Abstract
It is a rather involved task to describe the relations between isomorphisms between various functors which use coends. To make them accessible we introduce the monoidal bicategory of thick tangles as a kind of a free monoidal bicategory with duals generated by a self-dual object. An analogy would be to introduce the category of unoriented tangles as the free braided category, generated by a self-dual object. Naturally, the bicategory of thick tangles is much simpler than its braided counterpart— the category of 2-tangles proposed by Baez and Langford [BL98a]. They prove in [BL98b] that the category of 2-tangles is a free semistrict braided monoidal 2-category with duals on one unframed self-dual object. Very similar geometric 2-categories appear in the theory of knotted surfaces in 4-dim space as developed by Carter, Saito, Fischer, and others, see for example [CRS97], [CS98], and [Fi94].
Keywords
- Planar Graph
- Monoidal Category
- Abelian Category
- Isotopy Class
- Identity Morphism
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Bibliography
Abrams L., Two-dimensional topological quantum field theories and Frobenius algebras, J. Knot Theory Ramifications 5 (1996), no. 5, 569–587.
Alexander J., On the deformation of an n-cell, Proc. Nat. Acad. Sci. 9 (1923), 406–407.
Andersen H. H., Tensor products of quantized tilting modules, Commun. Math. Phys. 149 (1992), no. 1, 149–159.
Atiyah M., Patodi V, Singer I. M., Spectral asymmetry andRiemannian geometry. I, Math. Proc. Camb. Phil. Soc. 77 (1975), 43–69.
Atiyah M., Patodi V., Singer I. M., Spectral asymmetry and Riemannian geometry. II, Math. Proc. Camb. Phil. Soc. 78 (1975), no. 3, 405–432.
Atiyah M., Topological quantum field theories, List. Hautes Etudes Sci. Publ. Math. 68 (1988), 175–186.
Atiyah M., On framings ofZ-manifolds, Topology 29 (1990), 1–7.
Baez J. C, Higher-dimensional algebra II: 2-Hilbert spaces, Adv. Math. 127 (1997), 125–189.
Baez J. C, Dolan J., Higher-dimensional algebra and topological quantum field theory, Jour. Math. Phys. 36 (1995), 6073–6105.
Beliakova A., Spin topological quantum field theories, Internat. J. Math. 9 (1998), no. 2, 129–152.
Bénabou J., Introduction to bicategories, Lecture Notes in Math., vol. 47, Springer, New York, 1967, pp. 1–77.
Blanchet C, Habegger N., Masbaum G., Vogel P., Topological quantum field theories derived from the Kauffman bracket, Topology 34 (1995), no. 4, 883–927.
Birman J., On braid groups, Commun. Pure and Appl. Math. 22 (1969), 41–72.
Birman J., Mapping class groups and their relations to braid groups, Commun. Pure and Appl. Math. 22 (1969), 213–238.
Birman J., Braids, links and mapping class groups, Ann. of Math. Studies, vol. 82, Princeton Univ. Press, 1974.
Bröcker T., Jänich K., Einfuhrung in die Dijferentialtopologie, Heidelberger Taschenbücher, vol. 143, Springer-Verlag, 1973.
Bakalov B., Kirillov A., Jr., Lectures on tensor categories and modular functors, University Lecture Series, vol. 21, Amer. Math. Soc, Providence, RI, 2001.
Bespalov Yu. N., Kerler T., Lyubashenko V. V., Turaev V. G., Integrals for braided Hopf algebras, J. Pure and Appl. Algebra 148 (2000), no. 2, 113–164.
Baez J. C, Langford L., 2-tangles, Lett. Math. Phys. 43 (1998), no. 2, 187–197.
Baez J. C, Langford L., Higher-dimensional algebra IV: 2-tangles, math.QA/9811139, 1998.
Baez J. C, Neuchl M., Higher-dimensional algebra I: Braided monoidal 2-categories, Adv. Math. 121 (1996), no. 2, 196–244.
Breen L., On the classification of2-gerbes and 2-stacks, Asterisque, Soc. Math. de France 225 (1994), 1–160.
Birman J. S., Wenzl H., Braids, link polynomials and a new algebra, Trans. Amer. Phil. Soc. 313 (1989), no. 1, 249–273.
Barrett J. W., Westbury B. W., The equality ofZ-manifold invariants, Math. Proc. Camb. Phil. Soc. 118 (1995), 503–510.
Cerf J., La stratification naturelle de espace desfonctions differentiates reelles et le theoreme de la pseudoisotopie, Publ. Math. I.H.É.S. 39 (1970).
Crane L., Frenkel I. B., Four dimensional topological quantum field theory, Hopf categories, and the canonical bases, J. Math. Phys. 35 (1994), no. 10, 5136–5154.
Crane L., Kauffman L. H., Yetter D. N., State-sum invariants of A-manifolds, J. Knot Theory Ramifications 6 (1997), 177–234.
Carter J. S., Rieger J. H., Saito M., A combinatorial description of knotted surfaces and their isotopies, Adv. Math. 127 (1997), no. 1, 1–51.
Carter J. S., Saito M., Knotted surfaces and their diagrams, Math. Surveys and Monographs, vol. 55, Amer. Math. Soc, Providence, RI, 1998, xii+258 pp.
Crane L., Yetter D. N., A categorical construction of AD topological quantum field theories, Quantum topology, Ser. Knots Everything, no. 3, World Sci. Publishing, River Edge, NJ, 1993, pp. 120–130.
Deligne P., Catégories tannakiennes, The Grothendieck Festschrift, Progress in Mathematics, no. 87, Birkhauser, Boston, Basel, Berlin, 1991, pp. 111–195.
Durhuus B., Jonsson T, Classification and construction of unitary topological quantum field theories in two dimensions, J. Math. Phys. 35 (1994), 5306–5313.
Deligne P., Milne J. S., Tannakian categories, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math., no. 900, Springer-Verlag, Berlin, Heidelberg, New York, 1982, pp. 101–228.
Donaldson S. K., Topological field theories and formulae ofCasson and Meng-Taubes, Proceedings of the Kirbyfest (Joel Hass and Martin Scharlemann, eds.), Geometry and Topology Monographs, no. 2, 1999, pp. 87–102.
Dijkgraaf R., Pasquier V, Roche P., Quasi-quantum groups related to orbifold models, Nucl. Phys. B. Proc. Suppl. 18 (1990), 60–72.
Drinfeld V. G., Quantum Groups, Proceedings of the International Congress of Mathematicians (Berkeley 1986), Vol. 1 (Providence, RI) (A. Gleason, ed.), Amer. Math. Soc, 1987, pp. 798–820.
Drinfeld V. G., On almost cocommutative Hopf algebras, Leningrad Math. J. 1 (1990), no. 2, 321–342.
Day B. J., Street R. H., Monoidal bicategories and Hopf algebroids, Adv. Math. 129 (1997), no. 1, 99–157.
Dijkgraaf R., Witten E., Topological gauge theories and group cohomology, Commun. Math. Phys. 129 (1990), 393–429.
Ehresmann C, Categories doubles et categories structurees, C. R. Acad. Sc. 256 (1963), 1198–1201.
Ehresmann C, Categories doubles des quintettes, applications covariantes, C. R. Acad. Sc. 256 (1963), 1891–1894.
Evans D. E., Kawahigashi Y, From subfactors to 3-dimensional topological quantum field theories and back. A detailed account of Ocneanu’s theory., Interaat. J. Math. 6 (1995), 537–558.
Epstein D. B. A., Curves on 2-manifolds and isotopies, Acta Math. 115 (1966), 83–107.
Eilenberg S., Steenrod N. E., Foundations of algebraic topology, Princeton Mathematical Series, vol. 15, Princeton University Press, 1952.
Fischer J. E., 2-categories and 2-knots, Duke Math. J. 75 (1994), no. 2, 493–526.
Frohlich J., Kerler T, Quantum Groups, Quantum Categories and Quantum Field Theory, Lect. Notes Math., vol. 1542, Springer, Berlin, 1993.
Frohman C, Kania-Bartoszyńska J., SO(3)-topological quantum field theory, Comm. Anal. Geom. 4 (1996), no. 4, 589–679.
Fadell E., Neuwirth L., Configuration spaces, Math. Scand. 10 (1962), 111–118.
Frohman C, Nicas A., The Alexander polynomial via topological quantum field theory, Differential geometry, global analysis, and topology (Halifax, NS, 1990) (Providence, RI), CMS Conf. Proc, no. 12, Amer. Math. Soc, 1991, pp. 27–40.
Frohman C, Nicas A., An intersection homology invariant for knots in a rational homology 3-sphere, Topology 33 (1994), no. 1, 123–158.
Freedman M., Quinn R, Topology of4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, 1990.
Freed D. S., Quinn F., Chern-Simons theory with finite gauge group, Commun. Math. Phys. 156 (1993), no. 3, 435–472.
Fenn R., Rourke C, On Kirby’s calculus of links, Topology 18 (1979), 1–15.
Freyd P. J., Abelian categories, Harper and Row, New York, 1964.
Freed D. S., Higher algebraic structures and quantization, Commun. Math. Phys. 159 (1994), 343–398.
Fukaya K., Floer homology for 3 manifold with boundary, I, http://www.kusm.kyoto-u.ac.jp/~fukaya/bdrtl.pdf, March 1999.
Fadell E., Van Buskirk J., The braid groups of E 2 and S 2, Duke Math. J. 29 (1962), 243–257.
Freyd P. J., Yetter D. N., Coherence theorem via knot theory, J. Pure and Appl. Algebra 78 (1992), 49–76.
Gelca R., SL(2, C)-topological quantum field theory with corners, J. Knot Theory Ramifications 7 (1998), no. 7, 893–906.
Gelfand S. I., Kazhdan D., Invariants of three-dimensional manifolds, Geom. Funct. Anal. 6 (1996), 268–300.
Garoufalidis S., Levine J., Finite type 3-manifold invariants and the structure of the Torelligroup. I, Invent. Math. 131 (1998), 541–594.
Gordon R., Power A. J., Street R. H., Coherence for tricategories, vol. 117, Memoirs Amer. Math. Soc., no. 558, Amer. Math. Soc, Providence, Rhode Island, September 1995.
Gompf R. E., Stipsicz A. I., 4-manifolds and Kirby calculus, Graduate Studies in Mathematics, vol. 20, Amer. Math. Soc, Providence, Rhode Island, 1999, xvi+558 pp.
Hempel J., 3-manifolds, Ann. of Math. Studies, vol. 86, Princeton University Press, 1976.
Hennings M., Invariants of links and 3-manifolds obtained from Hopf algebras, J. London Math. Soc. (2) 54 (1996), no. 3, 594–624.
Hirsch M. W., Differential topology, GTM, vol. 33, Springer, Berlin, 1976.
Hatcher A., Wagoner J., Pseudo-isotopies of compact manifolds, Astérisque, vol. 6, Soc. Math, de France, 1973.
Jones V. F. R., Hecke algebra representations of braid groups and link polynomials, Ann. Math. 126 (1987), no. 2, 335–388.
Joyal A., Street R. H., Tortile Yang-Baxter operators in tensor categories, J. Pure Appl. Algebra 71 (1991), 43–51.
Karoubi M., K-théorie, Les Presses de l’Université de Montréal, 1971.
Kelly G. M., Basic concepts of enriched category theory, London Math. Soc. Lecture Notes, vol. 64, Cambridge Univ. Press, Cambridge, 1982.
Kerler T, Non-Tannakian categories in quantum field theory, New Symmetry Principles in Quantum Field Theory (Cargeése, 1991). NATO Adv. Sci. Inst. Ser. B Phys., vol. 295, Plenum Press, New York, 1992, pp. 449–482.
Kerler T., Mapping class group actions on quantum doubles, Commun. Math. Phys. 168 (1994), 353–388.
Kerler T., Genealogy of nonperturbative quantum-invariants of 3-manifolds: The surgical family, Geometry and Physics, (Aarhus, 1995) (New York), Lecture Notes in Pure and Appl. Math., no. 184, Marcel Dekker, 1997, pp. 503-547.
Kerler T., Equivalence of a bridged link calculus and Kirby’s calculus of links on non-simply connected 3-manifolds, Topology Appl. 87 (1998), 155–162.
Kerler T., On the connectivity of cobordisms and half-projective TQFT’s, Commun. Math. Phys. 198 (1998), no. 3, 535–590.
Kerler T., Bridged links and tangle presentations ofcobordism categories, Adv. Math. 141 (1999), 207–281.
Kerler T., Homology TQFT via Hopf algebras, math.GT/0008204, 2000.
Kirby R. C, A calculus for framed links in S 3, Invent. Math. 65 (1978), 35–56.
Kirby R. C, The topology of 4-manifolds, Lect. Notes in Math., vol. 1374, Springer-Verlag, Berlin, 1989.
Kirby R. C, Melvin P., The 3-manifold invariants of Witten and Reshetikhin-Turaevfor sl(2, C), Invent. Math. 105 (1991), 473–545.
Kneser H., Die Deformationssdtze der einfach zusammenhfigenden Fldchen, Math. Zeit. 25 (1926), 362–372.
Kontsevich M., Rational conformal field theory and invariants of 3-dimensional manifolds, Marseille preprint CPT-99/P2189.
Kauffman L. H., Radford D. E., A necessary and sufficient condition for a finite-dimensional Drinfel’’d double to be a ribbon Hopf algebra, J. Algebra 159 (1993), no. 1, 98–114.
Kauffman L. H., Radford D. E., Invariants of 3-manifolds derived from finite dimensional Hopf algebras, J. Knot Theory Ramifications 4 (1995), no. 1, 131-1997.
Kassel C, Rosso M., Turaev V, Quantum groups and knot invariants, Panoramas et Syntheses de la S.M.F., vol. 5, Soc. Math, de France, Paris, 1997.
Kelly G. M., Street R. H., Review of the elements of2-categories, Category Seminar (Proc. Sem., Sydney, 1972/1973), Lecture Notes in Mathematics, vol. 420, Springer-Verlag, 1974, pp. 75–103.
Karowski M., Schrader R., State sum invariants of three-manifolds: a combinatorial approach to topological quantum field theories, J. Geom. Phys. 11 (1993), 181’190.
Kauffman L. H., Saito M., Sullivan M. C, Quantum invariants of templates, available at http://www.math.usf.edu/~saito/preprints.html, 1997.
Kuperberg G., Involutory Hopf algebras and three-manifold invariants, Internat. J. Math. 2 (1991), 41–66.
Kuperberg G., Non-involutory Hopf algebras and 3-manifold invariants, Duke Math. J. 84 (1996), no. 1, 83–129.
Kapranov M. M., Voevodsky V. A., 2-categories and Zamolodchikov tetrahedra equations, Algebraic groups and their generalizations: quantum and infinite-dimensional methods, Proc. Symp. Pure Math., Vol. 56, Part 2 (Providence, RI) (William J. Haboush et al., eds.), Summer Research Institute on algebraic groups and their generalizations, July 6–26, 1991, Pennsylvania State University, University Park, PA, USA., Amer. Math. Soc, 1994, pp. 177-
Lawrence R. J., Triangulation, categories and extended topological field theories, Quantum Topology (R. A. Baadhio and L. H. Kauffman, eds.), Ser. Knots Everything, no. 3, World Sci. Publishing, River Edge, NJ, Singapore, 1993, pp. 191–208.
Lickorish W. B. R., A representation of orientable 3-manifolds, Ann. Math. 76 (1962), 531–540.
Lyubashenko V. V., Majid S., Braided groups and quantum Fourier transform, J. Algebra 166 (1994), 506–528.
Le T. Q. T., Murakami H., Murakami J., Ohtsuki T., A three-manifold invariant derived from the universal Vassiliev-Kontsevich invariant, Proc. Japan Acad., Ser. A 71, (1995), 125–127.
Larson R. G., Sweedier M. E., An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969), no. 1, 75–94.
Lyubashenko V. V, Tangles and Hopf algebras in braided categories, J. Pure and Applied Algebra 98 (1995), no. 3, 245–278.
Lyubashenko V. V., Modular transformations for tensor categories, J. Pure and Applied Algebra 98 (1995), no. 3, 279–327.
Lyubashenko V. V, Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity, Commun. Math. Phys.172 (1995), no. 3, 467–516.
Lyubashenko V. V, Ribbon abelian categories as modular categories, J. Knot Theory Ramifications 5 (1996), no. 3, 311–403.
Lyubashenko V. V., Squared Hopf algebras, Mem. Amer. Math. Soc. 142 (1999), no. 677, 184 pp.
Mac Lane S., Categories for the working mathematician, GTM, vol. 5, Springer Verlag, New York, 1971, 1988.
Majid S., Braided groups, J. Pure Appl. Algebra 86 (1993), no. 2, 187–221.
Masbaum G., Introduction to spin TQFT, Geometric topology (Athens, GA, 1993) (Providence, RI), AMS/IP Stud. Adv. Math., no. 2.1, Amer. Math. Soc, 1997, pp. 203–216.
Milnor J. W., Lectures on the h-cobordism theorem, Princeton Mathematical Notes, Princeton Univ. Press, Princeton, 1965.
Milnor J. W., Morse theory, Annals of Mathematical Studies, vol. 51, Princeton Univ. Press, Princeton, 1969.
Miller E. Y, The homology of the mapping class group, J. Diff. Geom. 24 (1986), 1–14.
Melvin P., Kazez W., Z-dimensional bordism, Michigan Math. J. 36 (1989), no. 2, 251–260.
Moise E. E., Affine structures in 3-manifolds II. positional properties ofi-spheres, Ann. of Math. (2) 55 (1952), 172–176.
Matveev S., Polyak M., A geometrical presentation of the surface mapping class group and surgery, Commun. Math. Phys. 160 (1994), no. 3, 537–550.
Moore G., Seiberg N., Classical and quantum conformal field theory, Commun. Math. Phys. 123 (1989), 177–254.
Murakami J., The Kauffinan polynomial of links and representation theory, Osaka J. Math. 24 (1987), no. 4, 745–758.
Murakami H., Quantum SU(2)-invariants dominate Casson’s SU(2)-invariant, Math. Proc. Camb. Phil. Soc. 115 (1994), 83–103.
Nikshych D., Vainerman L., A characterization of depth 2 subfactors of III factors, J. Funct. Anal. 171 (2000), no. 2, 278–307.
Ohtsuki T., Finite type invariants of integral homology 3-spheres, J. Knot Theory Ramifications 5 (1996), 101–115.
Ponzano G., Regge T., Semiclassical limit ofRacah coefficients, Spectroscopic and Group Theoretical Methods in Physics, North-Holland, Amsterdam, 1968, pp. 1–58.
Quinn R, Lectures on axiomatic topological quantum field theory, Geometry and Quantum Field Theory (Park City, UT, 1991), IAS/Park City Math. Ser., no. 1, Amer. Math. Soc., Providence, RI, 1995, pp. 323–453.
Radford D. E., The order ofantipode of a finite-dimensional Hopf algebra is finite, Amer. J. Math 98 (1976), 333–335.
Reshetikhin N. Yu., Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links, I and II, Preprints, LOMI, E-4-87, E-17-87, Leningrad, 1988.
Reshetikhin N. Yu., Quasitriangular Hopf algebras and invariants of tangles, Leningrad Math. J. 1 (1990), no. 2, 491–513, Russian: 1 (1989), no. 2,169–188.
Roberts J., Skein theory and Turaev-Viro invariants, Topology 34 (1995), no. 4, 771–787.
Roberts J., Kirby calculus in manifolds with boundary, Turkish J. Math. 21 (1997), no. 1, 111–117.
Rohlin V. A., A three dimensional manifold is the boundary of a four dimensional one, Dokl.Akad. Nauk. SSSR (N.S.) 114 (1951), 355–357.
Rolfsen D., Knots and links, Mathematics Lecture Series, vol. 7, Publish or Perish, 1976, 439 p.
Reshetikhin N. Yu., Turaev V. G., Ribbon graphs and their invariants derived from quantum groups, Commun. Math. Phys. 127 (1990), no. 1, 1–26.
Reshetikhin N. Yu., Turaev V. G., Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547–597.
Saavedra Rivano N., Categories Tannakiennes, Lecture Notes in Math., vol. 265, Springer, Berlin, Heidelberg, New York, 1972, 420p.
Sawin S., Three-dimensional 2-framed TQFTs and surgery, math.QA/9912065, 1999.
Schauenburg P., Tannaka duality for arbitrary Hopf algebras, Algebra Berichte, vol. 66, Verlag Reinhard Fischer, Miinchen, 1992.
Segal G. B., The definition of conformal field theory, Differential geometrical methods in theoretical physics (Como, 1987) (Dordrecht), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., no. 250, Kluwer Acad. Publ., 1988, pp. 165–171.
Segal G. B., Lecture Notes (unpublished) 1998.
Shum M. C, Tortile tensor categories, J. Pure Appl. Algebra 93 (1994), no. 1, 57–110.
Smale S., Diffeomorphisms of the 2-sphere, Proc. Amer. Math. Soc. 10 (1959), 621–626.
Sweedler M. E., Hopf algebras, W. A. Benjamin, New York, 1969.
Sweedler M. E., Integrals for Hopf algebras, Ann. of Math. 89 (1969), no. 2, 323–335.
Thorn R., Quelques proprietes globales des varietes differentiables, Comment. Math. Helv. 28 (1954), 17–86.
Tillman U., On the homotopy of the stable mapping class group, Oxford Preprint, 1995.
Turaev V. G., The Yang-Baxter equation and invariants for links, Invent. Math. 92 (1988), 527–553.
Turaev V. G., Quantum invariants of knots and 3-manifolds, de Gruyter Stud. Math., vol. 18, Walter de Gruyter & Co., Berlin, New York, 1994, 588 pp.
Turaev V. G., Viro O., State sum invariants of 3-manifolds and quantum 6-j-symbols, Topology 31 (1992), 865–902.
Turaev V. G., Wenzl H., Quantum invariants of 3-manifolds associated with classical simple Lie algebras, Internat. J. Math. 4 (1993), no. 2, 323–358.
Walker K., On Witten’s 3-manifold invariants, UCSD preprint, 1991.
Wallace A. H., Modifications and cobounding manifolds, Cam. J. Math. 12 (1960), 503–528.
Waldhausen F., On irreducible 3-manifolds which are sufficiently large, Ann. of Math. 87 (1968), 57–88.
Wall C. T. C, Non-additivity of signature, Invent. Math. 7 (1969), 269–274.
Witten E., Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989), no. 3, 351–399.
Xu F., 3-manifold invariants from cosets, Preprint available as math.GT/9907077.
Yetter D. N., Coalgebras, comodules, coends and reconstruction, Preprint.
Yetter D. N., Portrait of the handle as a Hopf algebra, Geometry and Physics, (Aarhus, 1995) (New York), Lecture Notes in Pure and Appl. Math., no. 184, Marcel Dekker, 1997, pp. 481–502.
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
(2001). Thick tangles. In: Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners. Lecture Notes in Mathematics, vol 1765. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44625-7_11
Download citation
DOI: https://doi.org/10.1007/3-540-44625-7_11
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42416-1
Online ISBN: 978-3-540-44625-5
eBook Packages: Springer Book Archive
