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IDEAL COSET INVARIANTS FOR SURFACE-LINKS IN ℝ⁴

YEWON JOUNG

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

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JIEON KIM

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/S0218216513500521Cited by:9 (Source: Crossref)

Abstract

In [Towards invariants of surfaces in 4-space via classical link invariants, Trans. Amer. Math. Soc.361 (2009) 237–265], Lee defined a polynomial [[D]] for marked graph diagrams D of surface-links in 4-space by using a state-sum model involving a given classical link invariant. In this paper, we deal with some obstructions to obtain an invariant for surface-links represented by marked graph diagrams D by using the polynomial [[D]] and introduce an ideal coset invariant for surface-links, which is defined to be the coset of the polynomial [[D]] in a quotient ring of a certain polynomial ring modulo some ideal and represented by a unique normal form, i.e. a unique representative for the coset of [[D]] that can be calculated from [[D]] with the help of a Gröbner basis package on computer.

Keywords:

AMSC: 57Q45, 57M25, 57M27

References

M. Asada, Kobe J. Math. 18, 163 (2001). Web of Science, Google Scholar
D. Cox , J. Little and D. O'Shea , Ideals, Varieties, and Algorithms ( Springer , 1997 ) . Google Scholar
R. H. Fox, Topology of 3-Manifolds and Related Topics (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1962) pp. 120–167. Google Scholar
S. Kamada, Osaka J. Math. 26, 367 (1989). Web of Science, Google Scholar
L. H. Kauffman, Trans. Amer. Math. Soc. 311, 697 (1989), DOI: 10.1090/S0002-9947-1989-0946218-0. 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), DOI: 10.2140/agt.2008.8.1223. Crossref, Web of Science, Google Scholar
S. A. King, Topol. Appl. 154, 1141 (2007), DOI: 10.1016/j.topol.2006.11.005. Crossref, Web of Science, Google Scholar
S. Y. Lee, Intelligence of Low Dimensional Topology 2006, Series on Knots and Everything 40 (World Scientific Publishing, Hackensack, NJ, 2007) pp. 189–196. Link, Google Scholar
S. Y. Lee, J. Knot Theory Ramifications 17, 439 (2008), DOI: 10.1142/S0218216508006221. Link, Web of Science, Google Scholar
S. Y. Lee, Trans. Amer. Math. Soc. 361, 237 (2009), DOI: 10.1090/S0002-9947-08-04568-6. Crossref, Web of Science, Google Scholar
L. Tr. Lomonaco, Pacific J. Math. 95, 349 (1981), DOI: 10.2140/pjm.1981.95.349. Crossref, Web of Science, Google Scholar
M. Soma, Topol. Appl. 121, 231 (2002), DOI: 10.1016/S0166-8641(01)00120-1. Crossref, Web of Science, Google Scholar
F. J. Swenton, J. Knot Theory and Its Ramifications 10, 1133 (2001), DOI: 10.1142/S0218216501001359. Link, Web of Science, Google Scholar
K. Yoshikawa, Osaka J. Math. 31, 497 (1994). Web of Science, Google Scholar