Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Homotopy on spatial graphs and the Sato-Levine invariant

Author(s): Thomas Fleming; Ryo Nikkuni
Journal: Trans. Amer. Math. Soc. 361 (2009), 1885-1902.
MSC (2000): Primary 57M15; Secondary 57M25
Posted: November 25, 2008
MathSciNet review: 2465822
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. We introduce some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the -component constituent algebraically split links and show examples of non-splittable spatial graphs up to edge (resp. vertex)-homotopy, all of whose constituent links are link-homotopically trivial.

References:

1.: T. D. Cochran, Concordance invariance of coefficients of Conway's link polynomial, Invent. Math. 82 (1985), 527-541. MR 87c:57002
2.: N. Habegger and X. -S. Lin, The classification of links up to link-homotopy, J. Amer. Math. Soc. 3 (1990), 389-419. MR 91e:57015
3.: R. Hartley, The Conway potential function for links, Comment. Math. Helv. 58 (1983), 365-378. MR 85h:57006
4.: F. Hosokawa, On $\nabla$ -polynomials of links, Osaka Math. J. 10 (1958), 273-282. MR 21:1606
5.: J. Hoste, The first coefficient of the Conway polynomial, Proc. Amer. Math. Soc. 95 (1985), 299-302. MR 86m:57009
6.: L. H. Kauffman, Formal knot theory, Mathematical Notes, 30, Princeton University Press, Princeton, NJ, 1983. MR 85b:57006
7.: J. P. Levine, An approach to homotopy classification of links, Trans. Amer. Math. Soc. 306 (1988), 361-387. MR 88m:57008
8.: J. Milnor, Link groups, Ann. of Math. (2) 59 (1954), 177-195. MR 17:70e
9.: T. Motohashi and K. Taniyama, Delta unknotting operation and vertex homotopy of graphs in $R\sp 3$ , KNOTS '96 (Tokyo), 185-200, World Sci. Publishing, River Edge, NJ, 1997. MR 99i:57021
10.: R. Nikkuni, Delta link-homotopy on spatial graphs, Rev. Mat. Complut. 15 (2002), 543-570. MR 2004d:57013
11.: R. Nikkuni, Edge-homotopy classification of spatial complete graphs on four vertices, J. Knot Theory Ramifications 13 (2004), 763-777. MR 2005f:57008
12.: R. Nikkuni, Sharp edge-homotopy on spatial graphs, Rev. Mat. Complut. 18 (2005), 181-207. MR 2135538
13.: Y. Ohyama and K. Taniyama, Vassiliev invariants of knots in a spatial graph, Pacific J. Math. 200 (2001), 191-205. MR 2003a:57025
14.: N. Sato, Cobordisms of semiboundary links, Topology Appl. 18 (1984), 225-234. MR 86d:57010
15.: T. Shibuya, Self $\sharp$ -unknotting operations of links, Mem. Osaka Inst. Tech. Ser. A 34 (1989), 9-17. MR 92a:57014
16.: T. Shibuya and A. Yasuhara, Classification of links up to self pass-move, J. Math. Soc. Japan 55 (2003), 939-946. MR 2004f:57016
17.: R. Sturm Beiss, The Arf and Sato link concordance invariants, Trans. Amer. Math. Soc. 322 (1990), 479-491. MR 91m:57006
18.: K. Taniyama, Link homotopy invariants of graphs in $R\sp 3$ , Rev. Mat. Univ. Complut. Madrid 7 (1994), 129-144. MR 95f:57023
19.: K. Taniyama, Cobordism, homotopy and homology of graphs in $R\sp 3$ , Topology 33 (1994), 509-523. MR 95h:57002

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57M15, 57M25

Retrieve articles in all Journals with MSC (2000): 57M15, 57M25

Additional Information:

Thomas Fleming
Affiliation: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093
Email: tfleming@math.ucsd.edu

Ryo Nikkuni
Affiliation: Institute of Human and Social Sciences, Faculty of Teacher Education, Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa, 920-1192, Japan
Email: nick@ed.kanazawa-u.ac.jp

DOI: 10.1090/S0002-9947-08-04510-8
PII: S 0002-9947(08)04510-8
Keywords: Spatial graph, edge-homotopy, vertex-homotopy, Sato-Levine invariant
Received by editor(s): August 31, 2005
Received by editor(s) in revised form: March 10, 2007
Posted: November 25, 2008
Additional Notes: The first author was supported by a Fellowship of the Japan Society for the Promotion of Science for Post-Doctoral Foreign Researchers (Short-Term) (No. PE05003).
The second author was partially supported by a Grant-in-Aid for Scientific Research (B) (2) (No. 15340019), Japan Society for the Promotion of Science.
Copyright of article: Copyright 2008, American Mathematical Society

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|