$2$-categories and $2$-knots
John E. Fischer, Jr.
Source: Duke Math. J. Volume 75, Number 2
(1994), 493-526.
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077287619
Mathematical Reviews number (MathSciNet): MR1290200
Zentralblatt MATH identifier: 0847.57008
Digital Object Identifier: doi:10.1215/S0012-7094-94-07514-5
References
[Ba1] R. J. Baxter, Exactly solved models in statistical mechanics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1982.
Mathematical Reviews (MathSciNet): MR86i:82002a
Zentralblatt MATH: 0538.60093
[Ba2] R. J. Baxter, On Zamolodchikov's solution of the tetrahedron equations, Comm. Math. Phys. 88 (1983), no. 2, 185–205.
Mathematical Reviews (MathSciNet): MR84i:81089
Digital Object Identifier: doi:10.1007/BF01209476
Project Euclid: euclid.cmp/1103922280
[Be] J. Benabou, Introduction to bicategories, Reports of the Midwest Category Seminar, Lecture Notes in Math., vol. 47, Springer, Berlin, 1967, pp. 1–77.
Mathematical Reviews (MathSciNet): MR36:3841
[CS1] J. S. Carter and M. Saito, Syzygies among elementary string interactions in $2+1$ dimensions, Lett. Math. Phys. 23 (1991), no. 4, 287–300.
Mathematical Reviews (MathSciNet): MR93e:57053
Zentralblatt MATH: 0744.57010
Digital Object Identifier: doi:10.1007/BF00398826
[CS2] J. S. Carter and M. Saito, Canceling branch points on projections of surfaces in $4$-spaces, preprint.
Mathematical Reviews (MathSciNet): MR1126191
Digital Object Identifier: doi:10.2307/2159319
[D] V. G. Drinfeld, Quantum Groups, Proc. Internat. Cong. Math., Vol 1, Amer. Math. Soc., Providence, 1986, pp. 798–820.
[E] C. Ehresmann, Catégories structurées, Ann. Sci. École Norm. Sup. (3) 80 (1963), 349–426.
Mathematical Reviews (MathSciNet): MR33:5694
Zentralblatt MATH: 0128.02002
[F] R. H. Fox, A quick trip through knot theory, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) ed. M. K. Fort, Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 120–167.
Mathematical Reviews (MathSciNet): MR25:3522
[FM] I. B. Frenkel and G. Moore, Simplex equations and their solutions, Comm. Math. Phys. 138 (1991), no. 2, 259–271.
Mathematical Reviews (MathSciNet): MR92e:17015
Zentralblatt MATH: 0749.17039
Digital Object Identifier: doi:10.1007/BF02099492
Project Euclid: euclid.cmp/1104202944
[FY] P. J. Freyd and D. N. Yetter, Braided compact closed categories with applications to low-dimensional topology, Adv. Math. 77 (1989), no. 2, 156–182.
Mathematical Reviews (MathSciNet): MR91c:57019
Zentralblatt MATH: 0679.57003
Digital Object Identifier: doi:10.1016/0001-8708(89)90018-2
[Gr] J. W. Gray, Formal category theory: adjointness for $2$-categories, Lecture Notes in Mathematics, vol. 391, Springer-Verlag, Berlin, 1974.
Mathematical Reviews (MathSciNet): MR51:8207
Zentralblatt MATH: 0285.18006
[GKZ] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Newton polytopes of the classical resultant and discriminant, Adv. Math. 84 (1990), no. 2, 237–254.
Mathematical Reviews (MathSciNet): MR92a:14060
Zentralblatt MATH: 0721.33002
Digital Object Identifier: doi:10.1016/0001-8708(90)90047-Q
[Ji] M. Jimbo, Introduction to the Yang-Baxter equation, Braid group, knot theory and statistical mechanics eds. C. N. Yang and M. L. Ge, Adv. Ser. Math. Phys., vol. 9, World Sci. Publishing, Teaneck, NJ, 1989, pp. 111–134.
Mathematical Reviews (MathSciNet): MR1062425
Zentralblatt MATH: 0716.17011
[Joh] M. Johnson, The geometry of $n$-categorical pasting, J. Pure Appl. Algebra 62 (1989), no. 3, 211–225.
Mathematical Reviews (MathSciNet): MR91a:18011
Zentralblatt MATH: 0694.18007
Digital Object Identifier: doi:10.1016/0022-4049(89)90136-9
[Jon] V. F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 103–111.
Mathematical Reviews (MathSciNet): MR86e:57006
Zentralblatt MATH: 0564.57006
Digital Object Identifier: doi:10.1090/S0273-0979-1985-15304-2
Project Euclid: euclid.bams/1183552338
[JS] A. Joyal and R. Street, Braided monoidal categories 850081, Macquarie Mathematics, 1986.
[Ka] L. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), no. 2, 417–471.
Mathematical Reviews (MathSciNet): MR90g:57007
Zentralblatt MATH: 0763.57004
Digital Object Identifier: doi:10.2307/2001315
[Ki] R. Kirby, A calculus for framed links in $S\sp3$, Invent. Math. 45 (1978), no. 1, 35–56.
Mathematical Reviews (MathSciNet): MR57:7605
Zentralblatt MATH: 0377.55001
Digital Object Identifier: doi:10.1007/BF01406222
[KM] R. Kirby and P. Melvin, The $3$-manifold invariants of Witten and Reshetikhin-Turaev for $\rm sl(2,\bf C)$, Invent. Math. 105 (1991), no. 3, 473–545.
Mathematical Reviews (MathSciNet): MR92e:57011
Zentralblatt MATH: 0745.57006
Digital Object Identifier: doi:10.1007/BF01232277
[KS] G. M. Kelly and R. Street, Review of the elements of $2$-categories, Category Seminar (Proc. Sem., Sydney, 1972/1973), Springer, Berlin, 1974, 75–103. Lecture Notes in Math., Vol. 420.
Mathematical Reviews (MathSciNet): MR50:10010
Zentralblatt MATH: 0334.18016
[KV] M. M. Kapranov and V. A. Voevodsky, $2$-categories and Zamolodchikov tetrahedra equations, to appear in Proc. Sympos. Pure Math.
Mathematical Reviews (MathSciNet): MR1278735
Zentralblatt MATH: 0809.18006
[M] S. MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1971.
Mathematical Reviews (MathSciNet): MR50:7275
Zentralblatt MATH: 0232.18001
[P] J. Power, A $2$-categorical pasting theorem, J. Algebra 129 (1990), no. 2, 439–445.
Mathematical Reviews (MathSciNet): MR91d:18009
Zentralblatt MATH: 0698.18005
Digital Object Identifier: doi:10.1016/0021-8693(90)90229-H
[Rei] K. Reidemeister, Knottentheorie, Springer-Verlag, Berlin, 1932.
[Res] N. Yu. Reshetikhin, Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links I, LOMI-preprint E-4-87, 1988; II, LOMI-preprint E-17-87, 1988.
[ReT] N. Yu. Reshetikhin and V. G. Turaev, Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547–597.
Mathematical Reviews (MathSciNet): MR92b:57024
Zentralblatt MATH: 0725.57007
Digital Object Identifier: doi:10.1007/BF01239527
[Ro] D. Roseman, Reidemeister-type moves for surfaces in four dimensional space, preprint.
Mathematical Reviews (MathSciNet): MR1634466
Zentralblatt MATH: 0906.57010
[S] M. Saito, Private communication.
[T1] V. G. Turaev, Operator invariants of tangles and $R$-matrices, Math. USSR-Izv. 35 (1990), no. 2, 411–444.
Zentralblatt MATH: 0707.57003
Mathematical Reviews (MathSciNet): MR1024455
[T2] V. G. Turaev, The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988), no. 3, 527–553.
Mathematical Reviews (MathSciNet): MR89e:57003
Zentralblatt MATH: 0648.57003
Digital Object Identifier: doi:10.1007/BF01393746
[Y1] D. N. Yetter, Markov algebras, Braids (Santa Cruz, CA, 1986) eds. J. S. Birman and A. Libgober, Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 705–730.
Mathematical Reviews (MathSciNet): MR90h:57011
Zentralblatt MATH: 0665.57004
[Y2] D. N. Yetter, Quantum groups and representations of monoidal categories, Math. Proc. Cambridge Philos. Soc. 108 (1990), no. 2, 261–290.
Mathematical Reviews (MathSciNet): MR91k:16028
Zentralblatt MATH: 0712.17014
Digital Object Identifier: doi:10.1017/S0305004100069139
[Z1] A. B. Zamolodchikov, Tetrahedra equations and integrable systems in three-dimensional space, Soviet. Phys. JETP 52 (1980), no. 2, 325–336.
Mathematical Reviews (MathSciNet): MR82e:81047
[Z2] A. B. Zamolodchikov, Tetrahedron equations and the relativistic $S$-matrix of straight-strings in $2+1$-dimensions, Comm. Math. Phys. 79 (1981), no. 4, 489–505.
Mathematical Reviews (MathSciNet): MR82k:81081
Digital Object Identifier: doi:10.1007/BF01209309
Project Euclid: euclid.cmp/1103909139
Duke Mathematical Journal
