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

DOI:
https://doi.org/10.1090/S0002-9939-2011-10893-1

Published electronically:
May 25, 2011

MathSciNet review:
2833546

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Abstract | References | Similar Articles | Additional Information

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.

**1.**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)****6.**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)****11.**H. R. Morton,*An irreducible -string braid with unknotted closure*, Math. Proc. Cambridge Philos. Soc.**93**(1983), 259-261. MR**691995 (84m:57006)****12.**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)****13.**R. Shinjo,*Non-conjugate braids whose closures result in the same knot*, J. Knot Theory Ramifications**19**(2010), 117-124. MR**2640995****14.**A. Stoimenow,*Lie groups, Burau representation, and non-conjugate braids with the same closure link*, preprint.**15.**A. Stoimenow,*The density of Lawrence-Krammer and non-conjugate braid representations of links*, preprint arXiv:0809.0033.**16.**K. Tanaka,*A note on CI-moves*, Intelligence of low dimensional topology 2006, 307-314, Ser. Knots Everything,**40**, World Sci. Publ., 2007. MR**2371740 (2009a:57017)****17.**O. Ya. Viro, Lecture given at Osaka City University, September, 1990.

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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:
https://doi.org/10.1090/S0002-9939-2011-10893-1

Received by editor(s):
July 16, 2010

Received by editor(s) in revised form:
November 11, 2010, and November 12, 2010

Published electronically:
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.