4904:
3075:
4916:
3504:
of knot Floer homology is the
Alexander polynomial. While the Alexander polynomial gives a lower bound on the genus of a knot, showed that knot Floer homology detects the genus. Similarly, while the Alexander polynomial gives an obstruction to a knot complement fibering over the circle, showed that
666:
Consider the entry corresponding to a particular region and crossing. If the region is not adjacent to the crossing, the entry is 0. If the region is adjacent to the crossing, the entry depends on its location. The following table gives the entry, determined by the location of the region at the
2629:
1177:
3479:
2886:
3202:
2969:
later rediscovered this in a different form and showed that the skein relation together with a choice of value on the unknot was enough to determine the polynomial. Conway's version is a polynomial in
3286:
2772:
2219:
1058:
2708:
2487:
1979:
Kauffman describes the first construction of the
Alexander polynomial via state sums derived from physical models. A survey of these topic and other connections with physics are given in.
2394:
2278:
1928:
3505:
knot Floer homology completely determines when a knot complement fibers over the circle. The knot Floer homology groups are part of the
Heegaard Floer homology family of invariants; see
946:
444:. It turns out that the Alexander polynomial of a knot is the same polynomial for the mirror image knot. In other words, it cannot distinguish between a knot and its mirror image.
2955:
1583:
1377:
2079:
2001:
Knots with symmetries are known to have restricted
Alexander polynomials. Nonetheless, the Alexander polynomial can fail to detect some symmetries, such as strong invertibility.
3069:
2657:
1837:
1753:
1717:
1681:
982:
442:
3117:
2434:
1424:
1338:
1291:
1244:
246:
3351:
3313:
661:
3500:
Using pseudo-holomorphic curves, Ozsváth-Szabó and
Rasmussen associated a bigraded abelian group, called knot Floer homology, to each isotopy class of knots. The graded
3001:
59:
in 1984. Soon after Conway's reworking of the
Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexander's paper on his polynomial.
820:
790:
399:
331:
2138:
192:
152:
2512:
2335:
545:
1783:
1534:
1484:
1454:
365:
2106:
2033:
1589:
From the point of view of the definition, this is an expression of the fact that the knot complement is a homology circle, generated by the covering transformation
1957:
611:
503:
883:
761:
692:
408:
Alexander proved that the
Alexander ideal is nonzero and always principal. Thus an Alexander polynomial always exists, and is clearly a knot invariant, denoted
2906:
2306:
1627:
1607:
1504:
1197:
860:
840:
736:
714:
585:
565:
477:
305:
285:
260:
2517:
1982:
There are other relations with surfaces and smooth 4-dimensional topology. For example, under certain assumptions, there is a way of modifying a smooth
1070:
3071:
are link diagrams resulting from crossing and smoothing changes on a local region of a specified crossing of the diagram, as indicated in the figure.
3764:
3356:
2780:
3525:
Alexander describes his skein relation toward the end of his paper under the heading "miscellaneous theorems", which is possibly why it got lost.
371:, take a generator; this is called an Alexander polynomial of the knot. Since this is only unique up to multiplication by the Laurent monomial
3600:, Theorem 11.5.3, p. 150. Kawauchi credits this result to Kondo, H. (1979), "Knots of unknotting number 1 and their Alexander polynomials",
3756:
3124:
3210:
1789:
Every integral
Laurent polynomial which is both symmetric and evaluates to a unit at 1 is the Alexander polynomial of a knot.
4281:
4207:
4000:
3979:
3957:
3933:
3741:
842:
is not necessarily the number of crossings in the knot. To resolve this ambiguity, divide out the largest possible power of
4849:
2713:
401:, one often fixes a particular unique form. Alexander's choice of normalization is to make the polynomial have a positive
2147:
1990:
that consists of removing a neighborhood of a two-dimensional torus and replacing it with a knot complement crossed with
995:
4768:
4124:
4062:
2439:
1976:; i.e., bounds a "locally-flat" topological disc in the 4-ball, if the Alexander polynomial of the knot is trivial.
103:
and gluing together infinitely many copies of the resulting manifold with boundary in a cyclic manner. There is a
2340:
2224:
1860:
4315:
2662:
902:
4007:(covers several different approaches, explains relations between different versions of the Alexander polynomial)
766:
Remove two columns corresponding to adjacent regions from the matrix, and work out the determinant of the new
4763:
4758:
4634:
4235:
4920:
4335:
4214:(explains classical approach using the Alexander invariant; knot and link table with Alexander polynomials)
1995:
4225:
2915:
1543:
1343:
452:
The following procedure for computing the
Alexander polynomial was given by J. W. Alexander in his paper.
4957:
4397:
4230:
3917:
3729:
2038:
3752:
3021:
2634:
36:
1800:
4947:
4467:
4462:
4403:
4274:
1722:
1686:
1632:
951:
411:
4962:
3830:
3087:
1382:
1296:
1249:
1202:
204:
4595:
4071:
3318:
3291:
616:
2977:
4809:
4778:
795:
769:
374:
310:
3529:
mentions in her paper that Mark
Kidwell brought her attention to Alexander's relation in 1970.
2111:
161:
121:
4639:
4129:
2399:
512:
1758:
1509:
1459:
1429:
505:
regions of the knot diagram. To work out the Alexander polynomial, first one must create an
344:
4952:
4942:
4908:
4267:
4183:
4090:
4030:
3877:
3849:
3501:
2084:
2015:
195:
1933:
948:, and introduced non-commutative differential calculus, which also permits one to compute
885:
if necessary, so that the constant term is positive. This gives the Alexander polynomial.
8:
4716:
4699:
3913:
1973:
1854:
1847:
590:
482:
104:
4187:
4094:
4034:
3853:
3604:
16: 551-559, and to Sakai, T. (1977), "A remark on the Alexander polynomials of knots",
3538:
Detailed exposition of this approach about higher Alexander polynomials can be found in
2492:
2315:
1065:
865:
743:
674:
4737:
4684:
4298:
4294:
4173:
4156:
4138:
4108:
4080:
4046:
4020:
3922:
3881:
3839:
3783:
2966:
2891:
2291:
1612:
1592:
1489:
1182:
845:
825:
792:
matrix. Depending on the columns removed, the answer will differ by multiplication by
721:
699:
570:
550:
462:
290:
270:
199:
72:
44:
3778:
4834:
4783:
4733:
4689:
4649:
4644:
4562:
4247:
4203:
4050:
3996:
3975:
3953:
3929:
3901:
3865:
3737:
4160:
4112:
4869:
4694:
4590:
4325:
4148:
4098:
4038:
3885:
3857:
3821:
3773:
2624:{\displaystyle \Delta _{K}(t)=\Delta _{f(S^{1}\times \{0\})}(t^{a})\Delta _{K'}(t)}
1969:
506:
56:
4120:
4058:
1172:{\displaystyle {\overline {H_{1}X}}\simeq \mathrm {Hom} _{\mathbb {Z} }(H_{1}X,G)}
4829:
4793:
4728:
4674:
4629:
4622:
4512:
4424:
4307:
3990:
3947:
3873:
3825:
2008:
fibers over the circle, then the Alexander polynomial of the knot is known to be
2005:
1994:. The result is a smooth 4-manifold homeomorphic to the original, though now the
1963:
368:
96:
88:
40:
4259:
1719:
defined as the order ideal of its infinite-cyclic covering space. In this case
4889:
4788:
4750:
4669:
4582:
4457:
4449:
4409:
4252:
3967:
3943:
3506:
2909:
2309:
1998:
has been modified by multiplication with the Alexander polynomial of the knot.
1987:
1839:
889:
84:
52:
28:
4103:
4066:
4042:
3474:{\displaystyle \Delta (L_{+})-\Delta (L_{-})=(t^{1/2}-t^{-1/2})\Delta (L_{0})}
2881:{\displaystyle \Delta _{K_{1}\#K_{2}}(t)=\Delta _{K_{1}}(t)\Delta _{K_{2}}(t)}
4936:
4824:
4612:
4605:
4600:
3869:
3734:
The Knot Book: An elementary introduction to the mathematical theory of knots
1843:
402:
334:
4256:. – knot and link tables with computed Alexander and Conway polynomials
3905:
4839:
4819:
4723:
4706:
4502:
4439:
4195:
3928:. Princeton Mathematical Series. Vol. 39. Princeton University Press.
4152:
3861:
4854:
4617:
4522:
4391:
4371:
4361:
4353:
4345:
4290:
3526:
3489:
20:
95:. This covering can be obtained by cutting the knot complement along a
55:, although its significance was not realized until the discovery of the
4874:
4859:
4814:
4711:
4664:
4659:
4654:
4484:
4381:
3787:
2965:
Alexander proved the Alexander polynomial satisfies a skein relation.
1983:
1064:
From the point of view of the definition, this is an expression of the
32:
3844:
4879:
4547:
4178:
4143:
4085:
4025:
3893:
3809:
2141:
2012:(the coefficients of the highest and lowest order terms are equal to
896:
613:
columns to the regions. The values for the matrix entries are either
3659:
4864:
4474:
456:
76:
3207:
The relationship to the standard Alexander polynomial is given by
1857:
knot, the Alexander polynomial satisfies the Fox–Milnor condition
667:
crossing from the perspective of the incoming undercrossing line.
3896:(1961). "A quick trip through knot theory". In Fort, M.K. (ed.).
3074:
1540:
Furthermore, the Alexander polynomial evaluates to a unit on 1:
4884:
4532:
4492:
4243:
3492:
for an example computing the Conway polynomial of the trefoil.
1755:
is, up to sign, equal to the order of the torsion subgroup of
114:. Consider the first homology (with integer coefficients) of
4773:
3748:(accessible introduction utilizing a skein relation approach)
3197:{\displaystyle \nabla (L_{+})-\nabla (L_{-})=z\nabla (L_{0})}
3900:. Englewood Cliffs. N. J.: Prentice-Hall. pp. 120–167.
1966:
is bounded below by the degree of the Alexander polynomial.
4844:
4011:
Ni, Yi (2007). "Knot Floer homology detects fibred knots".
3898:
Proceedings of the University of Georgia Topology Institute
1792:
3795:
Birman, Joan (1993). "New points of view in knot theory".
3647:
341:
and does not depend on choice of presentation matrix. If
3695:
3481:. Note that this relation gives a Laurent polynomial in
3671:
3611:
3281:{\displaystyle \Delta _{L}(t^{2})=\nabla _{L}(t-t^{-1})}
888:
The Alexander polynomial can also be computed from the
367:, set the ideal equal to 0. If the Alexander ideal is
929:
3974:(4th ed.). World Scientific Publishing Company.
3683:
3359:
3321:
3294:
3213:
3127:
3090:
3024:
3018:
Suppose we are given an oriented link diagram, where
2980:
2918:
2894:
2783:
2767:{\displaystyle H_{1}(S^{1}\times D^{2})=\mathbb {Z} }
2716:
2665:
2637:
2520:
2495:
2442:
2402:
2343:
2318:
2294:
2227:
2150:
2114:
2087:
2041:
2018:
1936:
1863:
1803:
1761:
1725:
1689:
1635:
1615:
1595:
1546:
1512:
1492:
1462:
1432:
1385:
1346:
1299:
1252:
1205:
1185:
1073:
998:
987:
954:
905:
868:
848:
828:
798:
772:
746:
724:
702:
677:
619:
593:
573:
553:
515:
485:
465:
414:
377:
347:
313:
293:
273:
207:
164:
124:
3567:
2283:
2214:{\displaystyle \Delta _{K}(t)={\rm {Det}}(tI-g_{*})}
287:, is less than or equal to the number of relations,
47:
showed a version of this polynomial, now called the
3635:
3623:
3921:
3473:
3345:
3315:must be properly normalized (by multiplication of
3307:
3280:
3196:
3111:
3063:
2995:
2949:
2900:
2880:
2766:
2702:
2651:
2623:
2506:
2481:
2428:
2388:
2329:
2300:
2272:
2213:
2132:
2100:
2073:
2027:
1951:
1922:
1831:
1777:
1747:
1711:
1675:
1621:
1601:
1577:
1528:
1498:
1478:
1448:
1418:
1371:
1332:
1285:
1238:
1191:
1171:
1053:{\displaystyle \Delta _{K}(t^{-1})=\Delta _{K}(t)}
1052:
976:
940:
877:
854:
834:
814:
784:
755:
730:
708:
686:
655:
605:
579:
559:
539:
497:
471:
436:
393:
359:
325:
299:
279:
240:
186:
146:
4289:
3828:(October 1998). "Knots, links, and 4-manifolds".
3765:Transactions of the American Mathematical Society
3555:
4934:
4127:(2004b). "Holomorphic disks and genus bounds".
3820:
3707:
3653:
3579:
2960:
2482:{\displaystyle S^{1}\times D^{2}\subset S^{3}}
899:considered a copresentation of the knot group
307:, then we consider the ideal generated by all
35:with integer coefficients to each knot type.
4275:
4119:
3912:
3701:
3617:
3495:
1842:the commutator subgroup of the knot group is
1199:is the quotient of the field of fractions of
4172:(Thesis). Harvard University. p. 6378.
4057:
3677:
2573:
2567:
2389:{\displaystyle f:S^{1}\times D^{2}\to S^{3}}
2273:{\displaystyle g_{*}\colon H_{1}S\to H_{1}S}
1923:{\displaystyle \Delta _{K}(t)=f(t)f(t^{-1})}
158:acts on the homology and so we can consider
3757:"Topological Invariants of Knots and Links"
2703:{\displaystyle K'\subset S^{1}\times D^{2}}
1959:is some other integral Laurent polynomial.
1456:ie: as an abelian group it is identical to
941:{\displaystyle \pi _{1}(S^{3}\backslash K)}
447:
4282:
4268:
3816:. Ginn and Co. after 1977 Springer Verlag.
3807:
3539:
4177:
4167:
4142:
4102:
4084:
4024:
3843:
3777:
3751:
3689:
3573:
2760:
2645:
1387:
1301:
1254:
1207:
1113:
333:minors of the matrix; this is the zeroth
209:
3988:
3966:
3942:
3665:
3641:
3629:
3597:
1797:Since the Alexander ideal is principal,
1793:Geometric significance of the polynomial
4194:
4067:"Holomorphic disks and knot invariants"
2489:is an unknotted solid torus containing
992:The Alexander polynomial is symmetric:
4935:
3794:
3561:
3119:(where O is any diagram of the unknot)
1972:proved that a knot in the 3-sphere is
259:The module is finitely presentable; a
4263:
3728:
4915:
2950:{\displaystyle \Delta _{K}(t)=\pm 1}
1578:{\displaystyle \Delta _{K}(1)=\pm 1}
1372:{\displaystyle {\overline {H_{1}X}}}
4202:(2nd ed.). Publish or Perish.
4170:Floer homology and knot complements
3892:
3585:
3081:Here are Conway's skein relations:
2974:with integer coefficients, denoted
2074:{\displaystyle S\to C_{K}\to S^{1}}
895:After the work of J. W. Alexander,
696:on the right before undercrossing:
13:
4010:
3713:
3452:
3382:
3360:
3296:
3244:
3215:
3175:
3150:
3128:
3091:
3073:
2981:
2920:
2853:
2827:
2799:
2785:
2598:
2544:
2522:
2181:
2178:
2175:
2152:
1865:
1805:
1727:
1691:
1548:
1107:
1104:
1101:
1032:
1000:
988:Basic properties of the polynomial
956:
740:on the right after undercrossing:
671:on the left before undercrossing:
416:
14:
4974:
4218:
3779:10.1090/S0002-9947-1928-1501429-1
3736:. American Mathematical Society.
3064:{\displaystyle L_{+},L_{-},L_{0}}
2652:{\displaystyle a\in \mathbb {Z} }
2284:Relations to satellite operations
718:on the left after undercrossing:
4914:
4903:
4902:
3353:) to satisfy the skein relation
2280:is the induced map on homology.
1832:{\displaystyle \Delta _{K}(t)=1}
1486:but the covering transformation
267:. If the number of generators,
4248:The Alexander-Conway Polynomial
2659:is the integer that represents
1683:it has an Alexander polynomial
4769:Dowker–Thistlethwaite notation
3591:
3532:
3519:
3468:
3455:
3449:
3404:
3398:
3385:
3376:
3363:
3275:
3253:
3237:
3224:
3191:
3178:
3166:
3153:
3144:
3131:
3100:
3094:
2990:
2984:
2935:
2929:
2875:
2869:
2849:
2843:
2820:
2814:
2753:
2727:
2618:
2612:
2594:
2581:
2576:
2551:
2537:
2531:
2423:
2412:
2373:
2254:
2208:
2186:
2167:
2161:
2124:
2058:
2045:
1946:
1940:
1917:
1901:
1895:
1889:
1880:
1874:
1820:
1814:
1748:{\displaystyle \Delta _{M}(1)}
1742:
1736:
1712:{\displaystyle \Delta _{M}(t)}
1706:
1700:
1676:{\displaystyle rank(H_{1}M)=1}
1664:
1648:
1563:
1557:
1413:
1391:
1327:
1305:
1280:
1258:
1233:
1211:
1166:
1144:
1139:
1117:
1047:
1041:
1025:
1009:
977:{\displaystyle \Delta _{K}(t)}
971:
965:
935:
916:
534:
516:
437:{\displaystyle \Delta _{K}(t)}
431:
425:
263:for this module is called the
235:
213:
181:
175:
141:
135:
1:
3549:
62:
3112:{\displaystyle \nabla (O)=1}
2777:Examples: For a connect-sum
2108:is the knot complement, let
1419:{\displaystyle \mathbb {Z} }
1364:
1333:{\displaystyle \mathbb {Z} }
1286:{\displaystyle \mathbb {Z} }
1239:{\displaystyle \mathbb {Z} }
1091:
1066:Poincaré Duality isomorphism
241:{\displaystyle \mathbb {Z} }
51:, could be computed using a
7:
4231:Encyclopedia of Mathematics
3814:Introduction to Knot Theory
3606:Math. Sem. Notes Kobe Univ.
3346:{\displaystyle \pm t^{n/2}}
3308:{\displaystyle \Delta _{L}}
3013:Conway–Alexander polynomial
3005:Alexander–Conway polynomial
2961:Alexander–Conway polynomial
2337:(there exists an embedding
656:{\displaystyle 0,1,-1,t,-t}
49:Alexander–Conway polynomial
39:discovered this, the first
10:
4979:
3722:
3654:Fintushel & Stern 1998
3496:Relation to Floer homology
2996:{\displaystyle \nabla (z)}
1629:is a 3-manifold such that
459:diagram of the knot with
37:James Waddell Alexander II
4898:
4802:
4759:Alexander–Briggs notation
4746:
4581:
4483:
4448:
4306:
4168:Rasmussen, Jacob (2003).
4104:10.1016/j.aim.2003.05.001
4043:10.1007/s00222-007-0075-9
3702:Ozsváth & Szabó 2004b
3618:Freedman & Quinn 1990
815:{\displaystyle \pm t^{n}}
785:{\displaystyle n\times n}
394:{\displaystyle \pm t^{n}}
326:{\displaystyle r\times r}
4013:Inventiones Mathematicae
3989:Kawauchi, Akio (2012) .
3831:Inventiones Mathematicae
3678:Ozsváth & Szabó 2004
3540:Crowell & Fox (1963)
3512:
3509:for further discussion.
2133:{\displaystyle g:S\to S}
2081:is a fiber bundle where
1996:Seiberg–Witten invariant
448:Computing the polynomial
187:{\displaystyle H_{1}(X)}
147:{\displaystyle H_{1}(X)}
4850:List of knots and links
4398:Kinoshita–Terasaka knot
4072:Advances in Mathematics
3992:A Survey of Knot Theory
3924:Topology of 4-manifolds
2429:{\displaystyle K=f(K')}
1846:(i.e. equal to its own
567:rows correspond to the
540:{\displaystyle (n,n+2)}
105:covering transformation
4226:"Alexander invariants"
3475:
3347:
3309:
3282:
3198:
3113:
3078:
3065:
2997:
2951:
2902:
2882:
2768:
2704:
2653:
2625:
2508:
2483:
2430:
2390:
2331:
2302:
2274:
2215:
2134:
2102:
2075:
2029:
1953:
1924:
1833:
1779:
1778:{\displaystyle H_{1}M}
1749:
1713:
1677:
1623:
1603:
1579:
1530:
1529:{\displaystyle t^{-1}}
1500:
1480:
1479:{\displaystyle H_{1}X}
1450:
1449:{\displaystyle H_{1}X}
1420:
1373:
1334:
1287:
1240:
1193:
1173:
1054:
978:
942:
879:
856:
836:
816:
786:
757:
732:
710:
688:
657:
607:
581:
561:
541:
499:
473:
438:
395:
361:
360:{\displaystyle r>s}
327:
301:
281:
248:. This is called the
242:
188:
154:. The transformation
148:
4640:Finite type invariant
4153:10.2140/gt.2004.8.311
4130:Geometry and Topology
3862:10.1007/s002220050268
3797:Bull. Amer. Math. Soc
3476:
3348:
3310:
3283:
3199:
3114:
3077:
3066:
2998:
2952:
2903:
2883:
2769:
2705:
2654:
2626:
2509:
2484:
2431:
2391:
2332:
2303:
2275:
2216:
2135:
2103:
2101:{\displaystyle C_{K}}
2076:
2030:
2028:{\displaystyle \pm 1}
1954:
1925:
1834:
1780:
1750:
1714:
1678:
1624:
1609:. More generally if
1604:
1580:
1531:
1501:
1481:
1451:
1421:
1374:
1335:
1288:
1241:
1194:
1174:
1055:
979:
943:
880:
857:
837:
822:, where the power of
817:
787:
758:
733:
711:
689:
658:
608:
582:
562:
542:
500:
479:crossings; there are
474:
439:
396:
362:
328:
302:
282:
243:
189:
149:
43:, in 1923. In 1969,
3914:Freedman, Michael H.
3502:Euler characteristic
3357:
3319:
3292:
3211:
3125:
3088:
3022:
2978:
2916:
2892:
2781:
2714:
2663:
2635:
2518:
2493:
2440:
2400:
2341:
2316:
2292:
2225:
2148:
2112:
2085:
2039:
2016:
1952:{\displaystyle f(t)}
1934:
1861:
1801:
1759:
1723:
1687:
1633:
1613:
1593:
1544:
1510:
1490:
1460:
1430:
1383:
1344:
1297:
1250:
1203:
1183:
1071:
996:
952:
903:
866:
846:
826:
796:
770:
744:
722:
700:
675:
617:
591:
571:
551:
513:
483:
463:
412:
375:
345:
311:
291:
271:
205:
162:
122:
25:Alexander polynomial
4810:Alexander's theorem
4188:2003math......6378R
4095:2002math......9056O
4035:2007InMat.170..577N
3854:1998InMat.134..363F
3668:, symmetry section.
1974:topologically slice
1855:topologically slice
1848:commutator subgroup
1340:-module, and where
606:{\displaystyle n+2}
587:crossings, and the
498:{\displaystyle n+2}
261:presentation matrix
250:Alexander invariant
200:Laurent polynomials
4958:John Horton Conway
3949:Formal Knot Theory
3808:Crowell, Richard;
3471:
3343:
3305:
3278:
3194:
3109:
3079:
3061:
2993:
2947:
2898:
2878:
2764:
2700:
2649:
2621:
2507:{\displaystyle K'}
2504:
2479:
2426:
2386:
2330:{\displaystyle K'}
2327:
2312:with pattern knot
2298:
2270:
2211:
2130:
2098:
2071:
2025:
1949:
1920:
1829:
1775:
1745:
1709:
1673:
1619:
1599:
1575:
1526:
1496:
1476:
1446:
1416:
1369:
1330:
1293:, considered as a
1283:
1236:
1189:
1169:
1050:
974:
938:
878:{\displaystyle -1}
875:
852:
832:
812:
782:
756:{\displaystyle -1}
753:
728:
706:
687:{\displaystyle -t}
684:
653:
603:
577:
557:
537:
495:
469:
434:
391:
357:
323:
297:
277:
238:
184:
144:
4930:
4929:
4784:Reidemeister move
4650:Khovanov homology
4645:Hyperbolic volume
4209:978-0-914098-16-4
4137:(2004): 311–334.
4002:978-3-0348-9227-8
3981:978-981-4383-00-4
3972:Knots and Physics
3959:978-0-486-45052-0
3935:978-0-691-08577-7
3822:Fintushel, Ronald
3743:978-0-8218-3678-1
3009:Conway polynomial
2901:{\displaystyle K}
2301:{\displaystyle K}
1622:{\displaystyle M}
1602:{\displaystyle t}
1499:{\displaystyle t}
1379:is the conjugate
1367:
1192:{\displaystyle G}
1094:
1060:for all knots K.
855:{\displaystyle t}
835:{\displaystyle n}
731:{\displaystyle t}
709:{\displaystyle 1}
580:{\displaystyle n}
560:{\displaystyle n}
472:{\displaystyle n}
300:{\displaystyle s}
280:{\displaystyle r}
198:over the ring of
4970:
4948:Diagram algebras
4918:
4917:
4906:
4905:
4870:Tait conjectures
4573:
4572:
4558:
4557:
4543:
4542:
4435:
4434:
4420:
4419:
4404:(−2,3,7) pretzel
4284:
4277:
4270:
4261:
4260:
4239:
4213:
4191:
4181:
4164:
4146:
4116:
4106:
4088:
4054:
4028:
4015:. Invent. Math.
4006:
3985:
3963:
3939:
3927:
3909:
3889:
3847:
3826:Stern, Ronald J.
3817:
3804:
3791:
3781:
3761:
3753:Alexander, J. W.
3747:
3717:
3711:
3705:
3699:
3693:
3687:
3681:
3675:
3669:
3663:
3657:
3651:
3645:
3639:
3633:
3627:
3621:
3615:
3609:
3595:
3589:
3583:
3577:
3571:
3565:
3559:
3543:
3536:
3530:
3523:
3480:
3478:
3477:
3472:
3467:
3466:
3448:
3447:
3443:
3424:
3423:
3419:
3397:
3396:
3375:
3374:
3352:
3350:
3349:
3344:
3342:
3341:
3337:
3314:
3312:
3311:
3306:
3304:
3303:
3287:
3285:
3284:
3279:
3274:
3273:
3252:
3251:
3236:
3235:
3223:
3222:
3203:
3201:
3200:
3195:
3190:
3189:
3165:
3164:
3143:
3142:
3118:
3116:
3115:
3110:
3070:
3068:
3067:
3062:
3060:
3059:
3047:
3046:
3034:
3033:
3002:
3000:
2999:
2994:
2956:
2954:
2953:
2948:
2928:
2927:
2910:Whitehead double
2908:is an untwisted
2907:
2905:
2904:
2899:
2887:
2885:
2884:
2879:
2868:
2867:
2866:
2865:
2842:
2841:
2840:
2839:
2813:
2812:
2811:
2810:
2798:
2797:
2773:
2771:
2770:
2765:
2763:
2752:
2751:
2739:
2738:
2726:
2725:
2709:
2707:
2706:
2701:
2699:
2698:
2686:
2685:
2673:
2658:
2656:
2655:
2650:
2648:
2630:
2628:
2627:
2622:
2611:
2610:
2609:
2593:
2592:
2580:
2579:
2563:
2562:
2530:
2529:
2513:
2511:
2510:
2505:
2503:
2488:
2486:
2485:
2480:
2478:
2477:
2465:
2464:
2452:
2451:
2435:
2433:
2432:
2427:
2422:
2395:
2393:
2392:
2387:
2385:
2384:
2372:
2371:
2359:
2358:
2336:
2334:
2333:
2328:
2326:
2307:
2305:
2304:
2299:
2279:
2277:
2276:
2271:
2266:
2265:
2250:
2249:
2237:
2236:
2220:
2218:
2217:
2212:
2207:
2206:
2185:
2184:
2160:
2159:
2139:
2137:
2136:
2131:
2107:
2105:
2104:
2099:
2097:
2096:
2080:
2078:
2077:
2072:
2070:
2069:
2057:
2056:
2034:
2032:
2031:
2026:
1986:by performing a
1970:Michael Freedman
1958:
1956:
1955:
1950:
1929:
1927:
1926:
1921:
1916:
1915:
1873:
1872:
1838:
1836:
1835:
1830:
1813:
1812:
1784:
1782:
1781:
1776:
1771:
1770:
1754:
1752:
1751:
1746:
1735:
1734:
1718:
1716:
1715:
1710:
1699:
1698:
1682:
1680:
1679:
1674:
1660:
1659:
1628:
1626:
1625:
1620:
1608:
1606:
1605:
1600:
1584:
1582:
1581:
1576:
1556:
1555:
1535:
1533:
1532:
1527:
1525:
1524:
1505:
1503:
1502:
1497:
1485:
1483:
1482:
1477:
1472:
1471:
1455:
1453:
1452:
1447:
1442:
1441:
1425:
1423:
1422:
1417:
1412:
1411:
1390:
1378:
1376:
1375:
1370:
1368:
1363:
1359:
1358:
1348:
1339:
1337:
1336:
1331:
1326:
1325:
1304:
1292:
1290:
1289:
1284:
1279:
1278:
1257:
1245:
1243:
1242:
1237:
1232:
1231:
1210:
1198:
1196:
1195:
1190:
1178:
1176:
1175:
1170:
1156:
1155:
1143:
1142:
1138:
1137:
1116:
1110:
1095:
1090:
1086:
1085:
1075:
1059:
1057:
1056:
1051:
1040:
1039:
1024:
1023:
1008:
1007:
983:
981:
980:
975:
964:
963:
947:
945:
944:
939:
928:
927:
915:
914:
884:
882:
881:
876:
862:and multiply by
861:
859:
858:
853:
841:
839:
838:
833:
821:
819:
818:
813:
811:
810:
791:
789:
788:
783:
762:
760:
759:
754:
737:
735:
734:
729:
715:
713:
712:
707:
693:
691:
690:
685:
662:
660:
659:
654:
612:
610:
609:
604:
586:
584:
583:
578:
566:
564:
563:
558:
546:
544:
543:
538:
507:incidence matrix
504:
502:
501:
496:
478:
476:
475:
470:
443:
441:
440:
435:
424:
423:
400:
398:
397:
392:
390:
389:
366:
364:
363:
358:
332:
330:
329:
324:
306:
304:
303:
298:
286:
284:
283:
278:
265:Alexander matrix
254:Alexander module
247:
245:
244:
239:
234:
233:
212:
193:
191:
190:
185:
174:
173:
153:
151:
150:
145:
134:
133:
83:be the infinite
57:Jones polynomial
31:which assigns a
4978:
4977:
4973:
4972:
4971:
4969:
4968:
4967:
4963:Knot invariants
4933:
4932:
4931:
4926:
4894:
4798:
4764:Conway notation
4748:
4742:
4729:Tricolorability
4577:
4571:
4568:
4567:
4566:
4556:
4553:
4552:
4551:
4541:
4538:
4537:
4536:
4528:
4518:
4508:
4498:
4479:
4458:Composite knots
4444:
4433:
4430:
4429:
4428:
4425:Borromean rings
4418:
4415:
4414:
4413:
4387:
4377:
4367:
4357:
4349:
4341:
4331:
4321:
4302:
4288:
4224:
4221:
4210:
4200:Knots and Links
4003:
3982:
3968:Kauffman, Louis
3960:
3944:Kauffman, Louis
3936:
3759:
3744:
3730:Adams, Colin C.
3725:
3720:
3712:
3708:
3700:
3696:
3688:
3684:
3676:
3672:
3664:
3660:
3652:
3648:
3640:
3636:
3628:
3624:
3616:
3612:
3596:
3592:
3584:
3580:
3572:
3568:
3560:
3556:
3552:
3547:
3546:
3537:
3533:
3524:
3520:
3515:
3498:
3462:
3458:
3439:
3432:
3428:
3415:
3411:
3407:
3392:
3388:
3370:
3366:
3358:
3355:
3354:
3333:
3329:
3325:
3320:
3317:
3316:
3299:
3295:
3293:
3290:
3289:
3266:
3262:
3247:
3243:
3231:
3227:
3218:
3214:
3212:
3209:
3208:
3185:
3181:
3160:
3156:
3138:
3134:
3126:
3123:
3122:
3089:
3086:
3085:
3055:
3051:
3042:
3038:
3029:
3025:
3023:
3020:
3019:
3007:(also known as
3003:and called the
2979:
2976:
2975:
2963:
2923:
2919:
2917:
2914:
2913:
2893:
2890:
2889:
2861:
2857:
2856:
2852:
2835:
2831:
2830:
2826:
2806:
2802:
2793:
2789:
2788:
2784:
2782:
2779:
2778:
2759:
2747:
2743:
2734:
2730:
2721:
2717:
2715:
2712:
2711:
2694:
2690:
2681:
2677:
2666:
2664:
2661:
2660:
2644:
2636:
2633:
2632:
2602:
2601:
2597:
2588:
2584:
2558:
2554:
2547:
2543:
2525:
2521:
2519:
2516:
2515:
2496:
2494:
2491:
2490:
2473:
2469:
2460:
2456:
2447:
2443:
2441:
2438:
2437:
2415:
2401:
2398:
2397:
2380:
2376:
2367:
2363:
2354:
2350:
2342:
2339:
2338:
2319:
2317:
2314:
2313:
2293:
2290:
2289:
2286:
2261:
2257:
2245:
2241:
2232:
2228:
2226:
2223:
2222:
2202:
2198:
2174:
2173:
2155:
2151:
2149:
2146:
2145:
2113:
2110:
2109:
2092:
2088:
2086:
2083:
2082:
2065:
2061:
2052:
2048:
2040:
2037:
2036:
2035:). In fact, if
2017:
2014:
2013:
2006:knot complement
1935:
1932:
1931:
1908:
1904:
1868:
1864:
1862:
1859:
1858:
1808:
1804:
1802:
1799:
1798:
1795:
1766:
1762:
1760:
1757:
1756:
1730:
1726:
1724:
1721:
1720:
1694:
1690:
1688:
1685:
1684:
1655:
1651:
1634:
1631:
1630:
1614:
1611:
1610:
1594:
1591:
1590:
1551:
1547:
1545:
1542:
1541:
1517:
1513:
1511:
1508:
1507:
1491:
1488:
1487:
1467:
1463:
1461:
1458:
1457:
1437:
1433:
1431:
1428:
1427:
1404:
1400:
1386:
1384:
1381:
1380:
1354:
1350:
1349:
1347:
1345:
1342:
1341:
1318:
1314:
1300:
1298:
1295:
1294:
1271:
1267:
1253:
1251:
1248:
1247:
1224:
1220:
1206:
1204:
1201:
1200:
1184:
1181:
1180:
1151:
1147:
1130:
1126:
1112:
1111:
1100:
1099:
1081:
1077:
1076:
1074:
1072:
1069:
1068:
1035:
1031:
1016:
1012:
1003:
999:
997:
994:
993:
990:
959:
955:
953:
950:
949:
923:
919:
910:
906:
904:
901:
900:
867:
864:
863:
847:
844:
843:
827:
824:
823:
806:
802:
797:
794:
793:
771:
768:
767:
745:
742:
741:
723:
720:
719:
701:
698:
697:
676:
673:
672:
618:
615:
614:
592:
589:
588:
572:
569:
568:
552:
549:
548:
514:
511:
510:
484:
481:
480:
464:
461:
460:
450:
419:
415:
413:
410:
409:
385:
381:
376:
373:
372:
346:
343:
342:
339:Alexander ideal
312:
309:
308:
292:
289:
288:
272:
269:
268:
226:
222:
208:
206:
203:
202:
169:
165:
163:
160:
159:
129:
125:
123:
120:
119:
97:Seifert surface
89:knot complement
65:
41:knot polynomial
17:
12:
11:
5:
4976:
4966:
4965:
4960:
4955:
4950:
4945:
4928:
4927:
4925:
4924:
4912:
4899:
4896:
4895:
4893:
4892:
4890:Surgery theory
4887:
4882:
4877:
4872:
4867:
4862:
4857:
4852:
4847:
4842:
4837:
4832:
4827:
4822:
4817:
4812:
4806:
4804:
4800:
4799:
4797:
4796:
4791:
4789:Skein relation
4786:
4781:
4776:
4771:
4766:
4761:
4755:
4753:
4744:
4743:
4741:
4740:
4734:Unknotting no.
4731:
4726:
4721:
4720:
4719:
4709:
4704:
4703:
4702:
4697:
4692:
4687:
4682:
4672:
4667:
4662:
4657:
4652:
4647:
4642:
4637:
4632:
4627:
4626:
4625:
4615:
4610:
4609:
4608:
4598:
4593:
4587:
4585:
4579:
4578:
4576:
4575:
4569:
4560:
4554:
4545:
4539:
4530:
4526:
4520:
4516:
4510:
4506:
4500:
4496:
4489:
4487:
4481:
4480:
4478:
4477:
4472:
4471:
4470:
4465:
4454:
4452:
4446:
4445:
4443:
4442:
4437:
4431:
4422:
4416:
4407:
4401:
4395:
4389:
4385:
4379:
4375:
4369:
4365:
4359:
4355:
4351:
4347:
4343:
4339:
4333:
4329:
4323:
4319:
4312:
4310:
4304:
4303:
4287:
4286:
4279:
4272:
4264:
4258:
4257:
4253:The Knot Atlas
4240:
4220:
4219:External links
4217:
4216:
4215:
4208:
4192:
4165:
4121:Ozsváth, Peter
4117:
4059:Ozsváth, Peter
4055:
4019:(3): 577–608.
4008:
4001:
3995:. Birkhäuser.
3986:
3980:
3964:
3958:
3940:
3934:
3910:
3890:
3838:(2): 363–400.
3818:
3805:
3792:
3772:(2): 275–306.
3749:
3742:
3724:
3721:
3719:
3718:
3706:
3694:
3690:Rasmussen 2003
3682:
3670:
3658:
3646:
3634:
3622:
3610:
3602:Osaka J. Math.
3590:
3578:
3574:Alexander 1928
3566:
3553:
3551:
3548:
3545:
3544:
3531:
3517:
3516:
3514:
3511:
3507:Floer homology
3497:
3494:
3470:
3465:
3461:
3457:
3454:
3451:
3446:
3442:
3438:
3435:
3431:
3427:
3422:
3418:
3414:
3410:
3406:
3403:
3400:
3395:
3391:
3387:
3384:
3381:
3378:
3373:
3369:
3365:
3362:
3340:
3336:
3332:
3328:
3324:
3302:
3298:
3277:
3272:
3269:
3265:
3261:
3258:
3255:
3250:
3246:
3242:
3239:
3234:
3230:
3226:
3221:
3217:
3205:
3204:
3193:
3188:
3184:
3180:
3177:
3174:
3171:
3168:
3163:
3159:
3155:
3152:
3149:
3146:
3141:
3137:
3133:
3130:
3120:
3108:
3105:
3102:
3099:
3096:
3093:
3058:
3054:
3050:
3045:
3041:
3037:
3032:
3028:
2992:
2989:
2986:
2983:
2962:
2959:
2946:
2943:
2940:
2937:
2934:
2931:
2926:
2922:
2897:
2877:
2874:
2871:
2864:
2860:
2855:
2851:
2848:
2845:
2838:
2834:
2829:
2825:
2822:
2819:
2816:
2809:
2805:
2801:
2796:
2792:
2787:
2762:
2758:
2755:
2750:
2746:
2742:
2737:
2733:
2729:
2724:
2720:
2697:
2693:
2689:
2684:
2680:
2676:
2672:
2669:
2647:
2643:
2640:
2620:
2617:
2614:
2608:
2605:
2600:
2596:
2591:
2587:
2583:
2578:
2575:
2572:
2569:
2566:
2561:
2557:
2553:
2550:
2546:
2542:
2539:
2536:
2533:
2528:
2524:
2502:
2499:
2476:
2472:
2468:
2463:
2459:
2455:
2450:
2446:
2425:
2421:
2418:
2414:
2411:
2408:
2405:
2383:
2379:
2375:
2370:
2366:
2362:
2357:
2353:
2349:
2346:
2325:
2322:
2310:satellite knot
2297:
2285:
2282:
2269:
2264:
2260:
2256:
2253:
2248:
2244:
2240:
2235:
2231:
2210:
2205:
2201:
2197:
2194:
2191:
2188:
2183:
2180:
2177:
2172:
2169:
2166:
2163:
2158:
2154:
2140:represent the
2129:
2126:
2123:
2120:
2117:
2095:
2091:
2068:
2064:
2060:
2055:
2051:
2047:
2044:
2024:
2021:
1948:
1945:
1942:
1939:
1919:
1914:
1911:
1907:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1871:
1867:
1840:if and only if
1828:
1825:
1822:
1819:
1816:
1811:
1807:
1794:
1791:
1787:
1786:
1774:
1769:
1765:
1744:
1741:
1738:
1733:
1729:
1708:
1705:
1702:
1697:
1693:
1672:
1669:
1666:
1663:
1658:
1654:
1650:
1647:
1644:
1641:
1638:
1618:
1598:
1574:
1571:
1568:
1565:
1562:
1559:
1554:
1550:
1538:
1537:
1523:
1520:
1516:
1495:
1475:
1470:
1466:
1445:
1440:
1436:
1415:
1410:
1407:
1403:
1399:
1396:
1393:
1389:
1366:
1362:
1357:
1353:
1329:
1324:
1321:
1317:
1313:
1310:
1307:
1303:
1282:
1277:
1274:
1270:
1266:
1263:
1260:
1256:
1235:
1230:
1227:
1223:
1219:
1216:
1213:
1209:
1188:
1168:
1165:
1162:
1159:
1154:
1150:
1146:
1141:
1136:
1133:
1129:
1125:
1122:
1119:
1115:
1109:
1106:
1103:
1098:
1093:
1089:
1084:
1080:
1049:
1046:
1043:
1038:
1034:
1030:
1027:
1022:
1019:
1015:
1011:
1006:
1002:
989:
986:
973:
970:
967:
962:
958:
937:
934:
931:
926:
922:
918:
913:
909:
890:Seifert matrix
874:
871:
851:
831:
809:
805:
801:
781:
778:
775:
764:
763:
752:
749:
738:
727:
716:
705:
694:
683:
680:
652:
649:
646:
643:
640:
637:
634:
631:
628:
625:
622:
602:
599:
596:
576:
556:
536:
533:
530:
527:
524:
521:
518:
494:
491:
488:
468:
449:
446:
433:
430:
427:
422:
418:
388:
384:
380:
356:
353:
350:
322:
319:
316:
296:
276:
237:
232:
229:
225:
221:
218:
215:
211:
183:
180:
177:
172:
168:
143:
140:
137:
132:
128:
64:
61:
53:skein relation
29:knot invariant
16:Knot invariant
15:
9:
6:
4:
3:
2:
4975:
4964:
4961:
4959:
4956:
4954:
4951:
4949:
4946:
4944:
4941:
4940:
4938:
4923:
4922:
4913:
4911:
4910:
4901:
4900:
4897:
4891:
4888:
4886:
4883:
4881:
4878:
4876:
4873:
4871:
4868:
4866:
4863:
4861:
4858:
4856:
4853:
4851:
4848:
4846:
4843:
4841:
4838:
4836:
4833:
4831:
4828:
4826:
4825:Conway sphere
4823:
4821:
4818:
4816:
4813:
4811:
4808:
4807:
4805:
4801:
4795:
4792:
4790:
4787:
4785:
4782:
4780:
4777:
4775:
4772:
4770:
4767:
4765:
4762:
4760:
4757:
4756:
4754:
4752:
4745:
4739:
4735:
4732:
4730:
4727:
4725:
4722:
4718:
4715:
4714:
4713:
4710:
4708:
4705:
4701:
4698:
4696:
4693:
4691:
4688:
4686:
4683:
4681:
4678:
4677:
4676:
4673:
4671:
4668:
4666:
4663:
4661:
4658:
4656:
4653:
4651:
4648:
4646:
4643:
4641:
4638:
4636:
4633:
4631:
4628:
4624:
4621:
4620:
4619:
4616:
4614:
4611:
4607:
4604:
4603:
4602:
4599:
4597:
4596:Arf invariant
4594:
4592:
4589:
4588:
4586:
4584:
4580:
4564:
4561:
4549:
4546:
4534:
4531:
4524:
4521:
4514:
4511:
4504:
4501:
4494:
4491:
4490:
4488:
4486:
4482:
4476:
4473:
4469:
4466:
4464:
4461:
4460:
4459:
4456:
4455:
4453:
4451:
4447:
4441:
4438:
4426:
4423:
4411:
4408:
4405:
4402:
4399:
4396:
4393:
4390:
4383:
4380:
4373:
4370:
4363:
4360:
4358:
4352:
4350:
4344:
4337:
4334:
4327:
4324:
4317:
4314:
4313:
4311:
4309:
4305:
4300:
4296:
4292:
4285:
4280:
4278:
4273:
4271:
4266:
4265:
4262:
4255:
4254:
4249:
4245:
4241:
4237:
4233:
4232:
4227:
4223:
4222:
4211:
4205:
4201:
4197:
4196:Rolfsen, Dale
4193:
4189:
4185:
4180:
4175:
4171:
4166:
4162:
4158:
4154:
4150:
4145:
4140:
4136:
4132:
4131:
4126:
4125:Szabó, Zoltán
4122:
4118:
4114:
4110:
4105:
4100:
4096:
4092:
4087:
4082:
4079:(1): 58–116.
4078:
4074:
4073:
4068:
4064:
4063:Szabó, Zoltán
4060:
4056:
4052:
4048:
4044:
4040:
4036:
4032:
4027:
4022:
4018:
4014:
4009:
4004:
3998:
3994:
3993:
3987:
3983:
3977:
3973:
3969:
3965:
3961:
3955:
3951:
3950:
3945:
3941:
3937:
3931:
3926:
3925:
3919:
3915:
3911:
3907:
3903:
3899:
3895:
3891:
3887:
3883:
3879:
3875:
3871:
3867:
3863:
3859:
3855:
3851:
3846:
3845:dg-ga/9612014
3841:
3837:
3833:
3832:
3827:
3823:
3819:
3815:
3811:
3806:
3803:(2): 253–287.
3802:
3798:
3793:
3789:
3785:
3780:
3775:
3771:
3767:
3766:
3758:
3754:
3750:
3745:
3739:
3735:
3731:
3727:
3726:
3715:
3710:
3703:
3698:
3691:
3686:
3679:
3674:
3667:
3666:Kawauchi 2012
3662:
3655:
3650:
3643:
3642:Kauffman 2012
3638:
3631:
3630:Kauffman 1983
3626:
3619:
3614:
3607:
3603:
3599:
3598:Kawauchi 2012
3594:
3587:
3582:
3575:
3570:
3563:
3558:
3554:
3541:
3535:
3528:
3522:
3518:
3510:
3508:
3503:
3493:
3491:
3486:
3484:
3463:
3459:
3444:
3440:
3436:
3433:
3429:
3425:
3420:
3416:
3412:
3408:
3401:
3393:
3389:
3379:
3371:
3367:
3338:
3334:
3330:
3326:
3322:
3300:
3270:
3267:
3263:
3259:
3256:
3248:
3240:
3232:
3228:
3219:
3186:
3182:
3172:
3169:
3161:
3157:
3147:
3139:
3135:
3121:
3106:
3103:
3097:
3084:
3083:
3082:
3076:
3072:
3056:
3052:
3048:
3043:
3039:
3035:
3030:
3026:
3016:
3014:
3010:
3006:
2987:
2973:
2968:
2958:
2944:
2941:
2938:
2932:
2924:
2911:
2895:
2872:
2862:
2858:
2846:
2836:
2832:
2823:
2817:
2807:
2803:
2794:
2790:
2775:
2756:
2748:
2744:
2740:
2735:
2731:
2722:
2718:
2695:
2691:
2687:
2682:
2678:
2674:
2670:
2667:
2641:
2638:
2615:
2606:
2603:
2589:
2585:
2570:
2564:
2559:
2555:
2548:
2540:
2534:
2526:
2500:
2497:
2474:
2470:
2466:
2461:
2457:
2453:
2448:
2444:
2419:
2416:
2409:
2406:
2403:
2381:
2377:
2368:
2364:
2360:
2355:
2351:
2347:
2344:
2323:
2320:
2311:
2295:
2281:
2267:
2262:
2258:
2251:
2246:
2242:
2238:
2233:
2229:
2203:
2199:
2195:
2192:
2189:
2170:
2164:
2156:
2143:
2127:
2121:
2118:
2115:
2093:
2089:
2066:
2062:
2053:
2049:
2042:
2022:
2019:
2011:
2007:
2002:
1999:
1997:
1993:
1989:
1985:
1980:
1977:
1975:
1971:
1967:
1965:
1960:
1943:
1937:
1912:
1909:
1905:
1898:
1892:
1886:
1883:
1877:
1869:
1856:
1851:
1849:
1845:
1841:
1826:
1823:
1817:
1809:
1790:
1772:
1767:
1763:
1739:
1731:
1703:
1695:
1670:
1667:
1661:
1656:
1652:
1645:
1642:
1639:
1636:
1616:
1596:
1588:
1587:
1586:
1572:
1569:
1566:
1560:
1552:
1521:
1518:
1514:
1493:
1473:
1468:
1464:
1443:
1438:
1434:
1408:
1405:
1401:
1397:
1394:
1360:
1355:
1351:
1322:
1319:
1315:
1311:
1308:
1275:
1272:
1268:
1264:
1261:
1228:
1225:
1221:
1217:
1214:
1186:
1163:
1160:
1157:
1152:
1148:
1134:
1131:
1127:
1123:
1120:
1096:
1087:
1082:
1078:
1067:
1063:
1062:
1061:
1044:
1036:
1028:
1020:
1017:
1013:
1004:
985:
968:
960:
932:
924:
920:
911:
907:
898:
893:
891:
886:
872:
869:
849:
829:
807:
803:
799:
779:
776:
773:
750:
747:
739:
725:
717:
703:
695:
681:
678:
670:
669:
668:
664:
650:
647:
644:
641:
638:
635:
632:
629:
626:
623:
620:
600:
597:
594:
574:
554:
531:
528:
525:
522:
519:
508:
492:
489:
486:
466:
458:
453:
445:
428:
420:
406:
404:
403:constant term
386:
382:
378:
370:
354:
351:
348:
340:
336:
335:Fitting ideal
320:
317:
314:
294:
274:
266:
262:
257:
255:
251:
230:
227:
223:
219:
216:
201:
197:
178:
170:
166:
157:
138:
130:
126:
117:
113:
109:
106:
102:
98:
94:
90:
86:
82:
78:
74:
70:
60:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
4919:
4907:
4835:Double torus
4820:Braid theory
4679:
4635:Crossing no.
4630:Crosscap no.
4316:Figure-eight
4251:
4229:
4199:
4179:math/0306378
4169:
4144:math/0311496
4134:
4128:
4086:math/0209056
4076:
4070:
4026:math/0607156
4016:
4012:
3991:
3971:
3948:
3923:
3918:Quinn, Frank
3897:
3835:
3829:
3813:
3800:
3796:
3769:
3763:
3733:
3709:
3697:
3685:
3673:
3661:
3649:
3637:
3625:
3613:
3605:
3601:
3593:
3581:
3569:
3557:
3534:
3521:
3499:
3487:
3482:
3206:
3080:
3017:
3012:
3008:
3004:
2971:
2964:
2776:
2287:
2009:
2003:
2000:
1991:
1981:
1978:
1968:
1961:
1852:
1796:
1788:
1539:
991:
894:
887:
765:
665:
454:
451:
407:
338:
264:
258:
253:
249:
155:
115:
111:
107:
100:
92:
85:cyclic cover
80:
68:
66:
48:
24:
18:
4953:Polynomials
4943:Knot theory
4670:Linking no.
4591:Alternating
4392:Conway knot
4372:Carrick mat
4326:Three-twist
4291:Knot theory
4246:" and "
3952:. Courier.
3608:5: 451~456.
3562:Birman 1993
3527:Joan Birman
3490:knot theory
2967:John Conway
1426:-module to
45:John Conway
21:mathematics
4937:Categories
4830:Complement
4794:Tabulation
4751:operations
4675:Polynomial
4665:Link group
4660:Knot group
4623:Invertible
4601:Bridge no.
4583:Invariants
4513:Cinquefoil
4382:Perko pair
4308:Hyperbolic
3894:Fox, Ralph
3810:Fox, Ralph
3550:References
2396:such that
2288:If a knot
1984:4-manifold
1964:knot genus
1962:Twice the
118:, denoted
110:acting on
63:Definition
33:polynomial
4724:Stick no.
4680:Alexander
4618:Chirality
4563:Solomon's
4523:Septafoil
4450:Satellite
4410:Whitehead
4336:Stevedore
4244:Main Page
4236:EMS Press
4051:119159648
3946:(2006) .
3870:0020-9910
3732:(2004) .
3453:Δ
3434:−
3426:−
3394:−
3383:Δ
3380:−
3361:Δ
3323:±
3297:Δ
3268:−
3260:−
3245:∇
3216:Δ
3176:∇
3162:−
3151:∇
3148:−
3129:∇
3092:∇
3044:−
2982:∇
2942:±
2921:Δ
2854:Δ
2828:Δ
2800:#
2786:Δ
2741:×
2688:×
2675:⊂
2642:∈
2599:Δ
2565:×
2545:Δ
2523:Δ
2467:⊂
2454:×
2374:→
2361:×
2255:→
2239::
2234:∗
2204:∗
2196:−
2153:Δ
2142:monodromy
2125:→
2059:→
2046:→
2020:±
1910:−
1866:Δ
1806:Δ
1728:Δ
1692:Δ
1570:±
1549:Δ
1519:−
1406:−
1365:¯
1320:−
1273:−
1226:−
1132:−
1097:≃
1092:¯
1033:Δ
1018:−
1001:Δ
957:Δ
930:∖
908:π
897:Ralph Fox
870:−
800:±
777:×
748:−
679:−
648:−
633:−
417:Δ
379:±
369:principal
318:×
228:−
4909:Category
4779:Mutation
4747:Notation
4700:Kauffman
4613:Brunnian
4606:2-bridge
4475:Knot sum
4406:(12n242)
4198:(1990).
4161:11374897
4113:11246611
4065:(2004).
3970:(2012).
3920:(1990).
3906:73203715
3812:(1963).
3755:(1928).
3586:Fox 1961
3288:. Here
2671:′
2631:, where
2607:′
2514:), then
2501:′
2436:, where
2420:′
2324:′
1506:acts by
509:of size
457:oriented
455:Take an
77:3-sphere
4921:Commons
4840:Fibered
4738:problem
4707:Pretzel
4685:Bracket
4503:Trefoil
4440:L10a140
4400:(11n42)
4394:(11n34)
4362:Endless
4238:, 2001
4184:Bibcode
4091:Bibcode
4031:Bibcode
3886:3752148
3878:1650308
3850:Bibcode
3799:. N.S.
3788:1989123
3723:Sources
3714:Ni 2007
2912:, then
2144:, then
2004:If the
1988:surgery
1844:perfect
547:. The
87:of the
79:. Let
75:in the
4885:Writhe
4855:Ribbon
4690:HOMFLY
4533:Unlink
4493:Unknot
4468:Square
4463:Granny
4206:
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2221:where
1930:where
1853:For a
1179:where
196:module
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4875:Twist
4860:Slice
4815:Berge
4803:Other
4774:Flype
4712:Prime
4695:Jones
4655:Genus
4485:Torus
4299:links
4295:knots
4174:arXiv
4157:S2CID
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4109:S2CID
4081:arXiv
4047:S2CID
4021:arXiv
3882:S2CID
3840:arXiv
3784:JSTOR
3760:(PDF)
3513:Notes
2888:. If
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4749:and
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