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

Carter, J.; Jelsovsky, Daniel; Kamada, Seiichi; Langford, Laurel; Saito, Masahico

doi:10.1090/S0002-9947-03-03046-0

Quandle cohomology and state-sum invariants of knotted curves and surfaces
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by J. Scott Carter, Daniel Jelsovsky, Seiichi Kamada, Laurel Langford and Masahico Saito PDF

Trans. Amer. Math. Soc. 355 (2003), 3947-3989 Request permission

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.

References

John C. Baez and Laurel Langford, $2$-tangles, Lett. Math. Phys. 43 (1998), no. 2, 187–197. MR 1607351, DOI 10.1023/A:1007449529401
Baez, J.; Langford, L., Higher-dimensional algebra IV: 2-Tangles, to appear in Adv. Math, preprint available at http://xxx.lanl.gov/abs/math.QA/9811139.
E. Brieskorn, Automorphic sets and braids and singularities, Braids (Santa Cruz, CA, 1986) Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 45–115. MR 975077, DOI 10.1090/conm/078/975077
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.
J. Scott Carter, Louis H. Kauffman, and Masahico Saito, Structures and diagrammatics of four-dimensional topological lattice field theories, Adv. Math. 146 (1999), no. 1, 39–100. MR 1706684, DOI 10.1006/aima.1998.1822
J. Scott Carter, Joachim H. Rieger, and Masahico Saito, A combinatorial description of knotted surfaces and their isotopies, Adv. Math. 127 (1997), no. 1, 1–51. MR 1445361, DOI 10.1006/aima.1997.1618
J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, Mathematical Surveys and Monographs, vol. 55, American Mathematical Society, Providence, RI, 1998. MR 1487374, DOI 10.1090/surv/055
J. Scott Carter and Masahico Saito, Canceling branch points on projections of surfaces in $4$-space, Proc. Amer. Math. Soc. 116 (1992), no. 1, 229–237. MR 1126191, DOI 10.1090/S0002-9939-1992-1126191-0
Robbert Dijkgraaf and Edward Witten, Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990), no. 2, 393–429. MR 1048699
M. Š. Farber, Linking coefficients and two-dimensional knots, Dokl. Akad. Nauk SSSR 222 (1975), no. 2, 299–301 (Russian). MR 0400246
M. Š. Farber, Duality in an infinite cyclic covering, and even-dimensional knots, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 4, 794–828, 959 (Russian). MR 0515677
Roger Fenn and Colin Rourke, Racks and links in codimension two, J. Knot Theory Ramifications 1 (1992), no. 4, 343–406. MR 1194995, DOI 10.1142/S0218216592000203
Roger Fenn, Colin Rourke, and Brian Sanderson, Trunks and classifying spaces, Appl. Categ. Structures 3 (1995), no. 4, 321–356. MR 1364012, DOI 10.1007/BF00872903
Fenn, R.; Rourke, C.; Sanderson, B., James bundles and applications, preprint available at http://www.maths.warwick.ac.uk/~cpr/ftp/james.ps.
R. H. Fox, A quick trip through knot theory, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 120–167. MR 0140099
M. Fukuma, S. Hosono, and H. Kawai, Lattice topological field theory in two dimensions, Comm. Math. Phys. 161 (1994), no. 1, 157–175. MR 1266073
Cole A. Giller, Towards a classical knot theory for surfaces in $\textbf {R}^{4}$, Illinois J. Math. 26 (1982), no. 4, 591–631. MR 674227
Greene, M. T. Some Results in Geometric Topology and Geometry, Ph.D. Dissertation, Warwick (1997).
V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335–388. MR 908150, DOI 10.2307/1971403
David Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982), no. 1, 37–65. MR 638121, DOI 10.1016/0022-4049(82)90077-9
Richard Hartley, Identifying noninvertible knots, Topology 22 (1983), no. 2, 137–145. MR 683753, DOI 10.1016/0040-9383(83)90024-1
Jonathan A. Hillman, Finite knot modules and the factorization of certain simple knots, Math. Ann. 257 (1981), no. 2, 261–274. MR 634467, DOI 10.1007/BF01458289
Seiichi Kamada, Surfaces in $\textbf {R}^4$ of braid index three are ribbon, J. Knot Theory Ramifications 1 (1992), no. 2, 137–160. MR 1164113, DOI 10.1142/S0218216592000082
Seiichi Kamada, A characterization of groups of closed orientable surfaces in $4$-space, Topology 33 (1994), no. 1, 113–122. MR 1259518, DOI 10.1016/0040-9383(94)90038-8
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).
M. M. Kapranov and V. A. Voevodsky, $2$-categories and Zamolodchikov tetrahedra equations, Algebraic groups and their generalizations: quantum and infinite-dimensional methods (University Park, PA, 1991) Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, pp. 177–259. MR 1278735, DOI 10.1016/0022-4049(94)90097-3
Louis H. Kauffman, Knots and physics, Series on Knots and Everything, vol. 1, World Scientific Publishing Co., Inc., River Edge, NJ, 1991. MR 1141156, DOI 10.1142/9789812796226
Akio Kawauchi, The invertibility problem on amphicheiral excellent knots, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 10, 399–402. MR 559040
Akio Kawauchi, Three dualities on the integral homology of infinite cyclic coverings of manifolds, Osaka J. Math. 23 (1986), no. 3, 633–651. MR 866269
Akio Kawauchi, The first Alexander modules of surfaces in $4$-sphere, Algebra and topology 1990 (Taejon, 1990) Korea Adv. Inst. Sci. Tech., Taejŏn, 1990, pp. 81–89. MR 1098722
Akio Kawauchi, A survey of knot theory, Birkhäuser Verlag, Basel, 1996. Translated and revised from the 1990 Japanese original by the author. MR 1417494
Ulrich Koschorke, A generalization of Milnor’s $\mu$-invariants to higher-dimensional link maps, Topology 36 (1997), no. 2, 301–324. MR 1415590, DOI 10.1016/0040-9383(96)00018-3
Langford, L., $2$-tangles as a free braided monoidal $2$-category with duals. Ph.D. dissertation, U.C. Riverside, 1997.
Jerome Levine, Knot modules. I, Trans. Amer. Math. Soc. 229 (1977), 1–50. MR 461518, DOI 10.1090/S0002-9947-1977-0461518-0
S. V. Matveev, Distributive groupoids in knot theory, Mat. Sb. (N.S.) 119(161) (1982), no. 1, 78–88, 160 (Russian). MR 672410
Kunio Murasugi, Knot theory and its applications, Birkhäuser Boston, Inc., Boston, MA, 1996. Translated from the 1993 Japanese original by Bohdan Kurpita. MR 1391727
Neuchl, M., Representation Theory of Hopf Categories, to appear in Adv. in Math. under the title Higher-dimensional algebra VI: Hopf categories, available at http://www.mathematik.uni-muenchen.de/~neuchl.
Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
Rourke, C., and Sanderson, B.J., There are two 2-twist-spun trefoils, Preprint at arXiv:math.GT/0006062.
Dennis Roseman, Reidemeister-type moves for surfaces in four-dimensional space, Knot theory (Warsaw, 1995) Banach Center Publ., vol. 42, Polish Acad. Sci. Inst. Math., Warsaw, 1998, pp. 347–380. MR 1634466
Witold Rosicki, Some simple invariants of the position of a surface in $\mathbf R^4$, Bull. Polish Acad. Sci. Math. 46 (1998), no. 4, 335–344. MR 1657152
Daniel Ruberman, Doubly slice knots and the Casson-Gordon invariants, Trans. Amer. Math. Soc. 279 (1983), no. 2, 569–588. MR 709569, DOI 10.1090/S0002-9947-1983-0709569-5
Lee Rudolph, Braided surfaces and Seifert ribbons for closed braids, Comment. Math. Helv. 58 (1983), no. 1, 1–37. MR 699004, DOI 10.1007/BF02564622
B. J. Sanderson, Bordism of links in codimension $2$, J. London Math. Soc. (2) 35 (1987), no. 2, 367–376. MR 881524, DOI 10.1112/jlms/s2-35.2.367
B. J. Sanderson, Triple links in codimension $2$, Topology. Theory and applications, II (Pécs, 1989) Colloq. Math. Soc. János Bolyai, vol. 55, North-Holland, Amsterdam, 1993, pp. 457–471. MR 1244387
Satoh, S., and Shima, A. The Two Twist Spun Trefiol has Triple Point Number Four, Preprint.
Mituhiro Sekine, Kawauchi’s second duality and knotted surfaces in $4$-sphere, Hiroshima Math. J. 19 (1989), no. 3, 641–651. MR 1035148
V. G. Turaev, The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988), no. 3, 527–553. MR 939474, DOI 10.1007/BF01393746
V. G. Turaev, Quantum invariants of knots and 3-manifolds, De Gruyter Studies in Mathematics, vol. 18, Walter de Gruyter & Co., Berlin, 1994. MR 1292673
Michihisa Wakui, On Dijkgraaf-Witten invariant for $3$-manifolds, Osaka J. Math. 29 (1992), no. 4, 675–696. MR 1192735