## Algebraic & Geometric Topology

### Knotted Legendrian surfaces with few Reeb chords

Georgios Dimitroglou Rizell

#### Abstract

For $g>0$, we construct $g+1$ Legendrian embeddings of a surface of genus $g$ into $J1(ℝ2)=ℝ5$ which lie in pairwise distinct Legendrian isotopy classes and which all have $g+1$ transverse Reeb chords ($g+1$ is the conjecturally minimal number of chords). Furthermore, for $g$ of the $g+1$ embeddings the Legendrian contact homology DGA does not admit any augmentation over $ℤ2$, and hence cannot be linearized. We also investigate these surfaces from the point of view of the theory of generating families. Finally, we consider Legendrian spheres and planes in $J1(S2)$ from a similar perspective.

#### Article information

Source
Algebr. Geom. Topol., Volume 11, Number 5 (2011), 2903-2936.

Dates
Revised: 23 June 2011
Accepted: 4 August 2011
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715309

Digital Object Identifier
doi:10.2140/agt.2011.11.2903

Mathematical Reviews number (MathSciNet)
MR2846915

Zentralblatt MATH identifier
1248.53073

Subjects
Primary: 53D42: Symplectic field theory; contact homology
Secondary: 53D12: Lagrangian submanifolds; Maslov index

#### Citation

Dimitroglou Rizell, Georgios. Knotted Legendrian surfaces with few Reeb chords. Algebr. Geom. Topol. 11 (2011), no. 5, 2903--2936. doi:10.2140/agt.2011.11.2903. https://projecteuclid.org/euclid.agt/1513715309

#### References

• P Albers, U Frauenfelder, A nondisplaceable Lagrangian torus in $T^*S^2$, Comm. Pure Appl. Math. 61 (2008) 1046–1051
• Y Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002) 441–483
• T Ekholm, Morse flow trees and Legendrian contact homology in $1$–jet spaces, Geom. Topol. 11 (2007) 1083–1224
• T Ekholm, J B Etnyre, Invariants of knots, embeddings and immersions via contact geometry, from: “Geometry and topology of manifolds”, (H U Boden, I Hambleton, A J Nicas, B D Park, editors), Fields Inst. Commun. 47, Amer. Math. Soc. (2005) 77–96
• T Ekholm, J Etnyre, M Sullivan, The contact homology of Legendrian submanifolds in ${\mathbb R}^{2n+1}$, J. Differential Geom. 71 (2005) 177–305
• T Ekholm, J Etnyre, M Sullivan, Non-isotopic Legendrian submanifolds in $\mathbb R^{2n+1}$, J. Differential Geom. 71 (2005) 85–128
• T Ekholm, J Etnyre, M Sullivan, Orientations in Legendrian contact homology and exact Lagrangian immersions, Int. J. Math. 16 (2005) 453–532
• T Ekholm, J Etnyre, M Sullivan, Legendrian contact homology in $P\times\mathbb R$, Trans. Amer. Math. Soc. 359 (2007) 3301–3335
• Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, from: “GAFA 2000 (Tel Aviv, 1999)”, (N Alon, J Bourgain, A Connes, M Gromov, V Milman, editors), Geom. Funct. Anal. Special Volume, Part II (2000) 560–673
• Y Eliashberg, N Mishachev, Introduction to the $h$–principle, Graduate Studies in Math. 48, Amer. Math. Soc. (2002)
• D Fuchs, D Rutherford, Generating families and Legendrian contact homology in the standard contact space
• 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.
• L Ng, Framed knot contact homology, Duke Math. J. 141 (2008) 365–406
• P E Pushkar$'$, Y V Chekanov, Combinatorics of fronts of Legendrian links, and Arnol$'$d's $4$–conjectures, Uspekhi Mat. Nauk 60 (2005) 99–154
• P Seidel, A long exact sequence for symplectic Floer cohomology, Topology 42 (2003) 1003–1063