An intrinsic nontriviality of graphs

Nikkuni, Ryo¹

¹Institute of Human and Social Sciences, Faculty of Teacher Education, Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa, 920-1192, Japan

Algebraic and Geometric Topology, Tome 9 (2009) no. 1, pp. 351-364

Résumé

We say that a graph is intrinsically nontrivial if every spatial embedding of the graph contains a nontrivial spatial subgraph. We prove that an intrinsically nontrivial graph is intrinsically linked, namely every spatial embedding of the graph contains a nonsplittable 2–component link. We also show that there exists a graph such that every spatial embedding of the graph contains either a nonsplittable 3–component link or an irreducible spatial handcuff graph whose constituent 2–component link is split.

Détail

DOI : 10.2140/agt.2009.9.351

Keywords: spatial graph, intrinsically linked, spatial handcuff graph

Bibliographie

[1] G Bowlin, J Foisy, Some new intrinsically $3$–linked graphs, J. Knot Theory Ramifications 13 (2004) 1021

[2] J H Conway, C M Gordon, Knots and links in spatial graphs, J. Graph Theory 7 (1983) 445

[3] E Flapan, R Naimi, J Pommersheim, Intrinsically triple linked complete graphs, Topology Appl. 115 (2001) 239

[4] E Flapan, J Pommersheim, J Foisy, R Naimi, Intrinsically $n$–linked graphs, J. Knot Theory Ramifications 10 (2001) 1143

[5] J Foisy, Graphs with a knot or $3$–component link in every spatial embedding, J. Knot Theory Ramifications 15 (2006) 1113

[6] L H Kauffman, Formal knot theory, Math. Notes 30, Princeton Univ. Press (1983)

[7] L H Kauffman, On knots, Annals of Math. Studies 115, Princeton Univ. Press (1987)

[8] S V Matveev, Generalized surgeries of three-dimensional manifolds and representations of homology spheres, Mat. Zametki 42 (1987) 268, 345

[9] T Motohashi, K Taniyama, Delta unknotting operation and vertex homotopy of graphs in $\mathbf R^3$, from: "KNOTS '96 (Tokyo)" (editor S Suzuki), World Sci. Publ. (1997) 185

[10] H Murakami, Y Nakanishi, On a certain move generating link-homology, Math. Ann. 284 (1989) 75

[11] R Nikkuni, Delta edge-homotopy invariants of spatial graphs via disk-summing the constituent knots, to appear in Illinois J. Math.

[12] Y Ohyama, T Tsukamoto, On Habiro's $C_n$–moves and Vassiliev invariants of order $n$, J. Knot Theory Ramifications 8 (1999) 15

[13] M Okada, Delta-unknotting operation and the second coefficient of the Conway polynomial, J. Math. Soc. Japan 42 (1990) 713

[14] N Robertson, P Seymour, R Thomas, Sachs' linkless embedding conjecture, J. Combin. Theory Ser. B 64 (1995) 185

[15] H Sachs, On spatial representations of finite graphs, from: "Finite and infinite sets, Vol. I, II (Eger, 1981)" (editors A Hajnal, L Lovász, V T Sòs), Colloq. Math. Soc. János Bolyai 37, North-Holland (1984) 649

[16] M Scharlemann, A Thompson, Detecting unknotted graphs in $3$–space, J. Differential Geom. 34 (1991) 539

[17] T Soma, H Sugai, A Yasuhara, Disk/band surfaces of spatial graphs, Tokyo J. Math. 20 (1997) 1

[18] K Taniyama, Cobordism, homotopy and homology of graphs in $\mathbf{R}^3$, Topology 33 (1994) 509

[19] K Taniyama, A Yasuhara, Realization of knots and links in a spatial graph, Topology Appl. 112 (2001) 87

[20] Y Q Wu, On planarity of graphs in $3$–manifolds, Comment. Math. Helv. 67 (1992) 635

[21] H Yamada, Delta distance and Vassiliev invariants of knots, J. Knot Theory Ramifications 9 (2000) 967