Special Issue: 2013 TAPU Workshop on Knot Theory and Related TopicsNo Access

On the Alexander biquandles of oriented surface-links via marked graph diagrams

Jieon Kim

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

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Yewon Joung

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

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, and

Sang Youl Lee

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

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https://doi.org/10.1142/S0218216514600074Cited by:7 (Source: Crossref)

Abstract

Carrell defined the fundamental biquandle of an oriented surface-link by a presentation obtained from its broken surface diagram, which is an invariant up to isomorphism of the fundamental biquandle. Ashihara gave a method to calculate the fundamental biquandle of an oriented surface-link from its marked graph diagram (ch-diagram). In this paper, we discuss the fundamental Alexander biquandles of oriented surface-links via marked graph diagrams, derived computable invariants and their applications to detect non-invertible oriented surface-links.

Keywords:

AMSC: 57Q45, 57M25

References

S. Ashihara , J. Knot Theory Ramifications 21 , 1250102 ( 2012 ) . Link, Web of Science, Google Scholar
T. Carrell , The Surface Biquandle ( Pomona College , 2009 ) . 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 , Mathematical Surveys and Monographs 55 ( American Mathematical Science , 1998 ) . Google Scholar
R. H. Crowell and R. H. Fox , Introduction to Knot Theory ( Springer , 1963 ) . Google Scholar
R. Fenn , M. Jordan-Santana and L. H. Kauffman , Topology Appl. 145 , 157 ( 2004 ) . Crossref, Web of Science, Google Scholar
R. H. Fox, Toplogy of 3-manifolds and Related Topics (Prentice-Hall, Englewood Cliffs, N.J., 1962) pp. 120–167. Google Scholar
Y. Joung, J. Kim and S. Y. Lee, J. Knot Theory Ramifications 22(9), 1350052 (2013). Link, Web of Science, Google Scholar
D. Joyce , J. Pure Appl. Algebra 23 , 37 ( 1982 ) . Crossref, Web of Science, Google Scholar
S. Kamada , Osaka J. Math. 26 , 367 ( 1989 ) . Web of Science, Google Scholar
L. H. Kauffman , Atti Sem. Mat. Fis. Univ. Modena (Suppl.) 49 , 241 ( 2001 ) . Google Scholar
L. H. Kauffman and D. E. Radford , Contemp. Math. 318 , 113 ( 2003 ) . Crossref, 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, On generating sets of Yoshikawa moves for marked graph diagrams of surface-links, preprint . Google Scholar
S. Y. Lee, Intelligence of Low Dimensional Topology 2006, Series on Knots Everything 40 (World Scientific Publisher, Hackensack, NJ, 2007) pp. 189–196. Link, 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
L. Tr. Lomonaco , 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 (in English) . 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