|
New points of view in knot theory
Author(s):
Joan
S.
Birman
Journal:
Bull. Amer. Math. Soc.
28
(1993),
253-287.
MSC (1991):
Primary 57M25;
Secondary 57N99
MathSciNet review:
1191478
Retrieve article in:
PDF
References |
Similar articles |
Additional information
References:
- [Al1]
- J. W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. U.S.A. \textbf{9} (1923), 93--95. MR
- [Al2]
- J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. \textbf{30} (1928), 275--306. MR 1501429
- [Arn1]
- V. I. Arnold (ed.), Theory of singularities and its applications, Adv. in Soviet Math., vol. 1, Amer. Math. Soc., Providence, RI, 1990. MR 1089667
- [Arn2]
- V. I. Arnold (ed.), The cohomology ring of the colored braid group, Notes Acad. Sci. USSR \textbf{5} (1969), 227--232. MR 242196
- [Art]
- E. Artin, Theorie der Z\"opfe, Hamburg Abh. \textbf{4} (1925), 47--72. MR
- [Bae]
- J. Baez, Link invariants of finite type and perturbation theory, Lett. Math. Phys. MR 1193625
- [Bax]
- R. J. Baxter, Exactly solvable models in statistical mechanics, Academic Press, London, 1982. MR 690578
- [BD]
- A. A. Belavin and V. G. Drinfeld, On classical Yang-Baxter equation for simple Lie algebras, Funct. Anal. Appl. \textbf{16} (1982), 1--10. MR 674005
- [Be]
- D. Bennequin, Entrlacements et \'equations de Pfaffe, Asterisque \textbf{107--108} (1983), 87--161. MR 753131
- [Bi]
- J. S. Birman, Braids, links and mapping class groups, Ann. of Math Stud., vol. 84, Princeton Univ. Press, Princeton, NJ, 1974. MR 375281
- [BL]
- J. S. Birman and X. S. Lin, Knot polynomials and Vassiliev invariants, Invent. Math. MR
- [BM1]
- J. S. Birman and W. W. Menasco, Studying links via closed braids {\rm III:} Classifying links which are closed $3$-braids, Pacific J. Math. MR 1237139
- [BM2]
- J. S. Birman and W. W. Menasco, Studying links via closed braids {\rm VI:} A non-finiteness theorem, Pacific J. Math., vol. 156 (1992), 265--285. MR 1186805
- [BN]
- D. Bar Natan, On the Vassiliev knot invariants, Harvard Univ., preprint 1992. MR 1318886
- [BW]
- J. S. Birman and H. Wenzl, Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc. \textbf{313} (1989), 249--273. MR 992598
- [BZ]
- G. Burde and H. Zieschang, Knots, de Gruyter, Berlin and New York, 1985. MR 808776
- [C]
- J. H. Conway, \emph{An enumeration of knots and links and some of their algebraic properties} (J. Leech, ed.), Pergamon Press, New York, 1970 pp.~329--358. MR 258014
- [CCG]
- D. Cooper, M. Culler, H. Gillet, D. D. Long, and P. B. Shalen, Plane curves associated to character varieties of $3$-manifolds, Univ. of California, Santa Barbara, preprint 1991. MR
- [CGL]
- M. Culler, C. McA. Gordon, J. Leucke and P. B. Shalen, Dehn surgery on knots, Ann. of Math. (2) \textbf{125} (1987), 237--300. MR 881270
- [De]
- M. Dehn, Die beiden Kleeblattschlingen, Math. Ann. \textbf{75} (1914), 402--413. MR 1511799
- [Dr1]
- V. G. Drinfeld, On a constant quasi-classical solution to the quantum Yang-Baxter equation, Doklady Acad. Nauk SSSR \textbf{273} (1983), 531--550. MR 725930
- [Dr2]
- V. G. Drinfeld, \emph{Quantum groups} vol.~1, Amer. Math. Soc., Providence, RI, 1987 pp.~798--820. MR 934283
- [FH]
- M. Freedman and Z. H. He, On the energy of knots and unknots, Univ. of California, San Diego, preprint 1992. MR
- [FHL]
- P. Freyd, J. Hoste, W. B. R. Lickorish, K. Millett, A. Ocneanu, and D. Yetter, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. \textbf{12} (1985), 239--246. MR 776477
- [FW]
- J. Franks and R. F. Williams, Braids and the Jones polynomial, Trans. Amer. Math. Soc. \textbf{303} (1987), 97--108. MR 896009
- [GL]
- C. McA. Gordon and J. Leucke, Knots are determined by their complements, Bull. Amer. Math. Soc. \textbf{20} (1989), 83--88. MR 972070
- [Go]
- L. Goeritz, Bermerkungen zur knotentheorie, Hamburg Abh. \textbf{10} (1934), 201--210. MR
- [Gu]
- M. Gusarov, A new form of the Conway-Jones polynomial for knots via von Neumann algebras, Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov. (LOMI) \textbf{193} (1991), 4--9. MR 1157140
- [Ha]
- W. Haken, \"Uber das Hom\"omorphieproblem der $3$-Mannigfaltigkeiten, Math. Z. \textbf{80} (1962), 89--120. MR 160196
- [He]
- G. Hemion, On the classification of homeomorphisms of $2$-manifolds and the classification of $3$-manifolds, Acta Math. \textbf{142} (1979), 123--155. MR 512214
- [Ho]
- J. Hoste, A polynomial invariant of knots and links, Pacific J. Math. \textbf{124} (1986), 295--320. MR 856165
- [Je]
- L. Jeffrey, Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Oxford Univ., preprint 1992. MR 1175494
- [Ji1]
- M. Jimbo, A $q$-difference analogue of $U(g)$ and the Yang-Baxter equation, Lett. Math. Phys. \textbf{10} (1985), 63--69. MR 797001
- [Ji2]
- M. Jimbo, Quantum $R$-matrix for the generalized Toda system, Comm. Math. Physics \textbf{102} (1986), 537--547. MR 824090
- [Jo1]
- V. F. R. Jones, Index for subfactors, Invent. Math. \textbf{72} (1983), 1--25. MR 696688
- [Jo2]
- V. F. R. Jones, \emph{Braid groups, Hecke algebras and type $\roman {II}_1$ factors}, Pitman Res. Notes Math. Ser. vol.~123, Longman Sci. Tech., Harlow, 1986 pp.~242--273. MR 866500
- [Jo3]
- V. F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. \textbf{12} (1985), 103--111. MR 766964
- [Jo4]
- V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) \textbf{126} (1987), 335--388. MR 908150
- [Jo5]
- V. F. R. Jones, On knot invariants related to some statistical mechanics models, Pacific J. Math. \textbf{137} (1989), 311--334. MR 990215
- [JR]
- V. Jones and M. Rosso, Invariants of torus knots derived from quantum groups, Abstracts Amer. Math. Soc., Abstract no. 874-16-124 (1992). MR 1209320
- [JT]
- W. Jaco and J. L. Tollefson, Algorithms for the complete decomposition of a closed $3$-manifold, Univ. of Conn. at Storres, preprint 1992. MR 1339832
- [Kal]
- E. Kalfagianni, The G$_2$ link invariant, Columbia Univ., preprint 1992. MR
- [Kan]
- T. Kanenobu, Infinitely many knots with the same polynomial, Proc. Amer. Math. Soc. \textbf{97} (1986), 158--162. MR 831406
- [Kau 1]
- L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. \textbf{318} (1990), 417--471. MR 958895
- [Kau 2]
- L. H. Kauffman, States models and the Jones polynomial, Topology \textbf{26} (1987), 395--407. MR 899057
- [KM]
- R. Kirby and P. Melvin, The $3$-manifold invariants of Witten and Reshetikhin-Turaev for $SL(2,\Bbb {C})$, Invent. Math. \textbf{105} (1991), 473--545. MR 1117149
- [Koh1]
- T. Kohno, Monodromy representations of braid groups and Yang-Baxter equations, Ann. Inst. Fourier (Grenoble) \textbf{37} (1987), 139--160. MR 927394
- [Koh 2]
- T. Kohno, \emph{Linear representations of braid groups and classical Yang-Baxter equations}, Amer. Math. Soc., Providence, RI, 1988 pp.~339--363. MR 975088
- [Kon]
- M. Kontsevich, Vassiliev's knot invariants, Adv. in Soviet Math. MR
- [Ku]
- G. Kuperberg, The $1,0,1,1,4,10$ Ansatz, preprint. MR 1995781
- [KV]
- L. Kauffman and P. Vogel, Link polynomials and a graphical calculus, J. Knot Theory Ramifications \textbf{1} (1992), 59--104. MR 1155094
- [La]
- R. Lawrence, Homological representations of the Hecke algebras, Comm. Math. Phys. \textbf{135} (1990), 141--191. MR 1086755
- [Li1]
- X. S. Lin, Vertex models, quantum groups and Vassiliev's knot invariants, Columbia Univ., preprint 1991. MR
- [Li2]
- X. S. Lin, Finite type link invariants of $3$-manifolds, Columbia Univ., preprint 1992. MR 1259514
- [LM]
- W. B. R. Lickorish and K. Millett, A polynomial invariant for knots and links, Topology \textbf{26} (1987), 107--141. MR 880512
- [Mak]
- G. S. Makanin, On an analogue of the Alexander-Markov theorem, Math. USSR Izv. \textbf{34} (1990), 201--211. MR 992984
- [Mar]
- A. A. Markov, \"Uber die freie \"Aquivalenz geschlossener Z\"opfe, Recueil Math. Moscou \textbf{1} (1935), 73--78. MR
- [Me]
- W. W. Menasco, A proof of the Bennequin-Milnor unknotting conjecture, SUNY at Buffalo, preprint 1992. MR 1273914
- [Mo1]
- H. Morton, Threading knot diagrams, Math. Proc. Cambridge Philos. Soc. \textbf{99} (1986), 247--260. MR 817666
- [Mo2]
- H. Morton, Seifert circles and knot polynomials, Math. Proc. Cambridge Philos. Soc. \textbf{99} (1986), 107--109. MR 809504
- [Mu]
- K. Murasugi, The Jones polynomial and classical conjectures in knot theory, Topology \textbf{26} (1987), 187--194. MR 895570
- [Pi]
- S. Piunikhin, Weights of Feynman diagrams and Vassiliev knot invariants, Moscow State Univ., preprint 1992. MR
- [Pr]
- C. Procesi, The invariant theory of n by n matrices, Adv. in Math. \textbf{19} (1976), 306--381. MR 419491
- [PT]
- J. Przytycki and P. Traczyk, Conway algebras and skein equivalence of links, Proc. Amer. Math. Soc. \textbf{100} (1987), 744--748. MR 894448
- [Re1]
- N. Yu. Reshetikhin, Quasitriangular Hopf algebras and invariants of tangles, Leningrad Math. J. \textbf{1} (1990), 491--513. MR 1025161
- [Re2]
- N. Yu. Reshetikhin, Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links, LOMI preprints \textbf{E-4-87, E-17-87} (1988). MR
- [Rol]
- D. Rolfsen, Knots and links, Publish or Perish, Berkeley, CA, 1976. MR 515288
- [Ros]
- M. Rosso, Groupes quantique et mod\`eles \`a vertex de V. Jones en th\'eorie des noeuds, C. R. Acad. Sci. Paris S\'er. I Math. \textbf{307} (1988), 207--210. MR 956807
- [RT]
- N. Reshetikhin and V. G. Turaev, Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. \textbf{103} (1991), 547--597. MR 1091619
- [Sei]
- H. Seifert, Verschlingungsinvarianten, Sber. Preuss. Akad. Wiss. \textbf{26} (1933), 811--828. MR
- [Sem]
- M. Semenov Tian-Shansky, What is a classical R-matrix\,{\rm ?}, Funct. Anal. Appl. \textbf{17} (1983), 259--270. MR
- [Si]
- J. Simon, How many knots have the same group\/{\rm ?}, Proc. Amer. Math. Soc. \textbf{80} (1980), 162--166. MR 574528
- [St1]
- T. Stanford, Finite type invariants of knots, links and graphs, Columbia Univ., preprint 1992. MR 1404922
- [St2]
- T. Stanford, Braid commutators and Vassiliev invariants, Columbia Univ., preprint 1992. MR 1398378
- [Ta]
- P. G. Tait, \emph{On knots {\rm I, II} and {\rm III }} vol.~1, Cambridge Univ. Press, Cambridge and New York, 1988 pp.~273--347. MR
- [Tr]
- H. Trotter, Non invertible knots exist, Topology \textbf{2} (1964), 275--280. MR 158395
- [Tu]
- V. Turaev, The Yang-Baxter equation and invariants of links, Invent. Math. \textbf{92} (1988), 527--553. MR 939474
- [V1]
- V. A. Vassiliev, \emph{Cohomology of knot spaces}, Amer. Math. Soc., Providence, RI, 1990 pp.~23--69. MR 1089670
- [V2]
- V. A. Vassiliev, \emph{Topology of complements to discriminants and loop spaces} (V. I. Arnold, ed.), Amer. Math. Soc., Providence, RI, 1990 pp.~9--21. MR 1089669
- [Wa]
- M. Wada, Group invariants of links, Topology \textbf{31} (1992), 399--406. MR 1167178
- [WAD]
- M. Wadati, Y. Akutsu, and T. Deguchi, \emph{Link polynomials and exactly solvable models}, Springer-Verlag, Berlin, 1990. MR 1118654
- [We1]
- H. Wenzl, Quantum groups and subfactors of type B, C and D, Comm. Math. Phys. \textbf{133} (1990), 383--432. MR 1090432
- [We2]
- H. Wenzl, Representations of braid groups and the quantum Yang-Baxter equation, Pacific J. Math. \textbf{145} (1990), 153--180. MR 1066402
- [Wi]
- E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. \textbf{121} (1989), 351--399. MR 990772
- [Wh]
- W. Whitten, Knot complements and groups, Topology \textbf{26} (1987), 41--44. MR 880506
- [WX]
- A Weinstein and P. Xu, Classical solutions of the quantum Yang-Baxter equation, Comm. Math. Phys. \textbf{148} (1992), 309--343. MR 1178147
- [Y1]
- S. Yamada, An invariant of spacial graphs, J. Graph Theory \textbf{13} (1989), 537--551. MR 1016274
- [Y2]
- S. Yamada, The minimum number of Seifert circles equals the braid index, Invent. Math. \textbf{89} (1987), 346--356. MR 894383
Similar Articles:
Retrieve articles in Bulletin of the American Mathematical Society
with MSC
(1991):
57M25, 57N99
Retrieve articles in all Journals with MSC
(1991):
57M25, 57N99
Additional Information:
DOI:
10.1090/S0273-0979-1993-00389-6
PII:
S 0273-0979(1993)00389-6
|
