State-sum invariants of knotted curves and surfaces from quandle cohomology
Authors:
J. Scott Carter, Daniel Jelsovsky, Seiichi Kamada, Laurel Langford and Masahico Saito
Journal:
Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 146-156
MSC (1991):
Primary 57M25, 57Q45; Secondary 55N99, 18G99
DOI:
https://doi.org/10.1090/S1079-6762-99-00073-6
Published electronically:
December 9, 1999
MathSciNet review:
1725613
Full-text PDF Free Access
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Abstract: State-sum invariants for classical knots and knotted surfaces in $4$-space are developed via the cohomology theory of quandles. Cohomology groups of quandles are computed to evaluate the invariants. Some twist spun torus knots are shown to be noninvertible using the invariants.
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- Michihisa Wakui, On Dijkgraaf-Witten invariant for $3$-manifolds, Osaka J. Math. 29 (1992), no. 4, 675–696. MR 1192735
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- Baez, J.C.; Langford, L., $2$-tangles, Lett. Math. Phys. 43 (1998), no. 2, 187–197.
- 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
- Brieskorn, E., Automorphic sets and singularities, Contemporary Math. 78 (1988), 45–115.
- Carter, J.S.; Jelsovsky, D.; Kamada, S.; Langford, L.; Saito, M., Quandle cohomology and state-sum invariants of knotted curves and surfaces, preprint at http://xxx.lanl.gov/abs/math.GT/9903135
- Carter, J.S.; Jelsovsky, D.; Kamada, S.; Saito, M., Computations of quandle cocycle invariants of knotted curves and surfaces, preprint at http://xxx.lanl.gov/abs/math.GT/9906115
- 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.
- Carter, J.S.; Saito, M., Knotted surfaces and their diagrams, the American Mathematical Society, 1998.
- Carter, J.S.; Saito, M., On formulations and solutions of simplex equations, Internat. J. Modern Phys. A 11 (1996), no. 24, 4453–4463.
- Carter, J.S.; Saito, M., Canceling branch points on the projections of surfaces in $4$-space, Proc. AMS 116, 1 (1992), 229–237.
- Dijkgraaf, R., and Witten, E., Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990), 393–429.
- Fenn, R.; Rourke, C., Racks and links in codimension two, Journal of Knot Theory and Its Ramifications 1 (1992), no. 4, 343–406.
- Fenn, R.; Rourke, C.; Sanderson, B., Trunks and classifying spaces, Appl. Categ. Structures 3 (1995), no. 4, 321–356.
- Fenn, R.; Rourke, C.; Sanderson, B., James bundles and applications, preprint found at http://www.maths.warwick.ac.uk/ bjs/
- Fox, R.H., A quick trip through knot theory, in Topology of $3$-Manifolds, Ed. M.K. Fort Jr., Prentice-Hall, 1962, pp. 120–167.
- Jones, V.F.R., Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1989), 335–388.
- Joyce, D., A classifying invariant of knots, the knot quandle, J. Pure Appl. Alg., 23, 37–65.
- Hartley, R., Identifying noninvertible knots, Topology 22 (1983), 137–145.
- Hillman, J.A., Finite knot modules and the factorization of certain simple knots, Math. Ann. 257 (1981), no. 2, 261–274.
- Kamada, S., Surfaces in $\mathbf {R}^4$ of braid index three are ribbon, Journal of Knot Theory and Its Ramifications 1 (1992), 137–160.
- Kamada, S., A characterization of groups of closed orientable surfaces in $4$-space, Topology 33 (1994), 113–122.
- 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.
- Kauffman, L.H., Knots and physics, World Scientific, Series on knots and everything, vol. 1, 1991.
- Kawauchi, A., A survey of knot theory, Birkhauser, 1996.
- Kawauchi, A., The invertibility problem on amphicheiral excellent knots, Proc. Japan Acad., Ser. A, Math. Sci. 55 (1979), 399–402.
- Matveev, S., Distributive groupoids in knot theory, Mat. Sb. (N.S.) 119(161) (1982), no. 1, 78–88, 160. (Russian)
- Murasugi, K., Knot theory and its applications, Translated from the 1993 Japanese original by Bohdan Kurpita. Birkhaauser Boston, Inc., Boston, MA, 1996.
- 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
- Ruberman, D., Doubly slice knots and the Casson-Gordon invariants, Trans. Amer. Math. Soc. 279 (1983), no. 2, 569–588.
- Trotter, H.F., Noninvertible knots exist, Topology 2 (1964), 341–358.
- Roseman, D., Reidemeister-type moves for surfaces in four dimensional space, Knot Theory, Banach Center Publications 42 (1998), 347–380.
- Rosicki, Witold, Some simple invariants of the position of a surface in $\mathbf {R}^4$, Bull. Polish Acad. Sci. Math. 46(4) (1998), 335–344.
- Rudolph, L., Braided surfaces and Seifert ribbons for closed braids, Comment. Math. Helv. 58 (1983), no. 1, 1–37.
- Rolfsen, D., Knots and links, Publish or Perish Press (Berkley 1976).
- Wakui, M., On Dijkgraaf-Witten invariant for $3$-manifolds, Osaka J. Math. 29 (1992), 675–696.
- Zeeman, E.C., Twisting spun knots, Trans. A.M.S. 115 (1965), 471–495.
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Additional Information
J. Scott Carter
Affiliation:
Department of Mathematics, University of South Alabama, Mobile, AL 36688
MR Author ID:
682724
Email:
carter@mathstat.usouthal.edu
Daniel Jelsovsky
Affiliation:
Department of Mathematics, University of South Florida, Tampa, FL 33620
Email:
jelsovsk@math.usf.edu
Seiichi Kamada
Affiliation:
Department of Mathematics, Osaka City University, Osaka 558-8585, JAPAN
Address at time of publication:
Department of Mathematics, University of South Alabama, Mobile, AL 36688
MR Author ID:
288529
Email:
kamada@sci.osaka-cu.ac.jp, skamada@mathstat.usouthal.edu
Laurel Langford
Affiliation:
Department of Mathematics, University of Wisconsin at River Falls, River Falls, WI 54022
Email:
laurel.langford@uwrf.edu
Masahico Saito
Affiliation:
Department of Mathematics, University of South Florida, Tampa, FL 33620
MR Author ID:
196333
Email:
saito@math.usf.edu
Keywords:
Knots,
knotted surfaces,
quandle cohomology,
state-sum invariants
Received by editor(s):
May 28, 1999
Published electronically:
December 9, 1999
Additional Notes:
The third author was supported by a Fellowship from the Japan Society for the Promotion of Science.
Communicated by:
Walter Neumann
Article copyright:
© Copyright 1999
American Mathematical Society
