1050:
1047:
1042:
1039:
1031:
1026:
1023:
1020:
1017:
1014:
987:
984:
976:
971:
968:
943:
923:
902:
899:
894:
888:
885:
853:
850:
844:
841:
829:
826:
823:
819:
813:
808:
805:
802:
798:
794:
791:
785:
782:
776:
773:
761:
756:
753:
750:
746:
740:
735:
732:
729:
725:
721:
718:
711:
710:
709:
695:
687:
679:
658:
653:
647:
637:
632:
621:
613:
603:
598:
585:
580:
570:
567:
562:
552:
547:
543:
537:
533:
526:
523:
519:
514:
511:
504:
503:
502:
488:
481:be points on
466:
437:
422:
406:
398:
378:
375:
372:
369:
366:
359:
358:
357:
355:
339:
336:
333:
325:
322:be its total
309:
301:
285:
263:
248:
244:
239:
237:
221:
214:. Its writhe
213:
212:knot diagrams
209:
191:
176:
172:
164:
162:
160:
156:
152:
144:
140:
135:
131:
121:
119:
114:
113:
109:
105:
100:
96:
95:
92:
90:
84:
82:
78:
70:
68:
66:
62:
58:
54:
50:
34:
26:
22:
3024:
3012:
2989:
2940:Double torus
2925:Braid theory
2740:Crossing no.
2735:Crosscap no.
2421:Figure-eight
2344:
2341:Adams, Colin
2278:
2272:
2262:
2237:
2231:
2179:
2173:
2167:
2150:
2146:
2130:
2103:
2099:
2086:
2053:math/0406148
2043:
2037:
2005:DNA Topology
2004:
1946:
1937:
1932:
1907:
1757:
1553:
1066:To evaluate
1065:
956:; note that
872:
683:
394:
240:
168:
154:
148:
142:
141:changes the
129:
85:
74:
65:real numbers
24:
18:
3048:Knot theory
2775:Linking no.
2696:Alternating
2497:Conway knot
2477:Carrick mat
2431:Three-twist
2396:Knot theory
2233:Biopolymers
236:real number
208:projections
77:knot theory
21:knot theory
2935:Complement
2899:Tabulation
2856:operations
2780:Polynomial
2770:Link group
2765:Knot group
2728:Invertible
2706:Bridge no.
2688:Invariants
2618:Cinquefoil
2487:Perko pair
2413:Hyperbolic
1984:References
1781:to obtain
2829:Stick no.
2785:Alexander
2723:Chirality
2668:Solomon's
2628:Septafoil
2555:Satellite
2515:Whitehead
2441:Stevedore
2184:CiteSeerX
1875:⋅
1857:×
1826:π
1817:∗
1813:Ω
1801:π
1794:Ω
1769:π
1725:⋅
1707:
1683:⋅
1665:
1641:⋅
1623:
1599:⋅
1581:
1570:∗
1566:Ω
1522:×
1493:×
1443:×
1414:×
1364:×
1335:×
1285:×
1256:×
1096:π
1075:Ω
1036:Ω
1011:Ω
981:Ω
965:Ω
903:π
882:Ω
854:π
838:Ω
820:∑
799:∑
786:π
770:Ω
747:∑
726:∑
638:−
604:−
586:⋅
568:×
544:∫
534:∫
527:π
376:−
337:−
137:A Type I
124:crossing
3042:Category
3014:Category
2884:Mutation
2852:Notation
2805:Kauffman
2718:Brunnian
2711:2-bridge
2580:Knot sum
2511:(12n242)
2343:(2004),
2254:10935971
2153:: 5–20.
2138:(1959).
2094:(1961).
2078:15850269
1952:See also
423:and let
397:integral
122:Negative
117:crossing
115:Positive
59:(or any
3026:Commons
2945:Fibered
2843:problem
2812:Pretzel
2790:Bracket
2608:Trefoil
2545:L10a140
2505:(11n42)
2499:(11n34)
2467:Endless
2323:5279522
2305:0278197
2283:Bibcode
2206:6195346
2159:0131846
2122:0149378
2070:1825964
298:be the
53:integer
2990:Writhe
2960:Ribbon
2795:HOMFLY
2638:Unlink
2598:Unknot
2573:Square
2568:Granny
2355:
2321:
2314:389050
2311:
2303:
2252:
2204:
2186:
2157:
2120:
2076:
2068:
2016:
1938:writhe
1704:arcsin
1662:arcsin
1620:arcsin
1578:arcsin
873:where
399:. Let
356:, and
354:ribbon
278:, let
247:ribbon
143:writhe
25:writhe
2980:Twist
2965:Slice
2920:Berge
2908:Other
2879:Flype
2817:Prime
2800:Jones
2760:Genus
2590:Torus
2404:links
2400:knots
2143:(PDF)
2074:S2CID
2048:arXiv
419:be a
324:twist
27:, or
2985:Wild
2950:Knot
2854:and
2841:and
2822:list
2653:Hopf
2402:and
2353:ISBN
2319:PMID
2250:PMID
2202:PMID
2014:ISBN
1833:sign
1129:and
1002:and
936:and
827:<
452:and
171:knot
145:by 1
81:link
49:link
2970:Sum
2491:161
2489:(10
2309:PMC
2291:doi
2242:doi
2194:doi
2180:170
2108:doi
2058:doi
1934:DNA
249:in
157:an
155:not
75:In
19:In
3044::
2670:(4
2655:(2
2640:(0
2630:(7
2620:(5
2610:(3
2600:(0
2532:(6
2517:(5
2481:18
2479:(8
2469:(7
2443:(6
2433:(5
2423:(4
2351:,
2347:,
2317:.
2307:.
2301:MR
2299:.
2289:.
2279:68
2277:.
2271:.
2248:.
2238:54
2236:.
2214:^
2200:.
2192:.
2178:.
2155:MR
2145:.
2118:MR
2116:.
2104:11
2098:.
2072:.
2066:MR
2064:.
2056:.
2044:10
2042:.
2028:^
2008:.
1992:^
1883:13
1865:12
1852:34
1530:13
1517:23
1501:13
1488:23
1451:23
1438:24
1422:23
1409:24
1372:24
1359:14
1343:24
1330:14
1293:14
1280:13
1264:14
1251:13
1063:.
719:Wr
512:Wr
379:Tw
373:Lk
367:Wr
340:Tw
334:Lk
310:Tw
286:Lk
222:Wr
177:,
91:.
35:Wr
2679:)
2675:1
2664:)
2660:1
2649:)
2645:1
2634:)
2632:1
2624:)
2622:1
2614:)
2612:1
2604:)
2602:1
2541:)
2537:2
2526:)
2522:1
2493:)
2483:)
2473:)
2471:4
2461:3
2459:6
2453:2
2451:6
2447:)
2445:1
2437:)
2435:2
2427:)
2425:1
2406:)
2398:(
2388:e
2381:t
2374:v
2325:.
2293::
2285::
2256:.
2244::
2208:.
2196::
2161:.
2151:4
2124:.
2110::
2080:.
2060::
2050::
2022:.
1893:.
1889:)
1879:r
1871:)
1861:r
1848:r
1843:(
1838:(
1823:4
1807:=
1798:4
1766:4
1743:.
1739:)
1733:1
1729:n
1720:4
1716:n
1711:(
1701:+
1697:)
1691:4
1687:n
1678:3
1674:n
1669:(
1659:+
1655:)
1649:3
1645:n
1636:2
1632:n
1627:(
1617:+
1613:)
1607:2
1603:n
1594:1
1590:n
1585:(
1575:=
1536:|
1526:r
1513:r
1508:|
1497:r
1484:r
1477:=
1472:4
1468:n
1463:,
1457:|
1447:r
1434:r
1429:|
1418:r
1405:r
1398:=
1393:3
1389:n
1384:,
1378:|
1368:r
1355:r
1350:|
1339:r
1326:r
1319:=
1314:2
1310:n
1305:,
1299:|
1289:r
1276:r
1271:|
1260:r
1247:r
1240:=
1235:1
1231:n
1207:q
1187:p
1165:q
1162:p
1158:r
1137:j
1117:i
1093:4
1088:/
1082:j
1079:i
1051:0
1048:=
1043:i
1040:i
1032:=
1027:1
1024:+
1021:i
1018:,
1015:i
988:i
985:j
977:=
972:j
969:i
944:j
924:i
900:4
895:/
889:j
886:i
869:,
851:4
845:j
842:i
830:i
824:j
814:N
809:2
806:=
803:i
795:2
792:=
783:4
777:j
774:i
762:N
757:1
754:=
751:j
741:N
736:1
733:=
730:i
722:=
696:N
675:.
659:3
654:|
648:2
643:r
633:1
628:r
622:|
614:2
609:r
599:1
594:r
581:2
576:r
571:d
563:1
558:r
553:d
548:C
538:C
524:4
520:1
515:=
489:C
467:2
462:r
438:1
433:r
407:C
391:.
370:=
264:3
259:R
192:3
187:R
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