Geodesic
On Legendrian graphs
O’Donnol, Danielle1 ; Pavelescu, Elena2
1Department of Mathematics and Statistics, Smith College, 44 College Lane, Northampton MA 01060, USA
2Department of Mathematics, Occidental College, 1600 Campus Road, Los Angeles CA 90041-3314, USA
Algebraic and Geometric Topology, Tome 12 (2012) no. 3, pp. 1273-1299
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We investigate Legendrian graphs in (ℝ3,ξstd). We extend the Thurston–Bennequin number and the rotation number to Legendrian graphs. We prove that a graph can be Legendrian realized with all its cycles Legendrian unknots with tb = −1 and rot = 0 if and only if it does not contain K4 as a minor. We show that the pair (tb,rot) does not characterize a Legendrian graph up to Legendrian isotopy if the graph contains a cut edge or a cut vertex. When we restrict to planar spatial graphs, a pair (tb,rot) determines two Legendrian isotopy classes of the lollipop graph and a pair (tb,rot) determines four Legendrian isotopy classes of the handcuff graph.

DOI : 10.2140/agt.2012.12.1273
Classification : 57M25, 57M50, 05C10
Keywords: Legendrian graph, Thurston–Bennequin number, rotation number, $K_4$

O’Donnol, Danielle  1   ; Pavelescu, Elena  2

1 Department of Mathematics and Statistics, Smith College, 44 College Lane, Northampton MA 01060, USA
2 Department of Mathematics, Occidental College, 1600 Campus Road, Los Angeles CA 90041-3314, USA 
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O’Donnol, Danielle; Pavelescu, Elena. On Legendrian graphs. Algebraic and Geometric Topology, Tome 12 (2012) no. 3, pp. 1273-1299. doi: 10.2140/agt.2012.12.1273

[1] S Baader, M Ishikawa, Legendrian graphs and quasipositive diagrams, Ann. Fac. Sci. Toulouse Math. 18 (2009) 285

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

[3] F Ding, H Geiges, Legendrian knots and links classified by classical invariants, Commun. Contemp. Math. 9 (2007) 135

[4] Y Eliashberg, Contact $3$–manifolds twenty years since J Martinet's work, Ann. Inst. Fourier (Grenoble) 42 (1992) 165

[5] Y Eliashberg, M Fraser, Topologically trivial Legendrian knots, J. Symplectic Geom. 7 (2009) 77

[6] J B Etnyre, K Honda, Knots and contact geometry I: Torus knots and the figure eight knot, J. Symplectic Geom. 1 (2001) 63

[7] H Geiges, An introduction to contact topology, Cambridge Studies in Adv. Math. 109, Cambridge Univ. Press (2008)

[8] E Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, from: "Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002)" (editor T Li), Higher Ed. Press (2002) 405

[9] L H Kauffman, Invariants of graphs in three-space, Trans. Amer. Math. Soc. 311 (1989) 697

[10] C Kuratowski, Sur le probléme des courbes gauches en topologie, Fund. Math. 15 (1930) 271

[11] K Mohnke, Legendrian links of topological unknots, from: "Topology, geometry, and algebra: interactions and new directions (Stanford, CA, 1999)" (editors A Adem, G Carlsson, R Cohen), Contemp. Math. 279, Amer. Math. Soc. (2001) 209

[12] N Robertson, P D Seymour, R Thomas, Linkless embeddings of graphs in $3$–space, Bull. Amer. Math. Soc. 28 (1993) 84

[13] 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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