1234:
249:
1224:
1214:
40:
2575:
2587:
1033:
512:
1206:, holds that 10 is the largest possible number of exceptional surgeries of any hyperbolic knot. However, it is not currently known whether the figure-eight knot is the only one that achieves the bound of 10. A well-known conjecture is that the bound (except for the two knots mentioned) is 6.
1073:. (Robert Riley and Troels Jørgensen, working independently of each other, had earlier shown that the figure-eight knot was hyperbolic by other means.) This construction, new at the time, led him to many powerful results and methods. For example, he was able to show that all but ten
1610:
736:
322:
1304:
725:
1139:
741:
327:
1613:
1618:
1616:
1612:
1611:
1551:
1617:
1385:
1449:
1028:{\displaystyle {\begin{aligned}G(x,y,z,t)=\ &(z(x^{2}+y^{2}+z^{2}+t^{2})+x(6x^{2}-2y^{2}-2z^{2}-2t^{2}),\\&\ tx{\sqrt {2}}+y(6x^{2}-2y^{2}-2z^{2}-2t^{2})).\end{aligned}}}
1664:
1615:
1169:
1604:
1574:
507:{\displaystyle {\begin{aligned}x&=\left(2+\cos {(2t)}\right)\cos {(3t)}\\y&=\left(2+\cos {(2t)}\right)\sin {(3t)}\\z&=\sin {(4t)}\end{aligned}}}
2638:
2643:
1148:
1088:
3-manifolds; these were the first such examples. Many more have been discovered by generalizing
Thurston's construction to other knots and links.
1614:
1256:
2658:
1952:
2618:
2520:
189:
2623:
2439:
1175:. From this perspective, the figure-eight knot can be considered the simplest hyperbolic knot. The figure eight knot complement is a
2628:
1085:
2633:
1876:
2429:
619:
2434:
2305:
269:
157:
107:
1098:
2006:
2068:
1464:
1245:
The figure-eight knot has genus 1 and is fibered. Therefore its complement fibers over the circle, the fibers being
1233:
2074:
1187:
2138:
2133:
1945:
1746:
1199:
24:
1842:
1324:
1092:
296:
in a rope and then joining the ends together, in the most natural way, gives a model of the mathematical knot.
1712:
578:
2266:
1203:
57:
2591:
1400:
1194:, Dehn surgeries resulting in a non-hyperbolic 3-manifold; they have 10 and 7, respectively. A theorem of
2480:
2449:
1809:
1056:
1043:
The figure-eight knot has played an important role historically (and continues to do so) in the theory of
248:
236:
2310:
2653:
2648:
2613:
2579:
2350:
1938:
1391:
1315:
1172:
606:
2387:
2370:
1082:
1067:
1223:
2408:
2355:
1969:
1965:
1642:
1180:
2505:
2454:
2404:
2360:
2320:
2315:
2233:
1913:
1154:
293:
147:
127:
50:
20:
1091:
The figure-eight knot is also the hyperbolic knot whose complement has the smallest possible
272:
of four. This makes it the knot with the third-smallest possible crossing number, after the
2540:
2365:
2261:
1996:
1742:
1579:
1455:
1253:
is then a homeomorphism of the 2-torus, which can be represented in this case by the matrix
1048:
544:. This follows from other, less simple (but very interesting) representations of the knot:
529:
220:
1849:
1829:
1816:
2500:
2464:
2399:
2345:
2300:
2293:
2183:
2095:
1978:
1846:
1826:
1813:
1246:
1060:
1052:
224:
179:
97:
304:
A simple parametric representation of the figure-eight knot is as the set of all points (
1213:
2560:
2459:
2421:
2340:
2253:
2128:
2120:
2080:
1904:
1784:
1559:
1176:
1078:
571:
39:
2607:
2495:
2283:
2276:
2271:
1916:
1854:
1834:
1195:
533:
87:
2510:
2490:
2394:
2377:
2173:
2110:
1866:
1250:
1074:
551:
541:
277:
228:
137:
77:
67:
2193:
2032:
2024:
2016:
1870:
1606:
in the Jones polynomial reflects the fact that the figure-eight knot is achiral.
2525:
2288:
2062:
2042:
1961:
1930:
1265:
1064:
594:
537:
257:
166:
1756:
2545:
2530:
2485:
2382:
2335:
2330:
2325:
2155:
2052:
1687:
1070:
1044:
598:
525:
281:
240:
232:
117:
2550:
2218:
1921:
1883:
521:
varying over the real numbers (see 2D visual realization at bottom right).
1183:, which has the smallest volume among non-compact hyperbolic 3-manifolds.
2535:
2145:
1801:
205:
198:
1299:{\displaystyle ({\begin{smallmatrix}2&1\\1&1\end{smallmatrix}})}
1750:
2555:
2203:
2163:
273:
1720:
2444:
1232:
1222:
1212:
19:
This article is about the mathematical concept. For the knot, see
1858:
2515:
1666:
either occurs always with positive or always with negative sign.
1934:
1249:
which are 2-dimensional tori with one boundary component. The
1823:
The orientable cusped hyperbolic 3-manifolds of minimum volume
1190:
are the only two hyperbolic knots known to have more than 6
1825:, Inventiones Mathematicae, 146 (2001), no. 3, 451–478.
1143:
252:
Figure-eight knot of practical knot-tying, with ends joined
1899:
1779:
1217:
Simple squared depiction of figure-eight configuration.
1227:
Symmetric depiction generated by parametric equations.
720:{\displaystyle F(x,y,z,t)=G(x,y,z^{2}-t^{2},2zt),\,\!}
1645:
1582:
1562:
1467:
1403:
1327:
1259:
1157:
1101:
739:
622:
574:
shows that any closed homogeneous braid is fibered.
325:
2473:
2417:
2252:
2154:
2119:
1977:
1237:
1134:{\displaystyle 6\Lambda (\pi /3)\approx 2.02988...}
214:
188:
178:
165:
156:
146:
136:
126:
116:
106:
96:
86:
76:
66:
56:
46:
32:
1658:
1598:
1568:
1545:
1443:
1379:
1298:
1163:
1133:
1027:
719:
506:
1888:, Princeton University lecture notes (1978–1981).
1639:A braid is called homogeneous if every generator
716:
1860:The maximal number of exceptional Dehn surgeries
1546:{\displaystyle V(q)=q^{2}-q+1-q^{-1}+q^{-2}.\ }
1946:
8:
1885:The Geometry and Topology of Three-Manifolds
1752:The Geometry and Topology of Three-Manifolds
1688:"Listing knot - Encyclopedia of Mathematics"
554:(namely, the closure of the 3-string braid σ
1953:
1939:
1931:
38:
16:Unique knot with a crossing number of four
1650:
1644:
1587:
1581:
1561:
1528:
1512:
1487:
1466:
1429:
1402:
1380:{\displaystyle \Delta (t)=-t+3-t^{-1},\ }
1362:
1326:
1263:
1258:
1198:and Meyerhoff, whose proof relies on the
1156:
1114:
1100:
1077:on the figure-eight knot resulted in non-
1006:
990:
974:
958:
935:
909:
893:
877:
861:
836:
823:
810:
797:
740:
738:
715:
691:
678:
621:
486:
452:
427:
382:
357:
326:
324:
292:The name is given because tying a normal
1608:
536:with an associated value of 5/3, and is
247:
1679:
1632:
1047:. Sometime in the mid-to-late 1970s,
577:(2) It is the link at (0,0,0,0) of an
29:
1444:{\displaystyle \nabla (z)=1-z^{2},\ }
7:
2586:
1872:Problems in low-dimensional topology
1806:Bounds on exceptional Dehn filling
1404:
1328:
1158:
1105:
540:. The figure-eight knot is also a
14:
2639:Fully amphichiral knots and links
1713:"Rational Knots with 4 crossings"
1264:
1051:showed that the figure-eight was
2644:Non-tricolorable knots and links
2585:
2574:
2573:
593:, so (according to a theorem of
1821:Chun Cao and Robert Meyerhoff,
2440:Dowker–Thistlethwaite notation
1477:
1471:
1413:
1407:
1337:
1331:
1293:
1260:
1186:The figure-eight knot and the
1122:
1108:
1015:
1012:
948:
915:
851:
842:
790:
784:
771:
747:
709:
659:
650:
626:
496:
487:
462:
453:
437:
428:
392:
383:
367:
358:
1:
1755:, p. 165, archived from
280:. The figure-eight knot is a
2659:Double torus knots and links
1908:. Accessed: 7 May 2013.
1875:, (see problem 1.77, due to
1845:140 (2000), no. 2, 243–282.
1839:Word hyperbolic Dehn surgery
1318:of the figure-eight knot is
268:) is the unique knot with a
2619:Alternating knots and links
1659:{\displaystyle \sigma _{i}}
2675:
2624:Hyperbolic knots and links
1747:"7. Computation of volume"
605:is actually a fibration.
18:
2569:
2430:Alexander–Briggs notation
1879:, for exceptional slopes)
1200:geometrization conjecture
581:of a real-polynomial map
524:The figure-eight knot is
219:
37:
1843:Inventiones Mathematicae
1164:{\displaystyle \Lambda }
2629:Fibered knots and links
2521:List of knots and links
2069:Kinoshita–Terasaka knot
1810:Geometry & Topology
1717:Rational Knots database
1039:Mathematical properties
613:for this knot, namely,
579:isolated critical point
1857:and Robert Meyerhoff,
1692:encyclopediaofmath.org
1660:
1623:
1600:
1599:{\displaystyle q^{-1}}
1570:
1547:
1445:
1381:
1300:
1238:
1228:
1218:
1165:
1135:
1029:
721:
508:
253:
23:. For other uses, see
2634:Prime knots and links
2311:Finite type invariant
1661:
1621:
1601:
1571:
1556:The symmetry between
1548:
1446:
1382:
1301:
1236:
1226:
1216:
1192:exceptional surgeries
1188:(−2,3,7) pretzel knot
1166:
1136:
1030:
722:
609:found the first such
509:
251:
1812:4 (2000), 431–449.
1643:
1580:
1560:
1465:
1401:
1325:
1316:Alexander polynomial
1257:
1173:Lobachevsky function
1155:
1099:
737:
620:
570:), and a theorem of
323:
2481:Alexander's theorem
1917:"Figure Eight Knot"
1204:computer assistance
1914:Weisstein, Eric W.
1882:William Thurston,
1656:
1624:
1596:
1566:
1543:
1441:
1377:
1296:
1291:
1290:
1239:
1229:
1219:
1181:Gieseking manifold
1161:
1131:
1025:
1023:
717:
504:
502:
254:
2601:
2600:
2455:Reidemeister move
2321:Khovanov homology
2316:Hyperbolic volume
1863:, arXiv:0808.1176
1711:Gruber, Hermann.
1622:Figure-eight knot
1619:
1569:{\displaystyle q}
1542:
1440:
1392:Conway polynomial
1376:
1243:
1242:
940:
928:
779:
294:figure-eight knot
262:figure-eight knot
246:
245:
237:fully amphichiral
128:Hyperbolic volume
51:Figure-eight knot
33:Figure-eight knot
21:Figure-eight knot
2666:
2589:
2588:
2577:
2576:
2541:Tait conjectures
2244:
2243:
2229:
2228:
2214:
2213:
2106:
2105:
2091:
2090:
2075:(−2,3,7) pretzel
1955:
1948:
1941:
1932:
1927:
1926:
1789:
1776:
1770:
1769:
1768:
1767:
1761:
1743:William Thurston
1739:
1733:
1732:
1730:
1728:
1719:. Archived from
1708:
1702:
1701:
1699:
1698:
1684:
1667:
1665:
1663:
1662:
1657:
1655:
1654:
1637:
1620:
1605:
1603:
1602:
1597:
1595:
1594:
1575:
1573:
1572:
1567:
1552:
1550:
1549:
1544:
1540:
1536:
1535:
1520:
1519:
1492:
1491:
1456:Jones polynomial
1450:
1448:
1447:
1442:
1438:
1434:
1433:
1386:
1384:
1383:
1378:
1374:
1370:
1369:
1305:
1303:
1302:
1297:
1292:
1247:Seifert surfaces
1209:
1208:
1170:
1168:
1167:
1162:
1146:
1140:
1138:
1137:
1132:
1118:
1049:William Thurston
1034:
1032:
1031:
1026:
1024:
1011:
1010:
995:
994:
979:
978:
963:
962:
941:
936:
926:
924:
914:
913:
898:
897:
882:
881:
866:
865:
841:
840:
828:
827:
815:
814:
802:
801:
777:
726:
724:
723:
718:
696:
695:
683:
682:
513:
511:
510:
505:
503:
499:
465:
445:
441:
440:
395:
375:
371:
370:
193:
42:
30:
2674:
2673:
2669:
2668:
2667:
2665:
2664:
2663:
2604:
2603:
2602:
2597:
2565:
2469:
2435:Conway notation
2419:
2413:
2400:Tricolorability
2248:
2242:
2239:
2238:
2237:
2227:
2224:
2223:
2222:
2212:
2209:
2208:
2207:
2199:
2189:
2179:
2169:
2150:
2129:Composite knots
2115:
2104:
2101:
2100:
2099:
2096:Borromean rings
2089:
2086:
2085:
2084:
2058:
2048:
2038:
2028:
2020:
2012:
2002:
1992:
1973:
1959:
1912:
1911:
1895:
1798:
1796:Further reading
1793:
1792:
1777:
1773:
1765:
1763:
1759:
1741:
1740:
1736:
1726:
1724:
1710:
1709:
1705:
1696:
1694:
1686:
1685:
1681:
1676:
1671:
1670:
1646:
1641:
1640:
1638:
1634:
1629:
1609:
1583:
1578:
1577:
1558:
1557:
1524:
1508:
1483:
1463:
1462:
1425:
1399:
1398:
1358:
1323:
1322:
1312:
1289:
1288:
1283:
1277:
1276:
1271:
1255:
1254:
1153:
1152:
1142:
1097:
1096:
1083:Seifert-fibered
1041:
1022:
1021:
1002:
986:
970:
954:
922:
921:
905:
889:
873:
857:
832:
819:
806:
793:
780:
735:
734:
687:
674:
618:
617:
612:
604:
584:
569:
565:
561:
557:
501:
500:
473:
467:
466:
414:
410:
403:
397:
396:
344:
340:
333:
321:
320:
302:
290:
270:crossing number
209:
202:
191:
180:Dowker notation
174:
158:Conway notation
28:
17:
12:
11:
5:
2672:
2670:
2662:
2661:
2656:
2651:
2646:
2641:
2636:
2631:
2626:
2621:
2616:
2606:
2605:
2599:
2598:
2596:
2595:
2583:
2570:
2567:
2566:
2564:
2563:
2561:Surgery theory
2558:
2553:
2548:
2543:
2538:
2533:
2528:
2523:
2518:
2513:
2508:
2503:
2498:
2493:
2488:
2483:
2477:
2475:
2471:
2470:
2468:
2467:
2462:
2460:Skein relation
2457:
2452:
2447:
2442:
2437:
2432:
2426:
2424:
2415:
2414:
2412:
2411:
2405:Unknotting no.
2402:
2397:
2392:
2391:
2390:
2380:
2375:
2374:
2373:
2368:
2363:
2358:
2353:
2343:
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2323:
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2313:
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2303:
2298:
2297:
2296:
2286:
2281:
2280:
2279:
2269:
2264:
2258:
2256:
2250:
2249:
2247:
2246:
2240:
2231:
2225:
2216:
2210:
2201:
2197:
2191:
2187:
2181:
2177:
2171:
2167:
2160:
2158:
2152:
2151:
2149:
2148:
2143:
2142:
2141:
2136:
2125:
2123:
2117:
2116:
2114:
2113:
2108:
2102:
2093:
2087:
2078:
2072:
2066:
2060:
2056:
2050:
2046:
2040:
2036:
2030:
2026:
2022:
2018:
2014:
2010:
2004:
2000:
1994:
1990:
1983:
1981:
1975:
1974:
1960:
1958:
1957:
1950:
1943:
1935:
1929:
1928:
1909:
1905:The Knot Atlas
1894:
1893:External links
1891:
1890:
1889:
1880:
1877:Cameron Gordon
1864:
1852:
1832:
1819:
1797:
1794:
1791:
1790:
1785:The Knot Atlas
1771:
1745:(March 2002),
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1127:
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1110:
1107:
1104:
1075:Dehn surgeries
1040:
1037:
1036:
1035:
1020:
1017:
1014:
1009:
1005:
1001:
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989:
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681:
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664:
661:
658:
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652:
649:
646:
643:
640:
637:
634:
631:
628:
625:
610:
607:Bernard Perron
602:
582:
572:John Stallings
567:
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555:
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288:Origin of name
286:
266:Listing's knot
244:
243:
217:
216:
212:
211:
207:
200:
196:
186:
185:
182:
176:
175:
172:
169:
163:
162:
160:
154:
153:
150:
148:Unknotting no.
144:
143:
140:
134:
133:
130:
124:
123:
120:
114:
113:
110:
104:
103:
100:
94:
93:
90:
84:
83:
80:
74:
73:
70:
64:
63:
60:
54:
53:
48:
44:
43:
35:
34:
15:
13:
10:
9:
6:
4:
3:
2:
2671:
2660:
2657:
2655:
2652:
2650:
2647:
2645:
2642:
2640:
2637:
2635:
2632:
2630:
2627:
2625:
2622:
2620:
2617:
2615:
2612:
2611:
2609:
2594:
2593:
2584:
2582:
2581:
2572:
2571:
2568:
2562:
2559:
2557:
2554:
2552:
2549:
2547:
2544:
2542:
2539:
2537:
2534:
2532:
2529:
2527:
2524:
2522:
2519:
2517:
2514:
2512:
2509:
2507:
2504:
2502:
2499:
2497:
2496:Conway sphere
2494:
2492:
2489:
2487:
2484:
2482:
2479:
2478:
2476:
2472:
2466:
2463:
2461:
2458:
2456:
2453:
2451:
2448:
2446:
2443:
2441:
2438:
2436:
2433:
2431:
2428:
2427:
2425:
2423:
2416:
2410:
2406:
2403:
2401:
2398:
2396:
2393:
2389:
2386:
2385:
2384:
2381:
2379:
2376:
2372:
2369:
2367:
2364:
2362:
2359:
2357:
2354:
2352:
2349:
2348:
2347:
2344:
2342:
2339:
2337:
2334:
2332:
2329:
2327:
2324:
2322:
2319:
2317:
2314:
2312:
2309:
2307:
2304:
2302:
2299:
2295:
2292:
2291:
2290:
2287:
2285:
2282:
2278:
2275:
2274:
2273:
2270:
2268:
2267:Arf invariant
2265:
2263:
2260:
2259:
2257:
2255:
2251:
2235:
2232:
2220:
2217:
2205:
2202:
2195:
2192:
2185:
2182:
2175:
2172:
2165:
2162:
2161:
2159:
2157:
2153:
2147:
2144:
2140:
2137:
2135:
2132:
2131:
2130:
2127:
2126:
2124:
2122:
2118:
2112:
2109:
2097:
2094:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2061:
2054:
2051:
2044:
2041:
2034:
2031:
2029:
2023:
2021:
2015:
2008:
2005:
1998:
1995:
1988:
1985:
1984:
1982:
1980:
1976:
1971:
1967:
1963:
1956:
1951:
1949:
1944:
1942:
1937:
1936:
1933:
1924:
1923:
1918:
1915:
1910:
1907:
1906:
1901:
1897:
1896:
1892:
1887:
1886:
1881:
1878:
1874:
1873:
1868:
1865:
1862:
1861:
1856:
1855:Marc Lackenby
1853:
1851:
1848:
1844:
1840:
1836:
1835:Marc Lackenby
1833:
1831:
1828:
1824:
1820:
1818:
1815:
1811:
1807:
1803:
1800:
1799:
1795:
1787:
1786:
1781:
1775:
1772:
1762:on 2020-07-27
1758:
1754:
1753:
1748:
1744:
1738:
1735:
1723:on 2006-02-09
1722:
1718:
1714:
1707:
1704:
1693:
1689:
1683:
1680:
1673:
1651:
1647:
1636:
1633:
1626:
1607:
1591:
1588:
1584:
1563:
1537:
1532:
1529:
1525:
1521:
1516:
1513:
1509:
1505:
1502:
1499:
1496:
1493:
1488:
1484:
1480:
1474:
1468:
1461:
1460:
1459:
1457:
1435:
1430:
1426:
1422:
1419:
1416:
1410:
1397:
1396:
1395:
1393:
1371:
1366:
1363:
1359:
1355:
1352:
1349:
1346:
1343:
1340:
1334:
1321:
1320:
1319:
1317:
1309:
1307:
1285:
1280:
1273:
1268:
1252:
1251:monodromy map
1248:
1235:
1231:
1225:
1221:
1215:
1211:
1210:
1207:
1205:
1201:
1197:
1193:
1189:
1184:
1182:
1178:
1174:
1150:
1145:
1128:
1125:
1119:
1115:
1111:
1102:
1094:
1089:
1087:
1084:
1080:
1076:
1072:
1069:
1066:
1062:
1058:
1054:
1050:
1046:
1038:
1018:
1007:
1003:
999:
996:
991:
987:
983:
980:
975:
971:
967:
964:
959:
955:
951:
945:
942:
937:
932:
929:
918:
910:
906:
902:
899:
894:
890:
886:
883:
878:
874:
870:
867:
862:
858:
854:
848:
845:
837:
833:
829:
824:
820:
816:
811:
807:
803:
798:
794:
787:
782:
774:
768:
765:
762:
759:
756:
753:
750:
744:
733:
732:
731:
712:
706:
703:
700:
697:
692:
688:
684:
679:
675:
671:
668:
665:
662:
656:
653:
647:
644:
641:
638:
635:
632:
629:
623:
616:
615:
614:
608:
600:
596:
592:
588:
580:
575:
573:
553:
550:
545:
543:
539:
535:
531:
527:
522:
520:
493:
490:
483:
480:
477:
475:
470:
459:
456:
449:
446:
442:
434:
431:
424:
421:
418:
415:
411:
407:
405:
400:
389:
386:
379:
376:
372:
364:
361:
354:
351:
348:
345:
341:
337:
335:
330:
319:
318:
317:
315:
311:
307:
299:
297:
295:
287:
285:
283:
279:
275:
271:
267:
264:(also called
263:
259:
250:
242:
238:
234:
230:
226:
222:
218:
213:
210:
203:
197:
195:
187:
183:
181:
177:
170:
168:
164:
161:
159:
155:
151:
149:
145:
141:
139:
135:
131:
129:
125:
121:
119:
115:
111:
109:
105:
101:
99:
95:
91:
89:
85:
81:
79:
75:
71:
69:
65:
61:
59:
58:Arf invariant
55:
52:
49:
45:
41:
36:
31:
26:
22:
2590:
2578:
2506:Double torus
2491:Braid theory
2306:Crossing no.
2301:Crosscap no.
1987:Figure-eight
1986:
1920:
1903:
1884:
1871:
1867:Robion Kirby
1859:
1838:
1822:
1805:
1783:
1774:
1764:, retrieved
1757:the original
1751:
1737:
1725:. Retrieved
1721:the original
1716:
1706:
1695:. Retrieved
1691:
1682:
1635:
1555:
1453:
1389:
1313:
1244:
1191:
1185:
1177:double-cover
1090:
1042:
729:
590:
586:
576:
552:closed braid
548:
547:(1) It is a
546:
542:fibered knot
523:
518:
516:
313:
309:
305:
303:
291:
278:trefoil knot
265:
261:
255:
167:A–B notation
108:Crossing no.
98:Crosscap no.
68:Braid length
2654:3-manifolds
2649:Twist knots
2614:Knot theory
2341:Linking no.
2262:Alternating
2063:Conway knot
2043:Carrick mat
1997:Three-twist
1962:Knot theory
1086:irreducible
1057:decomposing
1045:3-manifolds
595:John Milnor
549:homogeneous
530:alternating
300:Description
258:knot theory
221:alternating
47:Common name
2608:Categories
2501:Complement
2465:Tabulation
2422:operations
2346:Polynomial
2336:Link group
2331:Knot group
2294:Invertible
2272:Bridge no.
2254:Invariants
2184:Cinquefoil
2053:Perko pair
1979:Hyperbolic
1766:2020-10-19
1697:2020-06-25
1674:References
1310:Invariants
1141:(sequence
1129:2.02988...
1071:tetrahedra
1068:hyperbolic
1061:complement
1053:hyperbolic
599:Milnor map
282:prime knot
225:hyperbolic
184:4, 6, 8, 2
88:Bridge no.
2395:Stick no.
2351:Alexander
2289:Chirality
2234:Solomon's
2194:Septafoil
2121:Satellite
2081:Whitehead
2007:Stevedore
1922:MathWorld
1648:σ
1589:−
1530:−
1514:−
1506:−
1494:−
1423:−
1405:∇
1364:−
1356:−
1344:−
1329:Δ
1159:Λ
1151:), where
1126:≈
1112:π
1106:Λ
1063:into two
997:−
981:−
965:−
900:−
884:−
868:−
685:−
484:
450:
425:
380:
355:
138:Stick no.
78:Braid no.
2580:Category
2450:Mutation
2418:Notation
2371:Kauffman
2284:Brunnian
2277:2-bridge
2146:Knot sum
2077:(12n242)
1802:Ian Agol
1454:and the
1196:Lackenby
534:rational
316:) where
276:and the
25:Figure 8
2592:Commons
2511:Fibered
2409:problem
2378:Pretzel
2356:Bracket
2174:Trefoil
2111:L10a140
2071:(11n42)
2065:(11n34)
2033:Endless
1850:1756996
1830:1869847
1817:1799796
1179:of the
1171:is the
1147:in the
1144:A091518
538:achiral
229:fibered
204:/
132:2.02988
2556:Writhe
2526:Ribbon
2361:HOMFLY
2204:Unlink
2164:Unknot
2139:Square
2134:Granny
1541:
1439:
1375:
1093:volume
1081:, non-
927:
778:
730:where
597:) the
274:unknot
239:,
235:,
231:,
227:,
223:,
192:
190:Last /
2546:Twist
2531:Slice
2486:Berge
2474:Other
2445:Flype
2383:Prime
2366:Jones
2326:Genus
2156:Torus
1970:links
1966:knots
1760:(PDF)
1727:5 May
1627:Notes
1079:Haken
1065:ideal
1055:, by
526:prime
241:twist
233:prime
215:Other
118:Genus
2551:Wild
2516:Knot
2420:and
2407:and
2388:list
2219:Hopf
1968:and
1729:2022
1576:and
1390:the
1314:The
1202:and
1149:OEIS
1059:its
517:for
260:, a
194:Next
2536:Sum
2057:161
2055:(10
1902:",
1900:4_1
1782:",
1780:4_1
1458:is
1394:is
601:of
481:sin
447:sin
422:cos
377:cos
352:cos
256:In
2610::
2236:(4
2221:(2
2206:(0
2196:(7
2186:(5
2176:(3
2166:(0
2098:(6
2083:(5
2047:18
2045:(8
2035:(7
2009:(6
1999:(5
1989:(4
1919:.
1869:,
1847:MR
1841:,
1837:,
1827:MR
1814:MR
1808:,
1804:,
1749:,
1715:.
1690:.
1306:.
1095:,
585::
532:,
528:,
284:.
2245:)
2241:1
2230:)
2226:1
2215:)
2211:1
2200:)
2198:1
2190:)
2188:1
2180:)
2178:1
2170:)
2168:1
2107:)
2103:2
2092:)
2088:1
2059:)
2049:)
2039:)
2037:4
2027:3
2025:6
2019:2
2017:6
2013:)
2011:1
2003:)
2001:2
1993:)
1991:1
1972:)
1964:(
1954:e
1947:t
1940:v
1925:.
1898:"
1788:.
1778:"
1731:.
1700:.
1652:i
1592:1
1585:q
1564:q
1538:.
1533:2
1526:q
1522:+
1517:1
1510:q
1503:1
1500:+
1497:q
1489:2
1485:q
1481:=
1478:)
1475:q
1472:(
1469:V
1436:,
1431:2
1427:z
1420:1
1417:=
1414:)
1411:z
1408:(
1372:,
1367:1
1360:t
1353:3
1350:+
1347:t
1341:=
1338:)
1335:t
1332:(
1294:)
1286:1
1281:1
1274:1
1269:2
1261:(
1123:)
1120:3
1116:/
1109:(
1103:6
1019:.
1016:)
1013:)
1008:2
1004:t
1000:2
992:2
988:z
984:2
976:2
972:y
968:2
960:2
956:x
952:6
949:(
946:y
943:+
938:2
933:x
930:t
919:,
916:)
911:2
907:t
903:2
895:2
891:z
887:2
879:2
875:y
871:2
863:2
859:x
855:6
852:(
849:x
846:+
843:)
838:2
834:t
830:+
825:2
821:z
817:+
812:2
808:y
804:+
799:2
795:x
791:(
788:z
785:(
775:=
772:)
769:t
766:,
763:z
760:,
757:y
754:,
751:x
748:(
745:G
713:,
710:)
707:t
704:z
701:2
698:,
693:2
689:t
680:2
676:z
672:,
669:y
666:,
663:x
660:(
657:G
654:=
651:)
648:t
645:,
642:z
639:,
636:y
633:,
630:x
627:(
624:F
611:F
603:F
591:R
589:→
587:R
583:F
568:2
566:σ
564:1
562:σ
560:2
558:σ
556:1
519:t
497:)
494:t
491:4
488:(
478:=
471:z
463:)
460:t
457:3
454:(
443:)
438:)
435:t
432:2
429:(
419:+
416:2
412:(
408:=
401:y
393:)
390:t
387:3
384:(
373:)
368:)
365:t
362:2
359:(
349:+
346:2
342:(
338:=
331:x
314:z
312:,
310:y
308:,
306:x
208:1
206:5
201:1
199:3
173:1
171:4
152:1
142:7
122:1
112:4
102:2
92:2
82:3
72:4
62:1
27:.
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