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Infinite sequences of mutually non-conjugate surface braids representing same surface-links
Author:
Masahide Iwakiri
Journal:
Proc. Amer. Math. Soc. 140 (2012), 357-366
MSC (2010):
Primary 57Q45
Posted:
May 25, 2011
MathSciNet review:
2833546
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Abstract: We give an infinite sequence of mutually non-conjugate surface braids with same degree representing the trivial surface-link with at least two components and a pair of non-conjugate surface braids with same degree representing a spun  -torus knot for  . To give these examples, we introduce new invariants of conjugacy classes of surface braids via colorings by Alexander quandles or core quandles of groups.
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- J. S. Carter and M. Saito, Braids and movies, J. Knot Theory Ramifications 5 (1996), 589-608. MR 1414089 (97j:57028)
- 2.
- T. Fiedler, A small state sum for knots, Topology 32 (1993), 281-294. MR 1217069 (94c:57006)
- 3.
- E. Fukunaga, An infinite sequence of conjugacy classes in the -braid group representing a torus link of type , preprint.
- 4.
- I. Hasegawa, A certain linear representation of the classical braid group and its application to surface braids, Math. Proc. Cambridge Philos. Soc. 141 (2006), 287-301. MR 2265876 (2008e:57017)
- 5.
- F. Hosokawa and A. Kawauchi, Proposals for unknotted surfaces in four-spaces, Osaka J. Math. 16 (1979), 233-248. MR 527028 (81c:57018)
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- M. Iwakiri, The lower bound of the w-indices of surface links via quandle cocycle invariants, Trans. Amer. Math. Soc. 362 (2010), 1189-1210. MR 2563726 (2010j:57033)
- 7.
- S. Kamada, A characterization of groups of closed orientable surfaces in
 -space, Topology 33 (1994), 113-122. MR 1259518 (95a:57002)
- 8.
- S. Kamada, Alexander's and Markov's theorems in dimension four, Bull. Amer. Math. Soc. (N.S.) 31 (1994), 64-67. MR 1254074 (94j:57023)
- 9.
- S. Kamada, An observation of surface braids via chart description, J. Knot Theory Ramifications 4 (1996), 517-529. MR 1406718 (97j:57009)
- 10.
- S. Kamada, Braid and knot theory in dimension four, Math. Surveys Monogr. 95, Amer. Math. Soc., 2002. MR 1900979 (2003d:57050)
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- H. R. Morton, An irreducible
 -string braid with unknotted closure, Math. Proc. Cambridge Philos. Soc. 93 (1983), 259-261. MR 691995 (84m:57006)
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- R. Shinjo, An infinite sequence of non-conjugate
 -braids representing the same knot of braid index  , Intelligence of low dimensional topology 2006, 293-297, Ser. Knots Everything, 40, World Sci. Publ., 2007. MR 2371738 (2008m:57033)
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- R. Shinjo, Non-conjugate braids whose closures result in the same knot, J. Knot Theory Ramifications 19 (2010), 117-124. MR 2640995
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- A. Stoimenow, Lie groups, Burau representation, and non-conjugate braids with the same closure link, preprint.
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Additional Information
Masahide Iwakiri
Affiliation:
Graduate School of Science, Osaka City University, 3-3-138 Sugimoto Sumiyoshi-ku, Osaka 558-8585, Japan
Address at time of publication:
Graduate School of Science and Engineering, Saga University, 1 Honjo-machi, Saga City, Saga, 840-8502, Japan
Email:
iwakiri@sci.osaka-cu.ac.jp, iwakiri@ms.saga-u.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002-9939-2011-10893-1
PII:
S 0002-9939(2011)10893-1
Received by editor(s):
July 16, 2010
Received by editor(s) in revised form:
November 11, 2010, and November 12, 2010
Posted:
May 25, 2011
Communicated by:
Daniel Ruberman
Article copyright:
© Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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