244:
259:
214:
31:
229:
1857:
80:
1431:
701:
1052:
958:
357:, the boundary being the knot, i.e. homeomorphic to the unit circle. The genus of such a surface is defined to be the genus of the two-manifold, which is obtained by gluing the unit disk along the boundary.
1468:
156:
In layman's terms, the genus is the number of "holes" an object has ("holes" interpreted in the sense of doughnut holes; a hollow sphere would be considered as having zero holes in this sense). A
1717:. Classics in Mathematics. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel (Reprint of the 2nd, corr. print. of the 3rd ed.). Berlin:
775:
1518:. In particular, one may study the growth of the genus along the chain. Such a function (called the genus trace) shows the topological complexity and domain structure of biomolecules.
1111:
1224:
1302:
189:
has genus one, as does the surface of a coffee mug with a handle. This is the source of the joke "topologists are people who can't tell their donut from their coffee mug."
1500:
862:
1271:
1193:
1165:
1138:
1311:
1244:
833:
798:
622:
627:
1749:
Charles Rezk - Elliptic cohomology and elliptic curves (Felix Klein lectures, Bonn 2015. Department of
Mathematics, University of Illinois, Urbana, IL)
1844:
1726:
1638:
964:
243:
870:
1542:
1759:
Sułkowski, Piotr; Sulkowska, Joanna I.; Dabrowski-Tumanski, Pawel; Andersen, Ebbe Sloth; Geary, Cody; Zając, Sebastian (2018-12-03).
1919:
1865:
297:. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 −
1436:
1630:
258:
213:
1924:
442:
419:
1929:
1904:
1894:
1622:
709:
228:
1914:
1909:
1547:
601:
1058:
484:
1247:
54:) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a
354:
809:
313:
1201:
1772:
1710:
531:
112:
108:
1875:
incorrectly led you here, you may wish to change the link to point directly to the intended article.
1527:
1280:
554:
488:
377:
without rendering the resultant manifold disconnected. It is equal to the number of handles on it.
201:
138:
586:
385:
374:
339:
179:
127:
55:
434:). Thus, a planar graph has genus 0, because it can be drawn on a sphere without self-crossing.
289:
of a connected, non-orientable closed surface is a positive integer representing the number of
1899:
1889:
1840:
1806:
1788:
1722:
1684:
1634:
1476:
1305:
1196:
838:
813:
527:
1796:
1780:
1732:
1692:
1676:
1532:
538:
1256:
1178:
1143:
1116:
569:, then these definitions agree and coincide with the topological definition applied to the
30:
17:
1836:
1736:
1718:
1696:
1537:
1510:
Genus can be also calculated for the graph spanned by the net of chemical interactions in
1426:{\displaystyle \log _{\Phi }(x)=\int _{0}^{x}(1-2\delta t^{2}+\varepsilon t^{4})^{-1/2}dt}
570:
550:
542:
507:
409:
346:
334:
194:
92:
1776:
1801:
1760:
1229:
818:
783:
607:
592:
582:
558:
234:
1883:
1761:"Genus trace reveals the topological complexity and domain structure of biomolecules"
1680:
566:
276:
1552:
1511:
696:{\displaystyle s\in \Gamma (\mathbb {P} ^{2},{\mathcal {O}}_{\mathbb {P} ^{2}}(d))}
496:
320:
219:
79:
1830:
511:
39:
1784:
1868:
includes a list of related items that share the same name (or similar names).
1502:
is not a genus in this sense since it is not invariant concerning cobordisms.
370:
1792:
1688:
1627:
The Knot Book: An
Elementary Introduction to the Mathematical Theory of Knots
1168:
290:
1856:
1810:
1597:
472:
such that the graph can be drawn without crossing itself on a sphere with
449:
such that the graph can be drawn without crossing itself on a sphere with
426:
such that the graph can be drawn without crossing itself on a sphere with
164:
has 0. The green surface pictured above has 2 holes of the relevant sort.
578:
491:. Arthur T. White introduced the following concept. The genus of a group
373:
is an integer representing the maximum number of cuttings along embedded
104:
1515:
1047:{\displaystyle \Phi (M_{1}\times M_{2})=\Phi (M_{1})\cdot \Phi (M_{2})}
96:
1667:
Thomassen, Carsten (1989). "The graph genus problem is NP-complete".
1274:
294:
249:
172:
161:
83:
The coffee cup and donut shown in this animation both have genus one.
59:
453:
cross-caps (i.e. a non-orientable surface of (non-orientable) genus
99:
representing the maximum number of cuttings along non-intersecting
953:{\displaystyle \Phi (M_{1}\amalg M_{2})=\Phi (M_{1})+\Phi (M_{2})}
186:
157:
100:
78:
63:
29:
591:
connected non-singular projective curve of genus 1 with a given
661:
111:
on it. Alternatively, it can be defined in terms of the
1872:
800:
is the number of singularities when properly counted.
1479:
1463:{\displaystyle \delta ,\varepsilon \in \mathbb {C} .}
1439:
1314:
1283:
1259:
1232:
1204:
1181:
1146:
1119:
1061:
967:
873:
841:
821:
786:
712:
630:
610:
581:of complex points). For example, the definition of
1494:
1462:
1425:
1296:
1265:
1238:
1218:
1187:
1159:
1132:
1105:
1046:
952:
856:
827:
792:
769:
695:
616:
495:is the minimum genus of a (connected, undirected)
1277:manifolds with a connected compact structure if
487:there are several definitions of the genus of a
1862:Index of articles associated with the same name
430:handles (i.e. an oriented surface of the genus
1583:
1571:
8:
770:{\displaystyle g={\frac {(d-1)(d-2)}{2}}-s,}
353:. A Seifert surface of a knot is however a
107:disconnected. It is equal to the number of
624:given by the vanishing locus of a section
1800:
1715:Topological methods in algebraic geometry
1478:
1470:This genus is called an elliptic genus.
1453:
1452:
1438:
1407:
1400:
1390:
1374:
1349:
1344:
1319:
1313:
1288:
1282:
1258:
1231:
1212:
1211:
1203:
1180:
1151:
1145:
1124:
1118:
1106:{\displaystyle \Phi (M_{1})=\Phi (M_{2})}
1094:
1072:
1060:
1035:
1013:
991:
978:
966:
941:
919:
897:
884:
872:
840:
820:
785:
719:
711:
673:
669:
668:
666:
660:
659:
649:
645:
644:
629:
609:
1564:
604:, an irreducible plane curve of degree
345:is defined as the minimal genus of all
206:
1273:is multiplicative for all bundles on
522:There are two related definitions of
7:
457:). (This number is also called the
835:may be defined as a complex number
47:
1543:Genus of a multiplicative sequence
1320:
1289:
1260:
1182:
1084:
1062:
1025:
1003:
968:
931:
909:
874:
842:
637:
25:
1219:{\displaystyle R\to \mathbb {C} }
1855:
257:
252:: Double Toroidal graph: genus 2
242:
227:
212:
134:is the genus. For surfaces with
103:without rendering the resultant
1829:Popescu-Pampu, Patrick (2016).
476:cross-caps or on a sphere with
200:is given in the article on the
141:components, the equation reads
1489:
1483:
1397:
1355:
1334:
1328:
1208:
1100:
1087:
1078:
1065:
1041:
1028:
1019:
1006:
997:
971:
947:
934:
925:
912:
903:
877:
851:
845:
749:
737:
734:
722:
690:
687:
681:
640:
27:Number of "holes" of a surface
1:
1631:American Mathematical Society
1574:, p. xiii, Introduction.
1297:{\displaystyle \log _{\Phi }}
316:has a non-orientable genus 1.
305:is the non-orientable genus.
1681:10.1016/0196-6774(89)90006-0
1586:, p. xiv, Introduction.
208:Genus of orientable surfaces
122: = 2 − 2
323:has non-orientable genus 2.
95:, orientable surface is an
18:Genus of an algebraic curve
1946:
1854:
1785:10.1038/s41598-018-35557-3
864:subject to the conditions
407:
1548:Genus of a quadratic form
1473:The Euler characteristic
193:Explicit construction of
160:has 1 such hole, while a
1920:Topological graph theory
1495:{\displaystyle \chi (M)}
857:{\displaystyle \Phi (M)}
485:topological graph theory
1248:oriented cobordism ring
468:is the minimal integer
445:is the minimal integer
422:is the minimal integer
270:Non-orientable surfaces
118:, via the relationship
1496:
1464:
1427:
1298:
1267:
1240:
1220:
1189:
1161:
1134:
1107:
1048:
954:
858:
829:
794:
771:
697:
618:
355:manifold with boundary
264:Pretzel graph: genus 3
195:surfaces of the genus
145:= 2 − 2
84:
35:
1711:Hirzebruch, Friedrich
1669:Journal of Algorithms
1497:
1465:
1428:
1299:
1268:
1266:{\displaystyle \Phi }
1241:
1221:
1190:
1188:{\displaystyle \Phi }
1162:
1160:{\displaystyle M_{2}}
1135:
1133:{\displaystyle M_{1}}
1108:
1049:
955:
859:
830:
810:differential geometry
804:Differential geometry
795:
772:
698:
619:
314:real projective plane
182:both have genus zero.
82:
62:has genus 0, while a
33:
1477:
1437:
1312:
1281:
1257:
1230:
1202:
1179:
1144:
1117:
1059:
965:
871:
839:
819:
784:
710:
703:has geometric genus
628:
608:
602:Riemann–Roch theorem
439:non-orientable genus
113:Euler characteristic
101:closed simple curves
1925:Geometry processing
1777:2018NatSR...817537Z
1528:Group (mathematics)
1354:
508:graph genus problem
369:of a 3-dimensional
202:fundamental polygon
75:Orientable surfaces
1930:Set index articles
1905:Algebraic topology
1895:Geometric topology
1832:What is the Genus?
1765:Scientific Reports
1654:Graphs on surfaces
1584:Popescu-Pampu 2016
1572:Popescu-Pampu 2016
1492:
1460:
1423:
1340:
1294:
1263:
1236:
1216:
1185:
1157:
1130:
1103:
1044:
950:
854:
825:
790:
767:
693:
614:
587:algebraic geometry
557:of definition the
518:Algebraic geometry
85:
36:
1866:set index article
1846:978-3-319-42312-8
1728:978-3-540-58663-0
1640:978-0-8218-3678-1
1306:elliptic integral
1239:{\displaystyle R}
1197:ring homomorphism
828:{\displaystyle M}
814:oriented manifold
793:{\displaystyle s}
756:
617:{\displaystyle d}
34:A genus-2 surface
16:(Redirected from
1937:
1915:Graph invariants
1910:Algebraic curves
1876:
1859:
1850:
1815:
1814:
1804:
1756:
1750:
1747:
1741:
1740:
1707:
1701:
1700:
1664:
1658:
1657:
1650:
1644:
1643:
1619:
1613:
1612:
1610:
1608:
1596:Weisstein, E.W.
1593:
1587:
1581:
1575:
1569:
1533:Arithmetic genus
1501:
1499:
1498:
1493:
1469:
1467:
1466:
1461:
1456:
1432:
1430:
1429:
1424:
1416:
1415:
1411:
1395:
1394:
1379:
1378:
1353:
1348:
1324:
1323:
1303:
1301:
1300:
1295:
1293:
1292:
1272:
1270:
1269:
1264:
1245:
1243:
1242:
1237:
1225:
1223:
1222:
1217:
1215:
1194:
1192:
1191:
1186:
1175:In other words,
1166:
1164:
1163:
1158:
1156:
1155:
1139:
1137:
1136:
1131:
1129:
1128:
1112:
1110:
1109:
1104:
1099:
1098:
1077:
1076:
1053:
1051:
1050:
1045:
1040:
1039:
1018:
1017:
996:
995:
983:
982:
959:
957:
956:
951:
946:
945:
924:
923:
902:
901:
889:
888:
863:
861:
860:
855:
834:
832:
831:
826:
812:, a genus of an
799:
797:
796:
791:
776:
774:
773:
768:
757:
752:
720:
702:
700:
699:
694:
680:
679:
678:
677:
672:
665:
664:
654:
653:
648:
623:
621:
620:
615:
539:arithmetic genus
347:Seifert surfaces
261:
246:
231:
216:
49:
21:
1945:
1944:
1940:
1939:
1938:
1936:
1935:
1934:
1880:
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1869:
1863:
1853:
1847:
1837:Springer Verlag
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1729:
1719:Springer-Verlag
1709:
1708:
1704:
1666:
1665:
1661:
1652:
1651:
1647:
1641:
1621:
1620:
1616:
1606:
1604:
1595:
1594:
1590:
1582:
1578:
1570:
1566:
1561:
1538:Geometric genus
1524:
1508:
1475:
1474:
1435:
1434:
1396:
1386:
1370:
1315:
1310:
1309:
1284:
1279:
1278:
1255:
1254:
1228:
1227:
1200:
1199:
1177:
1176:
1147:
1142:
1141:
1120:
1115:
1114:
1090:
1068:
1057:
1056:
1031:
1009:
987:
974:
963:
962:
937:
915:
893:
880:
869:
868:
837:
836:
817:
816:
806:
782:
781:
721:
708:
707:
667:
658:
643:
626:
625:
606:
605:
571:Riemann surface
567:singular points
559:complex numbers
551:algebraic curve
543:geometric genus
520:
412:
410:Graph embedding
406:
363:
330:
272:
265:
262:
253:
247:
238:
232:
223:
217:
128:closed surfaces
77:
72:
28:
23:
22:
15:
12:
11:
5:
1943:
1941:
1933:
1932:
1927:
1922:
1917:
1912:
1907:
1902:
1897:
1892:
1882:
1881:
1861:
1860:
1852:
1851:
1845:
1825:
1823:
1820:
1817:
1816:
1751:
1742:
1727:
1702:
1675:(4): 568–576.
1659:
1645:
1639:
1614:
1588:
1576:
1563:
1562:
1560:
1557:
1556:
1555:
1550:
1545:
1540:
1535:
1530:
1523:
1520:
1507:
1504:
1491:
1488:
1485:
1482:
1459:
1455:
1451:
1448:
1445:
1442:
1422:
1419:
1414:
1410:
1406:
1403:
1399:
1393:
1389:
1385:
1382:
1377:
1373:
1369:
1366:
1363:
1360:
1357:
1352:
1347:
1343:
1339:
1336:
1333:
1330:
1327:
1322:
1318:
1291:
1287:
1262:
1235:
1214:
1210:
1207:
1184:
1173:
1172:
1154:
1150:
1127:
1123:
1102:
1097:
1093:
1089:
1086:
1083:
1080:
1075:
1071:
1067:
1064:
1054:
1043:
1038:
1034:
1030:
1027:
1024:
1021:
1016:
1012:
1008:
1005:
1002:
999:
994:
990:
986:
981:
977:
973:
970:
960:
949:
944:
940:
936:
933:
930:
927:
922:
918:
914:
911:
908:
905:
900:
896:
892:
887:
883:
879:
876:
853:
850:
847:
844:
824:
805:
802:
789:
778:
777:
766:
763:
760:
755:
751:
748:
745:
742:
739:
736:
733:
730:
727:
724:
718:
715:
692:
689:
686:
683:
676:
671:
663:
657:
652:
647:
642:
639:
636:
633:
613:
593:rational point
583:elliptic curve
519:
516:
408:Main article:
405:
402:
401:
400:
391:A solid torus
389:
380:For instance:
362:
359:
329:
326:
325:
324:
317:
308:For instance:
293:attached to a
277:non-orientable
271:
268:
267:
266:
263:
256:
254:
248:
241:
239:
235:Toroidal graph
233:
226:
224:
218:
211:
209:
191:
190:
183:
167:For instance:
76:
73:
71:
68:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1942:
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1926:
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1913:
1911:
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1885:
1874:
1873:internal link
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1599:
1592:
1589:
1585:
1580:
1577:
1573:
1568:
1565:
1558:
1554:
1551:
1549:
1546:
1544:
1541:
1539:
1536:
1534:
1531:
1529:
1526:
1525:
1521:
1519:
1517:
1513:
1512:nucleic acids
1505:
1503:
1486:
1480:
1471:
1457:
1449:
1446:
1443:
1440:
1420:
1417:
1412:
1408:
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1401:
1391:
1387:
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1380:
1375:
1371:
1367:
1364:
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1325:
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1205:
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1170:
1152:
1148:
1125:
1121:
1095:
1091:
1081:
1073:
1069:
1055:
1036:
1032:
1022:
1014:
1010:
1000:
992:
988:
984:
979:
975:
961:
942:
938:
928:
920:
916:
906:
898:
894:
890:
885:
881:
867:
866:
865:
848:
822:
815:
811:
803:
801:
787:
764:
761:
758:
753:
746:
743:
740:
731:
728:
725:
716:
713:
706:
705:
704:
684:
674:
655:
650:
634:
631:
611:
603:
598:
596:
594:
588:
584:
580:
576:
572:
568:
564:
560:
556:
552:
548:
544:
540:
536:
533:
529:
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517:
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513:
509:
504:
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498:
494:
490:
486:
481:
479:
475:
471:
467:
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460:
456:
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435:
433:
429:
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421:
417:
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403:
398:
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390:
387:
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381:
378:
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372:
368:
360:
358:
356:
352:
348:
344:
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315:
311:
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309:
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296:
292:
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1623:Adams, Colin
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1601:
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1567:
1553:Spinor genus
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1472:
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404:Graph theory
399:has genus 1.
396:
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388:has genus 0.
379:
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331:
321:Klein bottle
307:
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512:NP-complete
466:Euler genus
287:Euler genus
40:mathematics
1884:Categories
1822:References
1737:0843.14009
1697:0689.68071
1253:The genus
1246:is Thom's
530:algebraic
528:projective
371:handlebody
361:Handlebody
291:cross-caps
222:: genus 0
1793:2045-2322
1713:(1995) .
1689:0196-6774
1602:MathWorld
1559:Citations
1481:χ
1450:∈
1447:ε
1441:δ
1433:for some
1402:−
1384:ε
1368:δ
1362:−
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1290:Φ
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1169:cobordant
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1004:Φ
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969:Φ
932:Φ
910:Φ
891:⨿
875:Φ
843:Φ
759:−
744:−
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638:Γ
635:∈
561:, and if
480:handles.
459:demigenus
283:demigenus
237:: genus 1
93:connected
1900:Surfaces
1890:Topology
1811:30510290
1625:(2004),
1522:See also
1516:proteins
1308:such as
1226:, where
579:manifold
545:. When
541:and the
301:, where
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130:, where
105:manifold
70:Topology
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1773:Bibcode
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1506:Biology
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295:sphere
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178:and a
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