Geodesic
Intrinsically linked graphs in projective space
Bustamante, Jason1 ; Federman, Jared2 ; Foisy, Joel2 ; Kozai, Kenji3 ; Matthews, Kevin2 ; McNamara, Kristin4 ; Stark, Emily5 ; Trickey, Kirsten6
1Department of Mathematical Sciences, Montana Tech of The University of Montana, 1300 West Park Street, Butte, MT 59701, USA
2Department of Mathematics, SUNY Potsdam, Potsdam, NY 13676, USA
3Department of Mathematics, Stanford University, 450 Serra Mall, Building 380, Stanford, CA 94305, USA
4Department of Mathematics and Statistics, James Madison University, 305 Roop Hall MSC 1911, Harrisonburg, VA 22807, USA
5Department of Mathematics, Pomona College, 610 North College Avenue, Claremont, CA 91711, USA
6Department of Mathematics, Clarkson University, 8 Clarkson Avenue, Potsdam, NY 13699, USA
Algebraic and Geometric Topology, Tome 9 (2009) no. 3, pp. 1255-1274
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We examine graphs that contain a nontrivial link in every embedding into real projective space, using a weaker notion of unlink than was used in Flapan, et al [Algebr. Geom. Topol. 6 (2006) 1025–1035]. We call such graphs intrinsically linked in ℝP3. We fully characterize such graphs with connectivity 0, 1 and 2. We also show that only one Petersen-family graph is intrinsically linked in ℝP3 and prove that K7 minus any two edges is also minor-minimal intrinsically linked. In all, 597 graphs are shown to be minor-minimal intrinsically linked in ℝP3.

DOI : 10.2140/agt.2009.9.1255
Keywords: RP^3, projective, graph, link

Bustamante, Jason  1   ; Federman, Jared  2   ; Foisy, Joel  2   ; Kozai, Kenji  3   ; Matthews, Kevin  2   ; McNamara, Kristin  4   ; Stark, Emily  5   ; Trickey, Kirsten  6

1 Department of Mathematical Sciences, Montana Tech of The University of Montana, 1300 West Park Street, Butte, MT 59701, USA
2 Department of Mathematics, SUNY Potsdam, Potsdam, NY 13676, USA
3 Department of Mathematics, Stanford University, 450 Serra Mall, Building 380, Stanford, CA 94305, USA
4 Department of Mathematics and Statistics, James Madison University, 305 Roop Hall MSC 1911, Harrisonburg, VA 22807, USA
5 Department of Mathematics, Pomona College, 610 North College Avenue, Claremont, CA 91711, USA
6 Department of Mathematics, Clarkson University, 8 Clarkson Avenue, Potsdam, NY 13699, USA 
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     title = {Intrinsically linked graphs in projective space},
     journal = {Algebraic and Geometric Topology},
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     year = {2009},
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Bustamante, Jason; Federman, Jared; Foisy, Joel; Kozai, Kenji; Matthews, Kevin; McNamara, Kristin; Stark, Emily; Trickey, Kirsten. Intrinsically linked graphs in projective space. Algebraic and Geometric Topology, Tome 9 (2009) no. 3, pp. 1255-1274. doi: 10.2140/agt.2009.9.1255

[1] D Archdeacon, A Kuratowski theorem for the projective plane, J. Graph Theory 5 (1981) 243

[2] A Brouwer, R Davis, A Larkin, D Studenmund, C Tucker, Intrinsically $S^1$ $3$–linked graphs and other aspects of $S^1$ embeddings, Rose-Hulman Undergrad. Math. J. 8 (2007)

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

[4] Y V Drobotukhina, An analogue of the Jones polynomial for links in $\mathbf{R}\mathrm{P}^3$ and a generalization of the Kauffman–Murasugi theorem, Algebra i Analiz 2 (1990) 171

[5] E Flapan, H Howards, D Lawrence, B Mellor, Intrinsic linking and knotting of graphs in arbitrary $3$–manifolds, Algebr. Geom. Topol. 6 (2006) 1025

[6] H H Glover, J P Huneke, C S Wang, 103 graphs that are irreducible for the projective plane, J. Combin. Theory Ser. B 27 (1979) 332

[7] V Manturov, Knot theory, Chapman Hall/CRC (2004)

[8] R Motwani, A Raghunathan, H Saran, Constructive results from graph minors: linkless embeddings, from: "29th Annual Symposium on Foundations of Computer Science", IEEE Computer Soc. (1988) 398

[9] M Mroczkowski, Diagrammatic unknotting of knots and links in the projective space, J. Knot Theory Ramifications 12 (2003) 637

[10] J Nešetřil, R Thomas, A note on spatial representation of graphs, Comment. Math. Univ. Carolin. 26 (1985) 655

[11] D O’Donnol, Intrinsically $n$–linked complete bipartite graphs, J. Knot Theory Ramifications 17 (2008) 133

[12] M Ozawa, Y Tsutsumi, Primitive spatial graphs and graph minors, Rev. Mat. Complut. 20 (2007) 391

[13] N Robertson, P Seymour, Graph minors. XX. Wagner's conjecture, J. Combin. Theory Ser. B 92 (2004) 325

[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 Sós), Colloq. Math. Soc. János Bolyai 37, North-Holland (1984) 649

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