1699:
1218:
1694:{\displaystyle {\begin{aligned}\operatorname {link} (\gamma _{1},\gamma _{2})&={\frac {1}{4\pi }}\oint _{\gamma _{1}}\oint _{\gamma _{2}}{\frac {\mathbf {r} _{1}-\mathbf {r} _{2}}{|\mathbf {r} _{1}-\mathbf {r} _{2}|^{3}}}\cdot (d\mathbf {r} _{1}\times d\mathbf {r} _{2})\\&={\frac {1}{4\pi }}\int _{S^{1}\times S^{1}}{\frac {\det \left({\dot {\gamma }}_{1}(s),{\dot {\gamma }}_{2}(t),\gamma _{1}(s)-\gamma _{2}(t)\right)}{\left|\gamma _{1}(s)-\gamma _{2}(t)\right|^{3}}}\,ds\,dt\end{aligned}}}
724:
310:
4432:
158:
151:
213:
206:
199:
165:
3398:
337:
2186:
4444:
38:
2798:
3368:
1927:
3035:, it is clear that there will be terms describing the self-interaction of the particles, and these are uninteresting since they would be there even in the presence of just one loop. Therefore, we normalize the path integral by a factor precisely cancelling these terms. Going through the algebra, we obtain
2219:
is needed. Therefore, the topological invariance of right hand side ensures that the result of the path integral will be a topological invariant. The only thing left to do is provide an overall normalization factor, and a natural choice will present itself. Since the theory is
Gaussian and abelian,
3384:
The Chern-Simons gauge theory lives in 3 spacetime dimensions. More generally, there exists higher dimensional topological quantum field theories. There exists more complicated multi-loop/string-braiding statistics of 4-dimensional gauge theories captured by the link invariants of exotic
1186:
of Γ covers the sphere). Isotopy invariance of the linking number is automatically obtained as the degree is invariant under homotopic maps. Any other regular value would give the same number, so the linking number doesn't depend on any particular link diagram.
2615:
2325:
1067:
2181:{\displaystyle Z=\int {\mathcal {D}}A_{\mu }\exp \left({\frac {ik}{4\pi }}\int d^{3}x\varepsilon ^{\lambda \mu \nu }A_{\lambda }\partial _{\mu }A_{\nu }+i\int _{\gamma _{1}}dx^{\mu }\,A_{\mu }+i\int _{\gamma _{2}}dx^{\mu }\,A_{\mu }\right)}
3170:
2604:
3010:
282:
to a solid torus with a point removed (this can be seen by interpreting 3-space as the 3-sphere with the point at infinity removed, and the 3-sphere as two solid tori glued along the boundary), or the complement can be analyzed
892:
2465:
431:
630:
266:
This fact (that the linking number is the only invariant) is most easily proven by placing one circle in standard position, and then showing that linking number is the only invariant of the other circle. In detail:
3159:
251:, not just any map. However, this added condition does not change the definition of linking number (it does not matter if the curves are required to always be immersions or not), which is an example of an
1877:
1223:
3377:, where the path integral computes topological invariants. This also served as a hint that the nonabelian variant of Chern–Simons theory computes other knot invariants, and it was shown explicitly by
2514:
2793:{\displaystyle A_{\lambda }({\vec {x}})={\frac {1}{2k}}\int d^{3}{\vec {y}}\,{\frac {\varepsilon _{\lambda \mu \nu }\partial ^{\mu }J^{\nu }({\vec {y}})}{|{\vec {x}}-{\vec {y}}|}}}
3703:
Putrov, Pavel; Wang, Juven; Yau, Shing-Tung (September 2017). "Braiding
Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions".
1919:
2368:
546:
504:
2887:
2249:
934:
275:
is clear, since 3-space is contractible and thus all maps into it are homotopic, though the fact that this can be done through immersions requires some geometric argument.
1144:
1117:
713:
673:
2209:
3363:{\displaystyle \Phi ={\frac {1}{4\pi }}\int _{\gamma _{1}}dx^{\lambda }\int _{\gamma _{2}}dy^{\mu }\,{\frac {(x-y)^{\nu }}{|x-y|^{3}}}\varepsilon _{\lambda \mu \nu },}
918:
3464:
1755:
294:(either adding the point at infinity to get a solid torus, or adding the circle to get 3-space, allows one to compute the fundamental group of the desired space).
3033:
2841:
2821:
2522:
2388:
2238:
1804:
1781:
2895:
271:
A single curve is regular homotopic to a standard circle (any knot can be unknotted if the curve is allowed to pass through itself). The fact that it is
1158:
preserving or reversing orientation depending on the sign of the crossing. Thus in order to compute the linking number of the diagram corresponding to
829:
2396:
3530:
is an embedding into three-dimensional space such that every two cycles have zero linking number. The graphs that have a linkless embedding have a
353:
743:
Reversing the orientation of either of the curves negates the linking number, while reversing the orientation of both curves leaves it unchanged.
554:
3809:
4377:
4296:
2843:
field. To get the path integral for the Wilson loops, we substitute for a source describing two particles moving in closed loops, i.e.
3386:
2390:, we can get back the Wilson loops. Since we are in 3 dimensions, we can rewrite the equations of motion in a more familiar notation:
3041:
1812:
241:
Each curve may pass through itself during this motion, but the two curves must remain separated throughout. This is formalized as
3531:
3374:
3843:
1705:
736:
Any two unlinked curves have linking number zero. However, two curves with linking number zero may still be linked (e.g. the
291:
3489:
along the framing vectors. The self-linking number obtained by moving vertically (along the blackboard framing) is known as
3467:
1179:
300:
It is also true that regular homotopy classes are determined by linking number, which requires additional geometric argument.
4286:
4291:
4162:
3781:
3763:
3863:
3925:
3776:
3758:
814:
3931:
3995:
3990:
3802:
2476:
4470:
2212:
69:. Intuitively, the linking number represents the number of times that each curve winds around the other. In
4123:
3638:
represent the number of times that the red curve crosses in and out of the region bounded by the blue curve.
747:
246:
66:
58:
4448:
4337:
4306:
793:
3753:
4167:
2823:
is now easily done by substituting this into the Chern–Simons action to get an effective action for the
1889:
1883:
105:
2333:
3496:
The linking number is defined for two linked circles; given three or more circles, one can define the
2320:{\displaystyle \varepsilon ^{\lambda \mu \nu }\partial _{\mu }A_{\nu }={\frac {2\pi }{k}}J^{\lambda }}
4436:
4207:
3795:
3771:
3722:
3661:
3607:
3551:
3538:
1721:
1062:{\displaystyle \Gamma (s,t)={\frac {\gamma _{1}(s)-\gamma _{2}(t)}{|\gamma _{1}(s)-\gamma _{2}(t)|}}}
896:
509:
467:
290:
of 3-space minus a circle is the integers, corresponding to linking number. This can be seen via the
89:
1704:
This integral computes the total signed area of the image of the Gauss map (the integrand being the
4244:
4227:
3478:
2846:
1183:
4265:
4212:
3826:
3822:
3712:
3594:
3523:
3504:
3415:
3405:
generalize linking number to links with three or more components, allowing one to prove that the
1122:
1095:
678:
638:
136:
Any two closed curves in space, if allowed to pass through themselves but not each other, can be
101:
343:
The total number of positive crossings minus the total number of negative crossings is equal to
2194:
4362:
4311:
4261:
4217:
4177:
4172:
4090:
3516:
3497:
3402:
797:
754:
of link negates the linking number. The convention for positive linking number is based on a
287:
117:
78:
27:
Numerical invariant that describes the linking of two closed curves in three-dimensional space
1757:
903:
313:
With six positive crossings and two negative crossings, these curves have linking number two.
4397:
4222:
4118:
3853:
3730:
3685:
3669:
3527:
2220:
the path integral can be done simply by solving the theory classically and substituting for
297:
Thus homotopy classes of a curve in 3-space minus a circle are determined by linking number.
242:
121:
3681:
3437:
806:. In this case, the linking number is determined by the homology class of the other curve.
140:
into exactly one of the following standard positions. This determines the linking number:
4475:
4357:
4321:
4256:
4202:
4157:
4150:
4040:
3952:
3835:
3689:
3677:
3535:
3431:
3419:
3406:
2599:{\displaystyle \nabla ^{2}{\vec {A}}=-{\frac {2\pi }{k}}{\vec {\nabla }}\times {\vec {J}}}
2216:
1731:
755:
70:
3726:
3665:
4417:
4316:
4278:
4110:
3985:
3977:
3937:
3563:
3557:
3512:
3018:
3005:{\displaystyle J_{i}^{\mu }(x)=\int _{\gamma _{i}}dx_{i}^{\mu }\delta ^{3}(x-x_{i}(t))}
2826:
2806:
2373:
2223:
1789:
1766:
762:
737:
728:
723:
309:
17:
4464:
4352:
4140:
4133:
4128:
3378:
1205:
1175:
125:
4367:
4347:
4251:
4234:
4030:
3967:
3597:, though in this case we only label crossings that involve both curves of the link.
2471:
1760:
751:
279:
62:
4050:
3889:
3881:
3873:
3649:
1724:, Gauss's integral definition arises when computing the expectation value of the
4382:
4145:
3919:
3899:
3818:
3787:
3508:
3474:
1725:
925:
322:
252:
157:
150:
109:
97:
50:
3381:
that the nonabelian theory gives the invariant known as the Jones polynomial.
2211:
is the antisymmetric symbol. Since the theory is just
Gaussian, no ultraviolet
887:{\displaystyle \gamma _{1},\gamma _{2}\colon S^{1}\rightarrow \mathbb {R} ^{3}}
212:
205:
198:
164:
4402:
4387:
4342:
4239:
4192:
4187:
4182:
4012:
3909:
3734:
2460:{\displaystyle {\vec {\nabla }}\times {\vec {A}}={\frac {2\pi }{k}}{\vec {J}}}
82:
42:
31:
1763:. Explicitly, the abelian Chern–Simons action for a gauge potential one-form
4407:
4075:
3610:
if either curve is simple. For example, if the blue curve is simple, then
3397:
3373:
which is simply Gauss's linking integral. This is the simplest example of a
426:{\displaystyle {\text{linking number}}={\frac {n_{1}+n_{2}-n_{3}-n_{4}}{2}}}
336:
318:
3511:
is a far-reaching algebraic generalization of the linking number, with the
1076:, so that orthogonal projection of the link to the plane perpendicular to
464:
represent the number of crossings of each of the four types. The two sums
4392:
4002:
1784:
782:
137:
85:, where linking numbers can also be fractions or just not exist at all).
1092:
under the Gauss map corresponds to a crossing in the link diagram where
625:{\displaystyle {\text{linking number}}\,=\,n_{1}-n_{4}\,=\,n_{2}-n_{3}.}
3673:
810:
802:
113:
74:
4412:
4060:
4020:
3590:
3569:
2330:
Here, we have coupled the Chern–Simons field to a source with a term
675:
involves only the undercrossings of the blue curve by the red, while
3717:
548:
are always equal, which leads to the following alternative formula
37:
4301:
3396:
921:
789:
722:
308:
36:
258:(homotopy-principle), meaning that geometry reduces to topology.
4372:
3791:
3154:{\displaystyle Z=\exp {\left({\frac {2\pi i}{k}}\Phi \right)},}
3500:, which are a numerical invariant generalizing linking number.
2370:
in the
Lagrangian. Obviously, by substituting the appropriate
3409:
are linked, though any two components have linking number 0.
1971:
1872:{\displaystyle S_{CS}={\frac {k}{4\pi }}\int _{M}A\wedge dA}
1708:
of Γ) and then divides by the area of the sphere (which is 4
335:
3485:
with a new curve obtained by slightly moving the points of
1182:
of the Gauss map (i.e. the signed number of times that the
3554: – Study of curves from a differential point of view
321:
to compute the linking number of two curves from a link
30:"Link number" redirects here. For the logic puzzle, see
81:
of the two curves (this is not true for curves in most
3481:
obtained by computing the linking number of the knot
3440:
3173:
3044:
3021:
2898:
2849:
2829:
2809:
2618:
2525:
2479:
2399:
2376:
2336:
2252:
2226:
2197:
1930:
1892:
1815:
1792:
1769:
1734:
1221:
1154:) is mapped under the Gauss map to a neighborhood of
1125:
1098:
937:
906:
832:
681:
641:
557:
512:
470:
356:
3560: – Homotopy invariant of maps between n-spheres
3466:. Any such link has an associated Gauss map, whose
4330:
4274:
4109:
4011:
3976:
3834:
77:, but may be positive or negative depending on the
3458:
3362:
3153:
3027:
3004:
2881:
2835:
2815:
2792:
2598:
2508:
2459:
2382:
2362:
2319:
2232:
2203:
2180:
1913:
1871:
1798:
1775:
1749:
1693:
1138:
1111:
1061:
912:
886:
707:
667:
624:
540:
498:
425:
826:Given two non-intersecting differentiable curves
537:
495:
1496:
3650:"Quantum field theory and the Jones polynomial"
245:, which further requires that each curve be an
3589:This is the same labeling used to compute the
773:plane is equal to its linking number with the
3803:
1080:gives a link diagram. Observe that a point (
8:
3015:Since the effective action is quadratic in
2470:Taking the curl of both sides and choosing
788:More generally, if either of the curves is
142:
3810:
3796:
3788:
3470:is a generalization of the linking number.
2509:{\displaystyle \partial ^{\mu }A_{\mu }=0}
1190:This formulation of the linking number of
3716:
3439:
3345:
3332:
3327:
3312:
3304:
3285:
3284:
3278:
3263:
3258:
3248:
3233:
3228:
3209:
3197:
3184:
3172:
3133:
3120:
3092:
3086:
3068:
3055:
3043:
3020:
2984:
2965:
2955:
2950:
2935:
2930:
2908:
2903:
2897:
2873:
2860:
2848:
2828:
2808:
2782:
2771:
2770:
2756:
2755:
2750:
2734:
2733:
2724:
2714:
2698:
2691:
2690:
2679:
2678:
2672:
2650:
2633:
2632:
2623:
2617:
2585:
2584:
2570:
2569:
2554:
2537:
2536:
2530:
2524:
2494:
2484:
2478:
2446:
2445:
2430:
2416:
2415:
2401:
2400:
2398:
2375:
2354:
2344:
2335:
2311:
2292:
2283:
2273:
2257:
2251:
2225:
2196:
2167:
2162:
2156:
2141:
2136:
2120:
2115:
2109:
2094:
2089:
2073:
2063:
2053:
2037:
2024:
1997:
1980:
1970:
1969:
1954:
1941:
1929:
1905:
1901:
1900:
1891:
1851:
1832:
1820:
1814:
1791:
1768:
1733:
1680:
1673:
1665:
1645:
1623:
1592:
1570:
1548:
1537:
1536:
1517:
1506:
1505:
1493:
1485:
1472:
1467:
1448:
1429:
1424:
1411:
1406:
1387:
1382:
1375:
1370:
1360:
1355:
1349:
1341:
1336:
1326:
1321:
1317:
1309:
1304:
1292:
1287:
1268:
1252:
1239:
1222:
1220:
1130:
1124:
1103:
1097:
1051:
1036:
1014:
1005:
988:
966:
959:
936:
905:
878:
874:
873:
863:
850:
837:
831:
699:
686:
680:
659:
646:
640:
613:
600:
595:
591:
585:
572:
567:
563:
558:
556:
536:
530:
517:
511:
494:
488:
475:
469:
411:
398:
385:
372:
365:
357:
355:
96:. It is an important object of study in
1204:enables an explicit formula as a double
813:, the linking number is an example of a
3582:
278:The complement of a standard circle is
2243:The classical equations of motion are
2609:From electrostatics, the solution is
1166:number of times the Gauss map covers
88:The linking number was introduced by
7:
4443:
3515:being the algebraic analogs for the
3572: – Invariant of a knot diagram
821:
333:, according to the following rule:
108:, and has numerous applications in
3387:topological quantum field theories
3174:
3110:
2711:
2572:
2527:
2481:
2403:
2270:
2060:
1914:{\displaystyle M=\mathbb {R} ^{3}}
938:
907:
73:, the linking number is always an
61:that describes the linking of two
25:
2363:{\displaystyle -J_{\mu }A^{\mu }}
1072:Pick a point in the unit sphere,
715:involves only the overcrossings.
4442:
4431:
4430:
3532:forbidden minor characterization
3375:topological quantum field theory
2803:The path integral for arbitrary
1425:
1407:
1371:
1356:
1337:
1322:
211:
204:
197:
163:
156:
149:
1882:We are interested in doing the
781:-axis as a closed curve in the
541:{\displaystyle n_{2}+n_{4}\,\!}
499:{\displaystyle n_{1}+n_{3}\,\!}
4297:Dowker–Thistlethwaite notation
3491:Kauffman's self-linking number
3328:
3313:
3301:
3288:
3203:
3177:
3139:
3113:
3074:
3048:
2999:
2996:
2990:
2971:
2920:
2914:
2783:
2776:
2761:
2751:
2745:
2739:
2730:
2684:
2644:
2638:
2629:
2590:
2575:
2542:
2451:
2421:
2406:
1960:
1934:
1744:
1738:
1657:
1651:
1635:
1629:
1604:
1598:
1582:
1576:
1560:
1554:
1529:
1523:
1435:
1399:
1383:
1350:
1258:
1232:
1052:
1048:
1042:
1026:
1020:
1006:
1000:
994:
978:
972:
953:
941:
869:
347:the linking number. That is:
41:The two curves of this (2, 8)-
1:
3418:in three dimensions, any two
3414:Just as closed curves can be
2882:{\displaystyle J=J_{1}+J_{2}}
3389:in 4 spacetime dimensions.
1146:. Also, a neighborhood of (
765:of an oriented curve in the
305:Computing the linking number
3777:Encyclopedia of Mathematics
3759:Encyclopedia of Mathematics
1139:{\displaystyle \gamma _{2}}
1112:{\displaystyle \gamma _{1}}
822:Gauss's integral definition
708:{\displaystyle n_{2}-n_{3}}
668:{\displaystyle n_{1}-n_{4}}
4492:
3770:A.V. Chernavskii (2001) ,
3752:A.V. Chernavskii (2001) ,
815:topological quantum number
325:. Label each crossing as
292:Seifert–Van Kampen theorem
29:
4426:
4287:Alexander–Briggs notation
3735:10.1016/j.aop.2017.06.019
3566: – Geometric concept
2204:{\displaystyle \epsilon }
1162:it suffices to count the
731:have linking number zero.
45:have linking number four.
2516:, the equations become
1178:, this is precisely the
4378:List of knots and links
3926:Kinoshita–Terasaka knot
1716:In quantum field theory
913:{\displaystyle \Gamma }
777:-axis (thinking of the
719:Properties and examples
67:three-dimensional space
3606:This follows from the
3534:as the graphs with no
3460:
3410:
3364:
3155:
3029:
3006:
2883:
2837:
2817:
2794:
2600:
2510:
2461:
2384:
2364:
2321:
2234:
2205:
2182:
1915:
1873:
1800:
1777:
1751:
1695:
1210:Gauss linking integral
1140:
1113:
1063:
914:
888:
746:The linking number is
732:
727:The two curves of the
709:
669:
626:
542:
500:
427:
340:
314:
46:
18:Gauss linking integral
4168:Finite type invariant
3754:"Linking coefficient"
3461:
3459:{\displaystyle m+n+1}
3400:
3365:
3156:
3030:
3007:
2884:
2838:
2818:
2795:
2601:
2511:
2462:
2385:
2365:
2322:
2235:
2206:
2183:
1916:
1884:Feynman path integral
1874:
1801:
1778:
1752:
1696:
1141:
1114:
1064:
915:
889:
796:of its complement is
726:
710:
670:
627:
543:
501:
428:
339:
312:
106:differential geometry
40:
3608:Jordan curve theorem
3552:Differentiable curve
3438:
3171:
3042:
3019:
2896:
2847:
2827:
2807:
2616:
2523:
2477:
2397:
2374:
2334:
2250:
2224:
2195:
1928:
1890:
1886:for Chern–Simons in
1813:
1790:
1767:
1750:{\displaystyle U(1)}
1732:
1722:quantum field theory
1219:
1123:
1096:
935:
904:
830:
679:
639:
555:
510:
468:
354:
4338:Alexander's theorem
3727:2017AnPhy.384..254P
3666:1989CMaPh.121..351W
3648:Witten, E. (1989).
3479:self-linking number
3430:may be linked in a
2960:
2913:
124:, and the study of
92:in the form of the
3674:10.1007/bf01217730
3524:linkless embedding
3505:algebraic topology
3456:
3411:
3360:
3151:
3025:
3002:
2946:
2899:
2879:
2833:
2813:
2790:
2596:
2506:
2457:
2380:
2360:
2317:
2230:
2201:
2178:
1911:
1869:
1796:
1773:
1747:
1691:
1689:
1136:
1109:
1059:
910:
884:
733:
705:
665:
622:
538:
496:
423:
341:
315:
181:linking number −1
178:linking number −2
102:algebraic topology
47:
4458:
4457:
4312:Reidemeister move
4178:Khovanov homology
4173:Hyperbolic volume
3772:"Writhing number"
3705:Annals of Physics
3517:Milnor invariants
3498:Milnor invariants
3403:Milnor invariants
3339:
3222:
3108:
3028:{\displaystyle J}
2836:{\displaystyle J}
2816:{\displaystyle J}
2788:
2779:
2764:
2742:
2687:
2663:
2641:
2593:
2578:
2567:
2545:
2454:
2443:
2424:
2409:
2383:{\displaystyle J}
2305:
2233:{\displaystyle A}
2015:
1845:
1799:{\displaystyle M}
1776:{\displaystyle A}
1671:
1545:
1514:
1461:
1394:
1281:
1057:
792:, then the first
561:
421:
360:
288:fundamental group
239:
238:
233:linking number 3
230:linking number 2
227:linking number 1
184:linking number 0
118:quantum mechanics
16:(Redirected from
4483:
4446:
4445:
4434:
4433:
4398:Tait conjectures
4101:
4100:
4086:
4085:
4071:
4070:
3963:
3962:
3948:
3947:
3932:(−2,3,7) pretzel
3812:
3805:
3798:
3789:
3784:
3766:
3739:
3738:
3720:
3700:
3694:
3693:
3654:Comm. Math. Phys
3645:
3639:
3604:
3598:
3587:
3528:undirected graph
3465:
3463:
3462:
3457:
3420:closed manifolds
3369:
3367:
3366:
3361:
3356:
3355:
3340:
3338:
3337:
3336:
3331:
3316:
3310:
3309:
3308:
3286:
3283:
3282:
3270:
3269:
3268:
3267:
3253:
3252:
3240:
3239:
3238:
3237:
3223:
3221:
3210:
3202:
3201:
3189:
3188:
3160:
3158:
3157:
3152:
3147:
3146:
3142:
3138:
3137:
3125:
3124:
3109:
3104:
3093:
3073:
3072:
3060:
3059:
3034:
3032:
3031:
3026:
3011:
3009:
3008:
3003:
2989:
2988:
2970:
2969:
2959:
2954:
2942:
2941:
2940:
2939:
2912:
2907:
2888:
2886:
2885:
2880:
2878:
2877:
2865:
2864:
2842:
2840:
2839:
2834:
2822:
2820:
2819:
2814:
2799:
2797:
2796:
2791:
2789:
2787:
2786:
2781:
2780:
2772:
2766:
2765:
2757:
2754:
2748:
2744:
2743:
2735:
2729:
2728:
2719:
2718:
2709:
2708:
2692:
2689:
2688:
2680:
2677:
2676:
2664:
2662:
2651:
2643:
2642:
2634:
2628:
2627:
2605:
2603:
2602:
2597:
2595:
2594:
2586:
2580:
2579:
2571:
2568:
2563:
2555:
2547:
2546:
2538:
2535:
2534:
2515:
2513:
2512:
2507:
2499:
2498:
2489:
2488:
2466:
2464:
2463:
2458:
2456:
2455:
2447:
2444:
2439:
2431:
2426:
2425:
2417:
2411:
2410:
2402:
2389:
2387:
2386:
2381:
2369:
2367:
2366:
2361:
2359:
2358:
2349:
2348:
2326:
2324:
2323:
2318:
2316:
2315:
2306:
2301:
2293:
2288:
2287:
2278:
2277:
2268:
2267:
2239:
2237:
2236:
2231:
2210:
2208:
2207:
2202:
2187:
2185:
2184:
2179:
2177:
2173:
2172:
2171:
2161:
2160:
2148:
2147:
2146:
2145:
2125:
2124:
2114:
2113:
2101:
2100:
2099:
2098:
2078:
2077:
2068:
2067:
2058:
2057:
2048:
2047:
2029:
2028:
2016:
2014:
2006:
1998:
1985:
1984:
1975:
1974:
1959:
1958:
1946:
1945:
1920:
1918:
1917:
1912:
1910:
1909:
1904:
1878:
1876:
1875:
1870:
1856:
1855:
1846:
1844:
1833:
1828:
1827:
1805:
1803:
1802:
1797:
1782:
1780:
1779:
1774:
1756:
1754:
1753:
1748:
1711:
1700:
1698:
1697:
1692:
1690:
1672:
1670:
1669:
1664:
1660:
1650:
1649:
1628:
1627:
1612:
1611:
1607:
1597:
1596:
1575:
1574:
1553:
1552:
1547:
1546:
1538:
1522:
1521:
1516:
1515:
1507:
1494:
1492:
1491:
1490:
1489:
1477:
1476:
1462:
1460:
1449:
1441:
1434:
1433:
1428:
1416:
1415:
1410:
1395:
1393:
1392:
1391:
1386:
1380:
1379:
1374:
1365:
1364:
1359:
1353:
1347:
1346:
1345:
1340:
1331:
1330:
1325:
1318:
1316:
1315:
1314:
1313:
1299:
1298:
1297:
1296:
1282:
1280:
1269:
1257:
1256:
1244:
1243:
1145:
1143:
1142:
1137:
1135:
1134:
1118:
1116:
1115:
1110:
1108:
1107:
1068:
1066:
1065:
1060:
1058:
1056:
1055:
1041:
1040:
1019:
1018:
1009:
1003:
993:
992:
971:
970:
960:
919:
917:
916:
911:
893:
891:
890:
885:
883:
882:
877:
868:
867:
855:
854:
842:
841:
714:
712:
711:
706:
704:
703:
691:
690:
674:
672:
671:
666:
664:
663:
651:
650:
631:
629:
628:
623:
618:
617:
605:
604:
590:
589:
577:
576:
562:
559:
547:
545:
544:
539:
535:
534:
522:
521:
505:
503:
502:
497:
493:
492:
480:
479:
432:
430:
429:
424:
422:
417:
416:
415:
403:
402:
390:
389:
377:
376:
366:
361:
358:
243:regular homotopy
215:
208:
201:
167:
160:
153:
143:
126:DNA supercoiling
122:electromagnetism
94:linking integral
21:
4491:
4490:
4486:
4485:
4484:
4482:
4481:
4480:
4471:Knot invariants
4461:
4460:
4459:
4454:
4422:
4326:
4292:Conway notation
4276:
4270:
4257:Tricolorability
4105:
4099:
4096:
4095:
4094:
4084:
4081:
4080:
4079:
4069:
4066:
4065:
4064:
4056:
4046:
4036:
4026:
4007:
3986:Composite knots
3972:
3961:
3958:
3957:
3956:
3953:Borromean rings
3946:
3943:
3942:
3941:
3915:
3905:
3895:
3885:
3877:
3869:
3859:
3849:
3830:
3816:
3769:
3751:
3748:
3743:
3742:
3702:
3701:
3697:
3647:
3646:
3642:
3637:
3630:
3623:
3616:
3605:
3601:
3588:
3584:
3579:
3548:
3536:Petersen family
3513:Massey products
3436:
3435:
3432:Euclidean space
3407:Borromean rings
3395:
3393:Generalizations
3341:
3326:
3311:
3300:
3287:
3274:
3259:
3254:
3244:
3229:
3224:
3214:
3193:
3180:
3169:
3168:
3129:
3116:
3094:
3091:
3087:
3064:
3051:
3040:
3039:
3017:
3016:
2980:
2961:
2931:
2926:
2894:
2893:
2869:
2856:
2845:
2844:
2825:
2824:
2805:
2804:
2749:
2720:
2710:
2694:
2693:
2668:
2655:
2619:
2614:
2613:
2556:
2526:
2521:
2520:
2490:
2480:
2475:
2474:
2432:
2395:
2394:
2372:
2371:
2350:
2340:
2332:
2331:
2307:
2294:
2279:
2269:
2253:
2248:
2247:
2222:
2221:
2217:renormalization
2193:
2192:
2163:
2152:
2137:
2132:
2116:
2105:
2090:
2085:
2069:
2059:
2049:
2033:
2020:
2007:
1999:
1996:
1992:
1976:
1950:
1937:
1926:
1925:
1899:
1888:
1887:
1847:
1837:
1816:
1811:
1810:
1788:
1787:
1765:
1764:
1730:
1729:
1718:
1709:
1688:
1687:
1641:
1619:
1618:
1614:
1613:
1588:
1566:
1535:
1504:
1503:
1499:
1495:
1481:
1468:
1463:
1453:
1439:
1438:
1423:
1405:
1381:
1369:
1354:
1348:
1335:
1320:
1319:
1305:
1300:
1288:
1283:
1273:
1261:
1248:
1235:
1217:
1216:
1203:
1196:
1126:
1121:
1120:
1099:
1094:
1093:
1088:) that goes to
1032:
1010:
1004:
984:
962:
961:
933:
932:
902:
901:
872:
859:
846:
833:
828:
827:
824:
756:right-hand rule
721:
695:
682:
677:
676:
655:
642:
637:
636:
609:
596:
581:
568:
553:
552:
526:
513:
508:
507:
484:
471:
466:
465:
463:
456:
449:
442:
407:
394:
381:
368:
367:
352:
351:
307:
264:
134:
71:Euclidean space
57:is a numerical
35:
28:
23:
22:
15:
12:
11:
5:
4489:
4487:
4479:
4478:
4473:
4463:
4462:
4456:
4455:
4453:
4452:
4440:
4427:
4424:
4423:
4421:
4420:
4418:Surgery theory
4415:
4410:
4405:
4400:
4395:
4390:
4385:
4380:
4375:
4370:
4365:
4360:
4355:
4350:
4345:
4340:
4334:
4332:
4328:
4327:
4325:
4324:
4319:
4317:Skein relation
4314:
4309:
4304:
4299:
4294:
4289:
4283:
4281:
4272:
4271:
4269:
4268:
4262:Unknotting no.
4259:
4254:
4249:
4248:
4247:
4237:
4232:
4231:
4230:
4225:
4220:
4215:
4210:
4200:
4195:
4190:
4185:
4180:
4175:
4170:
4165:
4160:
4155:
4154:
4153:
4143:
4138:
4137:
4136:
4126:
4121:
4115:
4113:
4107:
4106:
4104:
4103:
4097:
4088:
4082:
4073:
4067:
4058:
4054:
4048:
4044:
4038:
4034:
4028:
4024:
4017:
4015:
4009:
4008:
4006:
4005:
4000:
3999:
3998:
3993:
3982:
3980:
3974:
3973:
3971:
3970:
3965:
3959:
3950:
3944:
3935:
3929:
3923:
3917:
3913:
3907:
3903:
3897:
3893:
3887:
3883:
3879:
3875:
3871:
3867:
3861:
3857:
3851:
3847:
3840:
3838:
3832:
3831:
3817:
3815:
3814:
3807:
3800:
3792:
3786:
3785:
3767:
3747:
3744:
3741:
3740:
3695:
3660:(3): 351–399.
3640:
3635:
3628:
3621:
3614:
3599:
3581:
3580:
3578:
3575:
3574:
3573:
3567:
3564:Kissing number
3561:
3558:Hopf invariant
3555:
3547:
3544:
3543:
3542:
3520:
3501:
3494:
3471:
3455:
3452:
3449:
3446:
3443:
3422:of dimensions
3394:
3391:
3371:
3370:
3359:
3354:
3351:
3348:
3344:
3335:
3330:
3325:
3322:
3319:
3315:
3307:
3303:
3299:
3296:
3293:
3290:
3281:
3277:
3273:
3266:
3262:
3257:
3251:
3247:
3243:
3236:
3232:
3227:
3220:
3217:
3213:
3208:
3205:
3200:
3196:
3192:
3187:
3183:
3179:
3176:
3162:
3161:
3150:
3145:
3141:
3136:
3132:
3128:
3123:
3119:
3115:
3112:
3107:
3103:
3100:
3097:
3090:
3085:
3082:
3079:
3076:
3071:
3067:
3063:
3058:
3054:
3050:
3047:
3024:
3013:
3012:
3001:
2998:
2995:
2992:
2987:
2983:
2979:
2976:
2973:
2968:
2964:
2958:
2953:
2949:
2945:
2938:
2934:
2929:
2925:
2922:
2919:
2916:
2911:
2906:
2902:
2876:
2872:
2868:
2863:
2859:
2855:
2852:
2832:
2812:
2801:
2800:
2785:
2778:
2775:
2769:
2763:
2760:
2753:
2747:
2741:
2738:
2732:
2727:
2723:
2717:
2713:
2707:
2704:
2701:
2697:
2686:
2683:
2675:
2671:
2667:
2661:
2658:
2654:
2649:
2646:
2640:
2637:
2631:
2626:
2622:
2607:
2606:
2592:
2589:
2583:
2577:
2574:
2566:
2562:
2559:
2553:
2550:
2544:
2541:
2533:
2529:
2505:
2502:
2497:
2493:
2487:
2483:
2468:
2467:
2453:
2450:
2442:
2438:
2435:
2429:
2423:
2420:
2414:
2408:
2405:
2379:
2357:
2353:
2347:
2343:
2339:
2328:
2327:
2314:
2310:
2304:
2300:
2297:
2291:
2286:
2282:
2276:
2272:
2266:
2263:
2260:
2256:
2229:
2213:regularization
2200:
2189:
2188:
2176:
2170:
2166:
2159:
2155:
2151:
2144:
2140:
2135:
2131:
2128:
2123:
2119:
2112:
2108:
2104:
2097:
2093:
2088:
2084:
2081:
2076:
2072:
2066:
2062:
2056:
2052:
2046:
2043:
2040:
2036:
2032:
2027:
2023:
2019:
2013:
2010:
2005:
2002:
1995:
1991:
1988:
1983:
1979:
1973:
1968:
1965:
1962:
1957:
1953:
1949:
1944:
1940:
1936:
1933:
1908:
1903:
1898:
1895:
1880:
1879:
1868:
1865:
1862:
1859:
1854:
1850:
1843:
1840:
1836:
1831:
1826:
1823:
1819:
1795:
1772:
1746:
1743:
1740:
1737:
1728:observable in
1717:
1714:
1702:
1701:
1686:
1683:
1679:
1676:
1668:
1663:
1659:
1656:
1653:
1648:
1644:
1640:
1637:
1634:
1631:
1626:
1622:
1617:
1610:
1606:
1603:
1600:
1595:
1591:
1587:
1584:
1581:
1578:
1573:
1569:
1565:
1562:
1559:
1556:
1551:
1544:
1541:
1534:
1531:
1528:
1525:
1520:
1513:
1510:
1502:
1498:
1488:
1484:
1480:
1475:
1471:
1466:
1459:
1456:
1452:
1447:
1444:
1442:
1440:
1437:
1432:
1427:
1422:
1419:
1414:
1409:
1404:
1401:
1398:
1390:
1385:
1378:
1373:
1368:
1363:
1358:
1352:
1344:
1339:
1334:
1329:
1324:
1312:
1308:
1303:
1295:
1291:
1286:
1279:
1276:
1272:
1267:
1264:
1262:
1260:
1255:
1251:
1247:
1242:
1238:
1234:
1231:
1228:
1225:
1224:
1201:
1194:
1133:
1129:
1106:
1102:
1070:
1069:
1054:
1050:
1047:
1044:
1039:
1035:
1031:
1028:
1025:
1022:
1017:
1013:
1008:
1002:
999:
996:
991:
987:
983:
980:
977:
974:
969:
965:
958:
955:
952:
949:
946:
943:
940:
909:
881:
876:
871:
866:
862:
858:
853:
849:
845:
840:
836:
823:
820:
819:
818:
807:
794:homology group
786:
763:winding number
759:
744:
741:
738:Whitehead link
729:Whitehead link
720:
717:
702:
698:
694:
689:
685:
662:
658:
654:
649:
645:
633:
632:
621:
616:
612:
608:
603:
599:
594:
588:
584:
580:
575:
571:
566:
560:linking number
533:
529:
525:
520:
516:
491:
487:
483:
478:
474:
461:
454:
447:
440:
434:
433:
420:
414:
410:
406:
401:
397:
393:
388:
384:
380:
375:
371:
364:
359:linking number
306:
303:
302:
301:
298:
295:
284:
276:
263:
260:
237:
236:
234:
231:
228:
225:
223:
220:
219:
216:
209:
202:
195:
193:
190:
189:
187:
185:
182:
179:
176:
173:
172:
170:
168:
161:
154:
147:
133:
130:
55:linking number
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4488:
4477:
4474:
4472:
4469:
4468:
4466:
4451:
4450:
4441:
4439:
4438:
4429:
4428:
4425:
4419:
4416:
4414:
4411:
4409:
4406:
4404:
4401:
4399:
4396:
4394:
4391:
4389:
4386:
4384:
4381:
4379:
4376:
4374:
4371:
4369:
4366:
4364:
4361:
4359:
4356:
4354:
4353:Conway sphere
4351:
4349:
4346:
4344:
4341:
4339:
4336:
4335:
4333:
4329:
4323:
4320:
4318:
4315:
4313:
4310:
4308:
4305:
4303:
4300:
4298:
4295:
4293:
4290:
4288:
4285:
4284:
4282:
4280:
4273:
4267:
4263:
4260:
4258:
4255:
4253:
4250:
4246:
4243:
4242:
4241:
4238:
4236:
4233:
4229:
4226:
4224:
4221:
4219:
4216:
4214:
4211:
4209:
4206:
4205:
4204:
4201:
4199:
4196:
4194:
4191:
4189:
4186:
4184:
4181:
4179:
4176:
4174:
4171:
4169:
4166:
4164:
4161:
4159:
4156:
4152:
4149:
4148:
4147:
4144:
4142:
4139:
4135:
4132:
4131:
4130:
4127:
4125:
4124:Arf invariant
4122:
4120:
4117:
4116:
4114:
4112:
4108:
4092:
4089:
4077:
4074:
4062:
4059:
4052:
4049:
4042:
4039:
4032:
4029:
4022:
4019:
4018:
4016:
4014:
4010:
4004:
4001:
3997:
3994:
3992:
3989:
3988:
3987:
3984:
3983:
3981:
3979:
3975:
3969:
3966:
3954:
3951:
3939:
3936:
3933:
3930:
3927:
3924:
3921:
3918:
3911:
3908:
3901:
3898:
3891:
3888:
3886:
3880:
3878:
3872:
3865:
3862:
3855:
3852:
3845:
3842:
3841:
3839:
3837:
3833:
3828:
3824:
3820:
3813:
3808:
3806:
3801:
3799:
3794:
3793:
3790:
3783:
3779:
3778:
3773:
3768:
3765:
3761:
3760:
3755:
3750:
3749:
3745:
3736:
3732:
3728:
3724:
3719:
3714:
3710:
3706:
3699:
3696:
3691:
3687:
3683:
3679:
3675:
3671:
3667:
3663:
3659:
3655:
3651:
3644:
3641:
3634:
3631: +
3627:
3620:
3617: +
3613:
3609:
3603:
3600:
3596:
3592:
3586:
3583:
3576:
3571:
3568:
3565:
3562:
3559:
3556:
3553:
3550:
3549:
3545:
3540:
3537:
3533:
3529:
3525:
3521:
3518:
3514:
3510:
3506:
3502:
3499:
3495:
3492:
3488:
3484:
3480:
3476:
3472:
3469:
3453:
3450:
3447:
3444:
3441:
3434:of dimension
3433:
3429:
3425:
3421:
3417:
3413:
3412:
3408:
3404:
3399:
3392:
3390:
3388:
3382:
3380:
3379:Edward Witten
3376:
3357:
3352:
3349:
3346:
3342:
3333:
3323:
3320:
3317:
3305:
3297:
3294:
3291:
3279:
3275:
3271:
3264:
3260:
3255:
3249:
3245:
3241:
3234:
3230:
3225:
3218:
3215:
3211:
3206:
3198:
3194:
3190:
3185:
3181:
3167:
3166:
3165:
3148:
3143:
3134:
3130:
3126:
3121:
3117:
3105:
3101:
3098:
3095:
3088:
3083:
3080:
3077:
3069:
3065:
3061:
3056:
3052:
3045:
3038:
3037:
3036:
3022:
2993:
2985:
2981:
2977:
2974:
2966:
2962:
2956:
2951:
2947:
2943:
2936:
2932:
2927:
2923:
2917:
2909:
2904:
2900:
2892:
2891:
2890:
2874:
2870:
2866:
2861:
2857:
2853:
2850:
2830:
2810:
2773:
2767:
2758:
2736:
2725:
2721:
2715:
2705:
2702:
2699:
2695:
2681:
2673:
2669:
2665:
2659:
2656:
2652:
2647:
2635:
2624:
2620:
2612:
2611:
2610:
2587:
2581:
2564:
2560:
2557:
2551:
2548:
2539:
2531:
2519:
2518:
2517:
2503:
2500:
2495:
2491:
2485:
2473:
2448:
2440:
2436:
2433:
2427:
2418:
2412:
2393:
2392:
2391:
2377:
2355:
2351:
2345:
2341:
2337:
2312:
2308:
2302:
2298:
2295:
2289:
2284:
2280:
2274:
2264:
2261:
2258:
2254:
2246:
2245:
2244:
2241:
2227:
2218:
2214:
2198:
2174:
2168:
2164:
2157:
2153:
2149:
2142:
2138:
2133:
2129:
2126:
2121:
2117:
2110:
2106:
2102:
2095:
2091:
2086:
2082:
2079:
2074:
2070:
2064:
2054:
2050:
2044:
2041:
2038:
2034:
2030:
2025:
2021:
2017:
2011:
2008:
2003:
2000:
1993:
1989:
1986:
1981:
1977:
1966:
1963:
1955:
1951:
1947:
1942:
1938:
1931:
1924:
1923:
1922:
1906:
1896:
1893:
1885:
1866:
1863:
1860:
1857:
1852:
1848:
1841:
1838:
1834:
1829:
1824:
1821:
1817:
1809:
1808:
1807:
1793:
1786:
1770:
1762:
1759:
1741:
1735:
1727:
1723:
1715:
1713:
1707:
1684:
1681:
1677:
1674:
1666:
1661:
1654:
1646:
1642:
1638:
1632:
1624:
1620:
1615:
1608:
1601:
1593:
1589:
1585:
1579:
1571:
1567:
1563:
1557:
1549:
1542:
1539:
1532:
1526:
1518:
1511:
1508:
1500:
1486:
1482:
1478:
1473:
1469:
1464:
1457:
1454:
1450:
1445:
1443:
1430:
1420:
1417:
1412:
1402:
1396:
1388:
1376:
1366:
1361:
1342:
1332:
1327:
1310:
1306:
1301:
1293:
1289:
1284:
1277:
1274:
1270:
1265:
1263:
1253:
1249:
1245:
1240:
1236:
1229:
1226:
1215:
1214:
1213:
1211:
1207:
1206:line integral
1200:
1193:
1188:
1185:
1181:
1177:
1176:regular value
1173:
1169:
1165:
1161:
1157:
1153:
1149:
1131:
1127:
1104:
1100:
1091:
1087:
1083:
1079:
1075:
1045:
1037:
1033:
1029:
1023:
1015:
1011:
997:
989:
985:
981:
975:
967:
963:
956:
950:
947:
944:
931:
930:
929:
927:
923:
900:
898:
894:, define the
879:
864:
860:
856:
851:
847:
843:
838:
834:
816:
812:
808:
805:
804:
799:
795:
791:
787:
784:
780:
776:
772:
768:
764:
760:
757:
753:
750:: taking the
749:
745:
742:
739:
735:
734:
730:
725:
718:
716:
700:
696:
692:
687:
683:
660:
656:
652:
647:
643:
619:
614:
610:
606:
601:
597:
592:
586:
582:
578:
573:
569:
564:
551:
550:
549:
531:
527:
523:
518:
514:
489:
485:
481:
476:
472:
460:
453:
446:
439:
418:
412:
408:
404:
399:
395:
391:
386:
382:
378:
373:
369:
362:
350:
349:
348:
346:
338:
334:
332:
328:
324:
320:
311:
304:
299:
296:
293:
289:
285:
281:
277:
274:
270:
269:
268:
261:
259:
257:
255:
250:
249:
244:
235:
232:
229:
226:
224:
222:
221:
217:
214:
210:
207:
203:
200:
196:
194:
192:
191:
188:
186:
183:
180:
177:
175:
174:
171:
169:
166:
162:
159:
155:
152:
148:
145:
144:
141:
139:
131:
129:
127:
123:
119:
115:
111:
107:
103:
99:
95:
91:
86:
84:
80:
76:
72:
68:
64:
63:closed curves
60:
56:
52:
44:
39:
33:
19:
4447:
4435:
4363:Double torus
4348:Braid theory
4197:
4163:Crossing no.
4158:Crosscap no.
3844:Figure-eight
3775:
3757:
3708:
3704:
3698:
3657:
3653:
3643:
3632:
3625:
3618:
3611:
3602:
3585:
3490:
3486:
3482:
3427:
3423:
3383:
3372:
3163:
3014:
2802:
2608:
2472:Lorenz gauge
2469:
2329:
2242:
2190:
1881:
1806:is given by
1761:gauge theory
1758:Chern–Simons
1719:
1703:
1209:
1198:
1191:
1189:
1171:
1167:
1163:
1159:
1155:
1151:
1147:
1089:
1085:
1081:
1077:
1073:
1071:
895:
825:
801:
778:
774:
770:
766:
752:mirror image
635:The formula
634:
458:
451:
444:
437:
435:
344:
342:
330:
326:
317:There is an
316:
280:homeomorphic
272:
265:
253:
247:
240:
135:
116:, including
93:
87:
54:
48:
4198:Linking no.
4119:Alternating
3920:Conway knot
3900:Carrick mat
3854:Three-twist
3819:Knot theory
3711:: 254–287.
3509:cup product
3475:framed knot
1783:on a three-
1726:Wilson loop
110:mathematics
98:knot theory
83:3-manifolds
79:orientation
51:mathematics
4465:Categories
4358:Complement
4322:Tabulation
4279:operations
4203:Polynomial
4193:Link group
4188:Knot group
4151:Invertible
4129:Bridge no.
4111:Invariants
4041:Cinquefoil
3910:Perko pair
3836:Hyperbolic
3746:References
3718:1612.09298
3690:0667.57005
798:isomorphic
256:-principle
132:Definition
43:torus link
32:Numberlink
4252:Stick no.
4208:Alexander
4146:Chirality
4091:Solomon's
4051:Septafoil
3978:Satellite
3938:Whitehead
3864:Stevedore
3782:EMS Press
3764:EMS Press
3353:ν
3350:μ
3347:λ
3343:ε
3321:−
3306:ν
3295:−
3280:μ
3261:γ
3256:∫
3250:λ
3231:γ
3226:∫
3219:π
3195:γ
3182:γ
3175:Φ
3131:γ
3118:γ
3111:Φ
3099:π
3084:
3066:γ
3053:γ
2978:−
2963:δ
2957:μ
2933:γ
2928:∫
2910:μ
2777:→
2768:−
2762:→
2740:→
2726:ν
2716:μ
2712:∂
2706:ν
2703:μ
2700:λ
2696:ε
2685:→
2666:∫
2639:→
2625:λ
2591:→
2582:×
2576:→
2573:∇
2561:π
2552:−
2543:→
2528:∇
2496:μ
2486:μ
2482:∂
2452:→
2437:π
2422:→
2413:×
2407:→
2404:∇
2356:μ
2346:μ
2338:−
2313:λ
2299:π
2285:ν
2275:μ
2271:∂
2265:ν
2262:μ
2259:λ
2255:ε
2199:ϵ
2169:μ
2158:μ
2139:γ
2134:∫
2122:μ
2111:μ
2092:γ
2087:∫
2075:ν
2065:μ
2061:∂
2055:λ
2045:ν
2042:μ
2039:λ
2035:ε
2018:∫
2012:π
1990:
1982:μ
1967:∫
1952:γ
1939:γ
1861:∧
1849:∫
1842:π
1643:γ
1639:−
1621:γ
1590:γ
1586:−
1568:γ
1543:˙
1540:γ
1512:˙
1509:γ
1479:×
1465:∫
1458:π
1418:×
1397:⋅
1367:−
1333:−
1307:γ
1302:∮
1290:γ
1285:∮
1278:π
1250:γ
1237:γ
1230:
1170:. Since
1128:γ
1101:γ
1034:γ
1030:−
1012:γ
986:γ
982:−
964:γ
939:Γ
920:from the
908:Γ
870:→
857::
848:γ
835:γ
693:−
653:−
607:−
579:−
405:−
392:−
319:algorithm
283:directly.
273:homotopic
248:immersion
59:invariant
4437:Category
4307:Mutation
4275:Notation
4228:Kauffman
4141:Brunnian
4134:2-bridge
4003:Knot sum
3934:(12n242)
3546:See also
2889:, with
1785:manifold
1706:Jacobian
1119:is over
783:3-sphere
331:negative
327:positive
4449:Commons
4368:Fibered
4266:problem
4235:Pretzel
4213:Bracket
4031:Trefoil
3968:L10a140
3928:(11n42)
3922:(11n34)
3890:Endless
3723:Bibcode
3682:0990772
3662:Bibcode
3164:where
924:to the
811:physics
323:diagram
114:science
75:integer
4476:Curves
4413:Writhe
4383:Ribbon
4218:HOMFLY
4061:Unlink
4021:Unknot
3996:Square
3991:Granny
3688:
3680:
3591:writhe
3570:Writhe
3526:of an
3507:, the
3477:has a
3468:degree
3416:linked
2191:Here,
1208:, the
1180:degree
1164:signed
926:sphere
790:simple
748:chiral
436:where
104:, and
53:, the
4403:Twist
4388:Slice
4343:Berge
4331:Other
4302:Flype
4240:Prime
4223:Jones
4183:Genus
4013:Torus
3827:links
3823:knots
3713:arXiv
3593:of a
3577:Notes
3539:minor
1184:image
1174:is a
922:torus
897:Gauss
345:twice
262:Proof
138:moved
90:Gauss
4408:Wild
4373:Knot
4277:and
4264:and
4245:list
4076:Hopf
3825:and
3709:384C
3624:and
3595:knot
3473:Any
3426:and
3401:The
1921::
1227:link
1197:and
761:The
506:and
286:The
112:and
4393:Sum
3914:161
3912:(10
3731:doi
3686:Zbl
3670:doi
3658:121
3503:In
3081:exp
2240:.
2215:or
1987:exp
1720:In
1712:).
1497:det
928:by
899:map
809:In
800:to
329:or
65:in
49:In
4467::
4093:(4
4078:(2
4063:(0
4053:(7
4043:(5
4033:(3
4023:(0
3955:(6
3940:(5
3904:18
3902:(8
3892:(7
3866:(6
3856:(5
3846:(4
3780:,
3774:,
3762:,
3756:,
3729:.
3721:.
3707:.
3684:.
3678:MR
3676:.
3668:.
3656:.
3652:.
3522:A
1212::
1150:,
1084:,
785:).
740:).
457:,
450:,
443:,
218:⋯
146:⋯
128:.
120:,
100:,
4102:)
4098:1
4087:)
4083:1
4072:)
4068:1
4057:)
4055:1
4047:)
4045:1
4037:)
4035:1
4027:)
4025:1
3964:)
3960:2
3949:)
3945:1
3916:)
3906:)
3896:)
3894:4
3884:3
3882:6
3876:2
3874:6
3870:)
3868:1
3860:)
3858:2
3850:)
3848:1
3829:)
3821:(
3811:e
3804:t
3797:v
3737:.
3733::
3725::
3715::
3692:.
3672::
3664::
3636:4
3633:n
3629:2
3626:n
3622:3
3619:n
3615:1
3612:n
3541:.
3519:.
3493:.
3487:C
3483:C
3454:1
3451:+
3448:n
3445:+
3442:m
3428:n
3424:m
3358:,
3334:3
3329:|
3324:y
3318:x
3314:|
3302:)
3298:y
3292:x
3289:(
3276:y
3272:d
3265:2
3246:x
3242:d
3235:1
3216:4
3212:1
3207:=
3204:]
3199:2
3191:,
3186:1
3178:[
3149:,
3144:)
3140:]
3135:2
3127:,
3122:1
3114:[
3106:k
3102:i
3096:2
3089:(
3078:=
3075:]
3070:2
3062:,
3057:1
3049:[
3046:Z
3023:J
3000:)
2997:)
2994:t
2991:(
2986:i
2982:x
2975:x
2972:(
2967:3
2952:i
2948:x
2944:d
2937:i
2924:=
2921:)
2918:x
2915:(
2905:i
2901:J
2875:2
2871:J
2867:+
2862:1
2858:J
2854:=
2851:J
2831:J
2811:J
2784:|
2774:y
2759:x
2752:|
2746:)
2737:y
2731:(
2722:J
2682:y
2674:3
2670:d
2660:k
2657:2
2653:1
2648:=
2645:)
2636:x
2630:(
2621:A
2588:J
2565:k
2558:2
2549:=
2540:A
2532:2
2504:0
2501:=
2492:A
2449:J
2441:k
2434:2
2428:=
2419:A
2378:J
2352:A
2342:J
2309:J
2303:k
2296:2
2290:=
2281:A
2228:A
2175:)
2165:A
2154:x
2150:d
2143:2
2130:i
2127:+
2118:A
2107:x
2103:d
2096:1
2083:i
2080:+
2071:A
2051:A
2031:x
2026:3
2022:d
2009:4
2004:k
2001:i
1994:(
1978:A
1972:D
1964:=
1961:]
1956:2
1948:,
1943:1
1935:[
1932:Z
1907:3
1902:R
1897:=
1894:M
1867:A
1864:d
1858:A
1853:M
1839:4
1835:k
1830:=
1825:S
1822:C
1818:S
1794:M
1771:A
1745:)
1742:1
1739:(
1736:U
1710:π
1685:t
1682:d
1678:s
1675:d
1667:3
1662:|
1658:)
1655:t
1652:(
1647:2
1636:)
1633:s
1630:(
1625:1
1616:|
1609:)
1605:)
1602:t
1599:(
1594:2
1583:)
1580:s
1577:(
1572:1
1564:,
1561:)
1558:t
1555:(
1550:2
1533:,
1530:)
1527:s
1524:(
1519:1
1501:(
1487:1
1483:S
1474:1
1470:S
1455:4
1451:1
1446:=
1436:)
1431:2
1426:r
1421:d
1413:1
1408:r
1403:d
1400:(
1389:3
1384:|
1377:2
1372:r
1362:1
1357:r
1351:|
1343:2
1338:r
1328:1
1323:r
1311:2
1294:1
1275:4
1271:1
1266:=
1259:)
1254:2
1246:,
1241:1
1233:(
1202:2
1199:γ
1195:1
1192:γ
1172:v
1168:v
1160:v
1156:v
1152:t
1148:s
1132:2
1105:1
1090:v
1086:t
1082:s
1078:v
1074:v
1053:|
1049:)
1046:t
1043:(
1038:2
1027:)
1024:s
1021:(
1016:1
1007:|
1001:)
998:t
995:(
990:2
979:)
976:s
973:(
968:1
957:=
954:)
951:t
948:,
945:s
942:(
880:3
875:R
865:1
861:S
852:2
844:,
839:1
817:.
803:Z
779:z
775:z
771:y
769:-
767:x
758:.
701:3
697:n
688:2
684:n
661:4
657:n
648:1
644:n
620:.
615:3
611:n
602:2
598:n
593:=
587:4
583:n
574:1
570:n
565:=
532:4
528:n
524:+
519:2
515:n
490:3
486:n
482:+
477:1
473:n
462:4
459:n
455:3
452:n
448:2
445:n
441:1
438:n
419:2
413:4
409:n
400:3
396:n
387:2
383:n
379:+
374:1
370:n
363:=
254:h
34:.
20:)
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