## Knotted Legendrian surfaces with few Reeb chords.(English)Zbl 1248.53073

In the article under review the author constructs $$g+1$$ (for $$g>0$$) Legendrian embeddings of a surface of genus $$g$$ into $$J^1(\mathbb R^2)=\mathbb R^5$$ which lie in pairwise distinct Legendrian isotopy classes and which all have $$g+1$$ transverse Reeb chords ($$g+1$$ is the conjectured minimal number of chords). Furthermore, the author shows that for $$g$$ of the $$g+1$$ embeddings the Legendrian contact homology DGA does not admit any augmentation over $$\mathbb Z/2 \mathbb Z$$, and hence cannot be linearized. The author also investigates these surfaces from the point of view of the theory of generating families. Finally, he considers Legendrian spheres and planes in $$J^1(S^2)$$ from a similar perspective.

### MSC:

 53D42 Symplectic field theory; contact homology 53D12 Lagrangian submanifolds; Maslov index
Full Text:

### References:

 [1] P Albers, U Frauenfelder, A nondisplaceable Lagrangian torus in $$T^*S^2$$, Comm. Pure Appl. Math. 61 (2008) 1046 · Zbl 1142.53067 [2] Y Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002) 441 · Zbl 1029.57011 [3] T Ekholm, Morse flow trees and Legendrian contact homology in $$1$$-jet spaces, Geom. Topol. 11 (2007) 1083 · Zbl 1162.53064 [4] T Ekholm, J B Etnyre, Invariants of knots, embeddings and immersions via contact geometry (editors H U Boden, I Hambleton, A J Nicas, B D Park), Fields Inst. Commun. 47, Amer. Math. Soc. (2005) 77 · Zbl 1118.57013 [5] T Ekholm, J Etnyre, M Sullivan, The contact homology of Legendrian submanifolds in $$\mathbbR^{2n+1}$$, J. Differential Geom. 71 (2005) 177 · Zbl 1103.53048 [6] T Ekholm, J Etnyre, M Sullivan, Non-isotopic Legendrian submanifolds in $$\mathbb R^{2n+1}$$, J. Differential Geom. 71 (2005) 85 · Zbl 1098.57013 [7] T Ekholm, J Etnyre, M Sullivan, Orientations in Legendrian contact homology and exact Lagrangian immersions, Int. J. Math. 16 (2005) 453 · Zbl 1076.53099 [8] T Ekholm, J Etnyre, M Sullivan, Legendrian contact homology in $$P\times\mathbb R$$, Trans. Amer. Math. Soc. 359 (2007) 3301 · Zbl 1119.53051 [9] Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory (editors N Alon, J Bourgain, A Connes, M Gromov, V Milman), Geom. Funct. Anal. Special Volume, Part II (2000) 560 · Zbl 0989.81114 [10] Y Eliashberg, N Mishachev, Introduction to the $$h$$-principle, Graduate Studies in Math. 48, Amer. Math. Soc. (2002) · Zbl 1008.58001 [11] D Fuchs, D Rutherford, Generating families and Legendrian contact homology in the standard contact space · Zbl 1237.57026 [12] K Fukaya, Y G Oh, H Ohta, K Ono, Toric degeneration and non-displaceable Lagrangian tori in $$S^2 \times S^2$$, to appear in Int. Math. Res. Not. · Zbl 1250.53077 [13] L Ng, Framed knot contact homology, Duke Math. J. 141 (2008) 365 · Zbl 1145.57010 [14] P E Pushkar$$^{\prime}$$‘, Y V Chekanov, Combinatorics of fronts of Legendrian links, and Arnol$$'`$$d’s $$4$$-conjectures, Uspekhi Mat. Nauk 60 (2005) 99 · Zbl 1085.57008 [15] P Seidel, A long exact sequence for symplectic Floer cohomology, Topology 42 (2003) 1003 · Zbl 1032.57035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.