Links, quantum groups and TQFTS

Bulletin of the American Mathematical Society

Volumn 33, Issue 4, 1996, Pages 413-445

  Sawin, Stephen ^a,b  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b FAIRFIELD UNIVERSITY   (United States)

Author keywords

Cobordisms; Hopf algebras; Jones polynomial; Knots; Link invariants; Quantum groups; Tangles; Three manifolds; Topological quantum field theory

Indexed keywords

EID: 0030489696     PISSN: 02730979     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0273-0979-96-00690-8     Document Type: Article

References (102)

1. H. H. Andersen. Tensor products of quantized tilting modules. Commun. Math. Phys., 149(1):149-159, 1992. MR 94b:17015
2. H. H. Andersen and J. Paradowski. Fusion categories arising from semisimple Lie algebras. Commun. Math. Phys., 169(3):563-588, 1995. MR 96e: 17026
3. S. Axelrod and I. M. Singer. Chern-Simons perturbation theory. In S. Catto and A. Rocha, editors, Proc. XXth DGM Conference, pages 3-45. World Scientific, Teaneck, 1992. MR 94g:58244
4. S. Axelrod and I. M. Singer. Chern-Simons perturbation theory, II. J. Differential Geom., 39(1):173-213, 1994. MR 95b:58163
5. M. F. Atiyah. New invariants of 3- and 4-dimensional manifolds. In R. Wells, editor, The mathematical heritage of Hermann Weyl (Durham, NC, 1987), volume 48 of Proc. Sympos. Pure Math, pages 285-299. Amer. Math. Soc., Providence, 1988. MR 89m:57034
6. M. F. Atiyah. Topological quantum field theories. Publ. Math. IHES, 68:175-186, 1989. MR 90e:57059
7. M. F. Atiyah. The Geometry and Physics of Knots. Lezioni Lincee. Cambridge University Press, 1990. ISBN 0-521-39521-6. MR 92b:57008
8. M. F. Atiyah. On framings of 3-manifolds. Topology, 29(1):1-7, 1990. MR 91g:57025
9. S. Axelrod. Geometric Quantization of Chern-Simons Gauge Theory. PhD thesis, Princeton University, 1991.
10. R. J. Baxter. Exactly Solved Models in Statistical Mechanics. Academic Press, New York, 1982. ISBN 0-12-083180-5. MR 86i:82002h
11. A. A. Belavin and V. G. Drinfel'd. Solutions of the classical Yang-Baxter equation for simple Lie algebras. Funct. Anal. Appl., 16:159-180, 1982. MR 84e:81034
12. C. Blanchet, N. Habegger, G. Massbaum, and P. Vogel. Three-manifold invariants derived from the Kauffman bracket. Topology, 31(4):685-699, 1992. MR 94a:57010
13. C. Blanchet, N. Habegger, G. Massbaum, and P. Vogel. Topological quantum field theories derived from the Kauffman bracket. Topology, 34(4):883-927, 1995. MR 1:362 791
14. J. S. Birman. New points of view in knot theory. Bull. Amer. Math. Soc. (N. S.), 28(2):253-287, 1993. MR 94b:57007
15. J. S. Birman and X.-S. Lin. Knot polynomials and Vassiliev's invariants. Invent. Math., 111(2):225-270, 1993. MR 94d:57010
16. D. Bar-Natan. On the Vassiliev knot invariants. Topology, 34(2):423-472, 1995. MR 1:318 886
17. G. Burde and H. Zieschang. Knots. Walter de Gruyter, New York, 1985. MR 87b:57004
18. J. Cerf. La stratification naturelle des espaces de fonctions différentiables réeles et la théorèm de la pseudoisotopie. Publ. Math. I.H.E.S., 39, 1970. MR 45:11716
19. V. Chari and A. Pressley. A Guide to Quantum Groups. Cambridge University Press, Cambridge, 1994. MR 95j:17010
20. L. Crane. 2-d physics and 3-d topology. Comm. Math. Phys., 135:615-640, 1991. MR 92h:57026
21. V. G. Drinfel'd. On constant quasiclassical solutions of the Yang-Baxter equation. Sov. Math. Dokl., 28:667-671, 1983. MR 85d:58040
22. V. G. Drinfel'd. Hopf algebras and the quantum Yang-Baxter equation. Sov. Math. Dokl., 32:254-258, 1985. MR 87h:58080
23. V. G. Drinfel'd. Quantum groups. In Proceedings of the International Congress of Mathematicians-86, pages 798-820. Amer. Math. Soc., Providence, 1987. MR 89f:17017
24. L. D. Faddeev. Integrable Models m (1 + 1)-Dimensional Quantum Field Theory (Lectures in Les Houches, 1982). Elsevier Science Publishers B.V., 1984. MR 87b:81002
25. R. Fenn and C. Rourke. On Kirby's calculus of framed links. Topology, 18(1):1-15, 1979. MR 80c:57005
26. L. Faddeev and L. Takhtajan. Hamiltonian Methods in the Theory of Solitons. Springer-Verlag, 1987. MR 89m:58103
27. P. Freyd and D. Yetter. Braided compact closed categories with applications to low-dimensional topology. Adv. in Math., 77(2):156-182, 1989. MR 91c:57019
28. P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, and A. Ocneanu. A new polynomial invariant of knots and links. Bull. Amer. Math. Soc., 12(2):239-246, 1985. MR 86e:57007
29. S. Garoufalidis. Applications of TQFT invariants to low-dimensional topology, (to appear in Topology).
30. S. Garoufalidis and J. Levine. On finite type 3-manifold invariants II. preprint, June 1995.
31. G. Hemion. The Classification of Knots and 3-Dimensional Spaces. Oxford University Press, New York, 1992. ISBN 0-19-859697-9. MR 94g:57015
32. J. E. Humphreys. Introduction to Lie Algebras and Representation Theory. Springer-Verlag, New York, 1972. MR 48:2197
33. M. Jimbo. A «j-difference analogue of U(g) and the Yang-Baxter equation. Lett. Math. Phys., 10:63-69, 1985. MR 86k: 17008
34. V. F. R. Jones. Index for subfactors. Invent. Math. 72(1):1-25, 1983. MR 84d:46097
35. V. F. R. Jones. A polynomial invariant for knots via von Neumann algebras. Bull. Amer. Math. Soc., 12(1):103-111, 1985. MR 86e:57006
36. A. Joyal and R. Street. Braided tensor categories. Adv. in Math., 102(1):20-78, 1993. MR 94m:18008
37. F. Jaeger, D.L. Vertigan, and D. J. A. Welsh. On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Cambridge Philos. Soc., 108(1):35-53, 1990. MR 91h:05038
38. C. Kassel. Quantum Groups, volume 155 of GTM. Springer-Verlag, New York, 1994. MR 96e:17041
39. L. H. Kauffman. An invariant of regular isotopy. Announcement-1985, preprint-1986.
40. L. H. Kauffman. On Knots, volume 115 of Annals of Mathematics Studies. Princeton University Press, Princeton, 1987. ISBN 0-691-08434-3. MR 89c:57005
41. L. H. Kauffman. Statistical mechanics and the Jones polynomial. Contemporary Mathematics, 78, 1988. MR 89:57002
42. L. H. Kauffman. Knots and Physics, volume 1 of Series on Knots and Everything. World Scientific, Teaneck, 1991. MR 93b:57010
43. W. T. Thompson (Lord Kelvin). On vortex atoms. Philosophical Magazine, 34:15-24, July 1867.
44. A. A. Kirillov. On inner product in modular tensor categories, I. q-alg/9508017, (to appear in J. Amer. Math. Soc.).
45. 3. Invent. Math., 45(1):35-56, 1978. MR 57:7605
46. L. H. Kauffman and S. L. Lins. Temperley-Lieb Recoupling Theory And Invariants of 3-Manifolds, volume 134 of Annals of Mathematics Studies. Princeton University Press, Princeton, N. J., 1994. ISBM 0-691-0364-3. MR 95c:57027
47. R. Kirby and P. Melvin. The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2,c). Invent. Math., 105(3):473-545, 1991. MR 92e:57011
48. T. Kohno. Topological invariants of three manifolds using representations of mapping class groups i. Topology, 31(2):203-230, 1992. MR 94c:57031
49. M. Kontsevich. Feynman diagrams and low-dimensional topology. In First European Congress of Mathematics, II (Paris, 1992), volume 120 of Prog. Math., pages 97-124. Birkäuser, Basel, 1994. MR 96h:5'7027
50. V. G. Kac and D. Petersen. Infinite dimensional Lie algebras, theta functions and modular forms. Adv. in Math., 53(2):125-264, 1984. MR 86a:17007
51. 2), q-orthogonal polynomials and invariants of links. In V. G. Kac, editor, Infinite Dimensional Lie Algebras and Groups (Luminyy-Marseille, 1988), volume 7 of Adv. Ser. in Math. Phys., pages 285-339. World Scientific, Teaneck, 1989. MR 90m:17022
52. A. N. Kirillov and N. Yu. Reshetikhin. q-Weyl group and a multiplicative formula for universal R-matrices. Comm. Math. Phys., 134(2):421-431, 1990. MR 92c: 17023
53. L. H. Kauffman and D. E. Radfcrd. Invariants of 3-manifolds derived from finite dimensional Hopf algebras. J. Knot Theory Ramifications, 4(1):131-162, 1995. MR 96d:57018
54. G. Kuperberg. Non-involutory Hopf algebras and 3-manifold invariants. (to appear in Duke Math. J.).
55. G. Kuperberg. Involutory Hopf algebras and 3-manifold invariants. Internat. J. Math., 2(1):41-66, 1991. MR 91m:57012
56. 2 link invariant. Internat. J. Math., 5(1):61-85, 1994. MR 95g:57013
57. M. M. Kapranov and V. A. Voevodsky. Braided monoidal 2-categories, 2-vector spaces and Zamalodchikov tetrahedra equations, (unpublished).
58. V. G. Kac and M. Wakimoto. Modular and conformal invariance constraints in representation theory of affine lie algebras. Adv. in Math., 70(2):156-236, 1988. MR 89h:17036
59. W. B. R. Lickorish. A representation of orientable combinatorial 3-manifolds. Ann. Math., 76:531-540, 1962. MR 27:1929
60. W. B. R. Lickorish. Invariants for 3-manifolds from the combinatorics of the Jones polynomial. Pacific. J. Math., 149:337-347, 1991. MR 92d:57007
61. X.-S. Lin. Finite type invariants of integral homology 3-spheres: A survey. q-alg/9510003, (to appear in Proceedings of Warsaw Workshop on Knot Theory).
62. W. B. R. Lickorish and K. C. Millet. The new polynomial invariants of knots and links. Math. Mag., 61(1):3-23, 1988. MR 89d:57006
63. R. G. Larson and D. E. Radford. Semisimple, cosemisimple Hopf algebras. Amer. J. Math., 110(1):187-195, 1988. MR 89a:16011
64. G. Lusztig. Quantum deformations of certain simple modules over enveloping algebras. Adv. in Math., 70(2):237-249, 1988. MR 89k:17029
65. G. Lusztig. Introduction To Quantum Groups. Number 110 in Progress in Mathematics. Birkhäuser, Boston, 1993. ISBN 0-8176-3712-5. MR 94m:17016
66. S. Mac Lane. Categories For the Working Mathematician. Graduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg-Berlin, 1971. MR 50:7275
67. S. Majid. Sequences of bicrossproduct and double crossproduct Hopf algebras. In International Conference on Hopf Algebras, Beer Sheva, Israel. 1989.
68. S. Majid. Quasitriangular Hopf algebras and Yang-Baxter equations. Internat. J. Modern Phys. A. 5:1-91, 1990. MR 90k: 16008
69. H. R. Morton. Invariants of links and 3-manifolds from skein theory and from quantum groups. In Topics in Knot Theory (Erzurum, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Kluwer Acad. Publ., Dordrecht, 1993. MR 1:257 908
70. T. Ohtsuki. A polynomial invariant of integral homology 3-spheres. Math. Proc. Cambridge Philos. Soc., 117(2):259-273. 1995. MR 951:57021
71. T. Ohtsuki. Finite type invariants of integral homology 3-spheres. J. Knot Theory Ramifications, 5(1):101-115, 1996. MR 1:373 813
72. J. Przytycki and P. Traczyk. Conway algebras and skein equivalence of links. Proc. Amer. Math. Soc., 100:744-748, 1987. MR 88m:57012
73. F. Quinn. Lectures on axiomatic topological quantum field theory. In D. Freed and K. Uhlenbeck, editors, Geometry and Quantum Field Theory (Park City, 1991), volume 1 of IAS/Park City Mathematics Series. Amer. Math. Soc., Providence, 1995. MR 96e:57021
74. N. Yu. Reshetikhin. Quantized universal enveloping algebras, the Yang-Baxter equation, and invariants of links, I and II. LOMI Preprints, E-4-87.E-17-87, 1988.
75. J. Roberts. The Kirby calculus for manifolds with boundary preprint.
76. J. Roberts. Skein theory and Turaev-Viro invariants. Topology, 34(4):771-787, 1995. MR 1:362 787
77. D. Rolfsen. Knots and Links. Publish or Perish Inc., 1976. ISBN 0-914098-16-0. MR 58:2423«
78. M. Rosso. Algébres enveloppantes quantifiées, groupes quantiques compacts de matrices et calcul differentiel non commutatif. Duke Math. J., 61:11-40, 1990. MR 921:17022
79. M. Rosso. Analogues de la forme de Killing et du théorèm d'Harish-Chandra pour les groupes quant iques. Ann. Sec. École Norm. Sup., 23(3):445-467, 1990. MR93e:17026
80. L. Rozanski. The trivial connection contribution to Witten's invariants of rational homology 3-spheres. preprint, 1995.
81. N. Yu Reshetikhin and V. G. Turaev. Ribbon graphs and their invariants derived from quantum groups. Comm. Math. Phys., 127:1-26, 1990. MR 91c:57016
82. N. Yu Reshetikhin and V. G. Turaev. Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math., 103:547-597, 1991. MR 92b:57024
83. S. Sawin. Links, tangles and three manifolds: An invitation to quantum topology, lecture notes.
84. S. Shnider and S. Sternberg. Quantum groups - from coalgebras to Drinfel'd algebra. International Press, Cambridge MA, 1994. MR 95e:17022
85. M. Sweedler. Hopf Algebras. W. A. Benjamin, Inc., New York, 1969. MR 40:5705
86. 1 and monodromy representations of braid groups. In A. Tsuchiyah, T. Eguchi, and M. Jimbo, editors, Conformal Field Theory and Solvable Lattice Models, (Kyoto, 1986), volume 16 of Adv. Studies in Pure Math., pages 297-372. Academic Press. Boston, 1988. MR 91h:81156
87. B. Trace. On the Reidemeister moves of a classical knot. Proc. Amer. Math. Soc., 89:722-724, 1983. MR 85f:57005
88. V. G. Turaev. The Yang-Baxter equation and invariants of links. Invent. Math., 92:527-553, 1988. MR 89e:57003
89. V. G. Turaev. Operator invariants of tangles and R-matrices. Izv. Akad. Nauk.. SSSR, 53(5), 1989. Eng. Trans.: Math. USSR-Izv. 35 (1990) 411-444. MR 91e:17011
90. V. G. Turaev. Operator invariants of tangles and R-matrices. Izv. Akad. Nauk.. SSSR, 53(5), 1989. Eng. Trans.: Math. USSR-Izv. 35 (1990) 411-444. MR 91e:17011
91. V. G. Turaev. Modular categories and 3-manifold invariants. Internat. J. Modern Phys. B, 6(11-12):1807-1824, 1992 MR 93k:57040
92. V. G. Turaev. Quantum Invariants of Knots and 3-Manifolds. Studies in Mathematics. Walter de Gruyter and Co., Berlin 1994. ISBN 3-11-013704-6. MR 95k:57014
93. V. Turaev and O. Viro. State sum invariants of 3-manifolds and quantum 6-j symbols. Topology, 31(4):865-902, 1992. MF: 94d:57044
94. V. Turaev and H. Wenzl. Quantum invariants of 3-manifolds associated with classical simple Lie algebras. Internat. J. Math., 4(2):323-358, 1993. MR 941:57019
95. V. A. Vassiliev. Cohomology of knot spaces. In V. I. Arnold, editor, Theory of Singularities and its Applications, volume 1 of Adv. Soviet Math., pages 23-69. Amer. Math. Soc., Providence, 1990. MR 92a:57016
96. V. A. Vassiliev. Complements of Discriminants of Smooth Maps: Topology and Applications, volume 98 of Translations of Math. Monographs. Amer. Math. Soc., Providence, 1992. MR 941:57020
97. E. Verlinde. Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B, 300:360-375, 1988. MR 89h:81238
98. K. Walker. On Witten's 3-manifold invariants, (preprint).
99. H. Wenzl. Braids and invariants of 3-manifolds. Invent. Math., 114(2):235-273. 1993. MR 941:57021
100. E. Witten. Quantum field theory and the Jones polynomial. Comm. Math. Phys, 121(3):351-399, 1989. MR 90h:57009
101. E. Witten. Gauge theories and integrable lattice models. Nuclear Phys. B, 322(3):G29-697, 1989. MR 901:81096
102. D. N. Yetter. Markov algebras. In Braids (Santa Cruz, 1986), volume 78 of Contemp. Math., pages 705-730. Amer. Math. Soc., 1988. MR 90h:57011