Special issue: 2014 KOOK-TAPU Workshop on Knot Theory and Related TopicsNo Access

Computations of quandle cocycle invariants of surface-links using marked graph diagrams

Department of Mathematics, Osaka City University, Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

Department of Mathematics, Graduate School of Natural Sciences, Pusan National University, Busan 609-735, Republic of Korea

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Sang Youl Lee

Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea

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https://doi.org/10.1142/S0218216515400106Cited by:4 (Source: Crossref)

Abstract

By using the cohomology theory of quandles, quandle cocycle invariants and shadow quandle cocycle invariants are defined for oriented links and surface-links via broken surface diagrams. By using symmetric quandles, symmetric quandle cocycle invariants are also defined for unoriented links and surface-links via broken surface diagrams. A marked graph diagram is a link diagram possibly with 4-valent vertices equipped with markers. Lomonaco, Jr. and Yoshikawa introduced a method of describing surface-links by using marked graph diagrams. In this paper, we give interpretations of these quandle cocycle invariants in terms of marked graph diagrams, and introduce a method of computing them from marked graph diagrams.

Keywords:

AMSC: 57M25, 57M27

References

S. Ashihara, J. Knot Theory Ramifications 21(10), 23 (2012). Link, Web of Science, Google Scholar
J. S. Carteret al., Adv. Math. 157(1), 36 (2001). Crossref, Web of Science, Google Scholar
J. S. Carteret al., J. Pure Appl. Algebra 157(2), 135 (2001). Crossref, Web of Science, Google Scholar
J. S. Carter et al. , Trans. Amer. Math. Soc. 355 , 3947 ( 2003 ) . Crossref, Web of Science, Google Scholar
J. S. Carter, S. Kamada and M. Saito, J. Knot Theory Ramifications 10(3), 345 (2001). Link, Web of Science, Google Scholar
J. S. Carter , S. Kamada and M. Saito , Surfaces in 4-Space ( Springer , 2004 ) . Crossref, Google Scholar
J. S. Carter and M. Saito , Knotted Surfaces and Their Diagrams ( American Mathematical Society , 1998 ) . Google Scholar
R. Fenn, C. Rourke and B. Sanderson, Appl. Categ. Structures 3(4), 321 (1995). Crossref, Web of Science, Google Scholar
R. Fenn, C. Rourke and B. Sanderson, James bundles and applications, preprint (1996), http://www.maths.warwick.ac.uk/cpr/ftp/james.ps . Google Scholar
J. Flower, Cyclic bordism and rack spaces, Ph.D. Dissertation, Warwick (1995) . Google Scholar
R. H. Fox, Topology of 3-Manifolds and Related Topics (Prentice-Hall, Englewood Cliffs, N.J., 1962) pp. 120–167. Google Scholar
M. T. Greene, Some results in geometric topology and geometry, Ph.D. Dissertation, Warwick (1997) . Google Scholar
Y. Joung, S. Kamada and S. Y. Lee, J. Knot Theory Ramifications 24(10), 1540003-1 (2015). Link, Web of Science, Google Scholar
Y. Joung, J. Kim and S. Y. Lee, J. Knot Theory Ramifications 22(9), 25 (2013). Link, Web of Science, Google Scholar
D. Joyce , J. Pure Appl. Algebra 23 , 37 ( 1982 ) . Crossref, Web of Science, Google Scholar
S. Kamada , Fund. Math. 168 , 105 ( 2001 ) . Crossref, Web of Science, Google Scholar
S. Kamada, Intelligence of Low Dimensional Topology 2006, eds. J. S. Carteret al. (World Scientific Publishing, 2007) pp. 101–108. Link, Google Scholar
S. Kamada and K. Oshiro, Trans. Amer. Math. Soc. 362(10), 5501 (2010). Crossref, Web of Science, Google Scholar
A. Kawauchi , A Survey of Knot Theory ( Birkhäuser , 1996 ) . Google Scholar
A. Kawauchi , T. Shibuya and S. Suzuki , Math. Sem. Notes Kobe Univ. 10 , 75 ( 1982 ) . Google Scholar
C. Kearton and V. Kurlin , Algebr. Geom. Topol. 8 , 1223 ( 2008 ) . Crossref, Web of Science, Google Scholar
J. Kim, Y. Joung and S. Y. Lee, J. Knot Theory Ramifications 23(7), 26 (2014). Google Scholar
J. Kim, Y. Joung and S. Y. Lee, J. Knot Theory Ramifications 24(4), 1550018-1 (2015). Google Scholar
S. Y. Lee, Invariants of surface links in R4 via classical link invariants, Intelligence of Low Dimensional Topology 200640, Series on Knots Everything (World Scientific Publishing, Hackensack, NJ, 2007) pp. 189–196. Google Scholar
S. Y. Lee , J. Knot Theory Ramifications 17 , 439 ( 2008 ) . Link, Web of Science, Google Scholar
S. Y. Lee , Trans. Amer. Math. Soc. 361 , 237 ( 2009 ) . Crossref, Web of Science, Google Scholar
S. J. Lomonaco Jr. , Pacific J. Math. 95 , 349 ( 1981 ) . Crossref, Web of Science, Google Scholar
S. V. Matveev, Distributive groupoids in knot theory, Mat. Sb. (N.S.) 119 (161)(1) (1982) 78–88 (in Russian); Math. USSR-Sb. 47 (1984) 73–83 . Google Scholar
D. Roseman , Banach Center Publ. 42 , 347 ( 1998 ) . Crossref, Google Scholar
M. Soma , Topology Appl. 121 , 231 ( 2002 ) . Crossref, Web of Science, Google Scholar
F. J. Swenton , J. Knot Theory Ramifications 10 , 1133 ( 2001 ) . Link, Web of Science, Google Scholar
K. Yoshikawa , Osaka J. Math. 31 , 497 ( 1994 ) . Web of Science, Google Scholar