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Quandle cohomology and state-sum invariants of knotted curves and surfaces

Authors: J. Scott Carter, Daniel Jelsovsky, Seiichi Kamada, Laurel Langford and Masahico Saito
Journal: Trans. Amer. Math. Soc. 355 (2003), 3947-3989
MSC (2000): Primary 57M25, 57Q45; Secondary 55N99, 18G99
Published electronically: June 24, 2003
MathSciNet review: 1990571
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Abstract: The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. A proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed. Our proof is based on a state-sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles (also known as distributive groupoids).

A quandle is a set with a binary operation -- the axioms of which model the Reidemeister moves in classical knot theory. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define state-sum invariants for knotted circles in $3$-space and knotted surfaces in $4$-space.

Cohomology groups of various quandles are computed herein and applied to the study of the state-sum invariants. Non-triviality of the invariants is proved for a variety of knots and links, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles.

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  • 1. Baez, J. C.; Langford, L., $2$-tangles, Lett. Math. Phys. 43 (1998), no. 2, 187-197. MR 2000c:57054
  • 2. Baez, J.; Langford, L., Higher-dimensional algebra IV: 2-Tangles, to appear in Adv. Math, preprint available at
  • 3. Brieskorn, E., Automorphic sets and singularities, Contemporary math., 78 (1988), 45-115. MR 90a:32024
  • 4. Carter, J.S., Jelsovsky, D., Kamada, S., and Saito, M., Quandle Homology Groups, Their Betti Numbers, and Virtual Knots, Journal of Pure and Applied Algebra, 157 (2001), 135-155.
  • 5. Carter, J.S.; Kauffman, L.H.; Saito, M., Structures and diagrammatics of four dimensional topological lattice field theories, Advances in Math. 146 (1999), 39-100. MR 2000j:57064
  • 6. Carter, J.S.; Rieger, J.H.; Saito, M., A combinatorial description of knotted surfaces and their isotopies, Advances in Mathematics, 127, No. 1, April 15 (1997), 1-51. MR 98c:57023
  • 7. Carter, J.S.; Saito, M., Knotted surfaces and their diagrams, the American Mathematical Society, 1998. MR 98m:57027
  • 8. Carter, J.S.; Saito, M., Canceling branch points on the projections of surfaces in 4-space, Proc. AMS 116, 1, (1992) 229-237. MR 93i:57029
  • 9. Dijkgraaf, R., Witten, E., Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990), 393-429. MR 91g:81133
  • 10. Farber, M.S., Linking coefficients and two-dimensional knots, Soviet. Math. Doklady 16 (1975), 647-650. MR 53:4081
  • 11. Farber, M.S., Duality in an infinite cyclic covering and even-dimensional knots, Math. USSR-Izv. 11 (1977), 749-781. MR 58:24279
  • 12. Fenn, R.; Rourke, C., Racks and links in codimension two. Journal of Knot Theory and Its Ramifications Vol. 1 No. 4 (1992), 343-406. MR 94e:57006
  • 13. Fenn, R.; Rourke, C.; Sanderson, B., Trunks and classifying spaces, Appl. Categ. Structures 3 (1995), no. 4, 321-356. MR 96i:57023
  • 14. Fenn, R.; Rourke, C.; Sanderson, B., James bundles and applications, preprint available at$\sim$cpr/ftp/
  • 15. Fox, R.H., A quick trip through knot theory, in Topology of $3$-Manifolds, Ed. M.K. Fort Jr., Prentice-Hall (1962) 120-167. MR 25:3522
  • 16. Fukuma, M., Hosono, S., and Kawai, H., Lattice topological field theory in two dimensions, Comm. Math. Phys., 161 (1994), 151-175. MR 95b:81179
  • 17. Giller, C., Towards a classical knot theory for surfaces in $\mathbf{R}^4$, Illinois Journal of Mathematics 26, No. 4, (Winter 1982), 591-631. MR 84c:57011
  • 18. Greene, M. T. Some Results in Geometric Topology and Geometry, Ph.D. Dissertation, Warwick (1997).
  • 19. Jones, V.F.R., Hecke algebra representations of braid groups and link polynomials, Ann. of Math., 126 (1989), 335-388. MR 89c:46092
  • 20. Joyce, D., A classifying invariant of knots, the knot quandle, J. Pure Appl. Alg., 23, 37-65. MR 83m:57007
  • 21. Hartley, R., Identifying non-invertible knots, Topology, 22 (1983), 137-145. MR 85c:57003
  • 22. Hillman, J.A., Finite knot modules and the factorization of certain simple knots, Math. Ann. 257 (1981), no. 2, 261-274. MR 83c:57009
  • 23. Kamada, S., Surfaces in $\mathbf{R}^4$ of braid index three are ribbon, Journal of Knot Theory and its Ramifications 1 (1992), 137-160. MR 93h:57039
  • 24. Kamada, S., A characterization of groups of closed orientable surfaces in 4-space, Topology 33 (1994), 113-122. MR 95a:57002
  • 25. Kamada, S., $2$-dimensional braids and chart descriptions, ``Topics in Knot Theory (Erzurum, 1992),'' 277-287, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 399, Kluwer Acad. Publ., (Dordrecht, 1993).
  • 26. Kapranov, M.; Voevodsky, V., 2-Categories and Zamolodchikov tetrahedra equations. Proc. Symp. Pure Math., 56 (1994), Part 2, 177-259. MR 95f:18011
  • 27. L. H. Kauffman, Knots and Physics, World Scientific, Series on knots and everything, vol. 1, 1991. MR 93b:57010
  • 28. Kawauchi, A., The invertibility problem on amphicheiral excellent knots, Proc. Japan Acad., Ser.A, Math. Sci. (1979), 55, 399-402. MR 81b:57003
  • 29. Kawauchi, A., Three dualities on the integral homology of infinite cyclic coverings of manifolds, Osaka J. Math. 23 (1986), 633-651. MR 88e:57021
  • 30. Kawauchi, A., The first Alexander modules of surfaces in 4-sphere, ``Algebra and Topology (Taejon, 1990),'' 81-89, Proc. KAIST Math. Workshop, 5, KAIST, Taejon, Korea, 1990. MR 92b:57032
  • 31. Kawauchi, A., A survey of knot theory, Birkhauser, 1996. MR 97k:57011
  • 32. Koschorke, U., A generalization of Milnor's $\mu$-invariants to higher-dimensional link maps, Topology 36, 2 (1997), 301-324. MR 2000a:57063
  • 33. Langford, L., $2$-tangles as a free braided monoidal $2$-category with duals. Ph.D. dissertation, U.C. Riverside, 1997.
  • 34. Levine, J., Knot modules I, Trans. Amer. Math. Soc. 229 (1977), 1-50. MR 57:1503
  • 35. Matveev, S., Distributive groupoids in knot theory, (Russian) Mat. Sb. (N.S.) 119(161) (1982), no. 1, 78-88, 160. MR 84e:57008
  • 36. Murasugi, K., Knot theory and its applications, Translated from the 1993 Japanese original by Bohdan Kurpita. Birkheuser Boston, Inc., Boston, MA, 1996. MR 97g:57011
  • 37. Neuchl, M., Representation Theory of Hopf Categories, to appear in Adv. in Math. under the title Higher-dimensional algebra VI: Hopf categories, available at$\sim$neuchl.
  • 38. Rolfsen, D., Knots and Links. Publish or Perish Press, (Berkley 1976). MR 58:24236
  • 39. Rourke, C., and Sanderson, B.J., There are two 2-twist-spun trefoils, Preprint at arxiv:math.GT/0006062.
  • 40. Roseman, D., Reidemeister-type moves for surfaces in four dimensional space, in Banach Center Publications 42 Knot theory, (1998) 347-380. MR 99f:57029
  • 41. Rosicki, Witold, Some Simple Invariants of the Position of a Surface in $\mathbf{R}^4$, Bull.of the Pol. Ac.of Sci. Math. 46(4), 1998, 335-344. MR 99h:57050
  • 42. Ruberman, D., Doubly slice knots and the Casson-Gordon invariants, Trans. Amer. Math. Soc. 279 (1983), no. 2, 569-588. MR 85e:57025
  • 43. Rudolph, L., Braided surfaces and Seifert ribbons for closed braids. Comment. Math. Helv. 58 (1983), no. 1, 1-37. MR 84j:57006
  • 44. Sanderson, B. J., Bordism of links in codimension $2$, J. London Math. Soc. (2) 35 (1987), no. 2, 367-376. MR 88d:57023
  • 45. Sanderson, B. J., Triple links in codimension $2$, Topology. Theory and applications, II (Pécs, 1989), 457-471, Colloq. Math. Soc. János Bolyai, 55, North-Holland, Amsterdam, 1993. MR 96a:57056
  • 46. Satoh, S., and Shima, A. The Two Twist Spun Trefiol has Triple Point Number Four, Preprint.
  • 47. Sekine, M., Kawauchi's second duality and knotted surfaces in 4-sphere, Hiroshima Math. J. 19 (1989), 641-651. MR 91c:57030
  • 48. Turaev, V., The Yang-Baxter equation and invariants of links, Invent. math. 92 (1988) 527-553. MR 89e:57003
  • 49. Turaev, V., ``Quantum invariants of knots and 3-manifolds,'' de Gruyter Studies in Mathematics, 18. Walter de Gruyter & Co., (Berlin, 1994). MR 95k:57014
  • 50. Wakui, M., On Dijkgraaf-Witten invariant for $3$-manifolds, Osaka J. Math. 29 (1992), 675-696. MR 95e:57033

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Additional Information

J. Scott Carter
Affiliation: Department of Mathematics, University of South Alabama, Mobile, Alabama 36688

Daniel Jelsovsky
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620
Address at time of publication: Department of Mathematics, Florida Southern College, Lakeland, Florida 33801

Seiichi Kamada
Affiliation: Department of Mathematics, Osaka City University, Osaka 558-8585, Japan
Address at time of publication: Department of Mathematics, Hiroshima University, Hiroshima 739-8526, Japan

Laurel Langford
Affiliation: Department of Mathematics, University of Wisconsin at River Falls, River Falls, Wisconsin 54022

Masahico Saito
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620

Keywords: Knots, links, knotted surfaces, quandle, rack, quandle cohomology, state-sum invariants, non-invertibility
Received by editor(s): August 21, 2001
Received by editor(s) in revised form: February 20, 2002
Published electronically: June 24, 2003
Dedicated: Dedicated to Professor Kunio Murasugi for his 70th birthday
Article copyright: © Copyright 2003 American Mathematical Society

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