## Algebraic & Geometric Topology

### Equivalence classes of augmentations and Morse complex sequences of Legendrian knots

#### Abstract

Let $L$ be a Legendrian knot in $ℝ3$ with the standard contact structure. In earlier work of Henry, a map was constructed from equivalence classes of Morse complex sequences for $L$, which are combinatorial objects motivated by generating families, to homotopy classes of augmentations of the Legendrian contact homology algebra of $L$. Moreover, this map was shown to be a surjection. We show that this correspondence is, in fact, a bijection. As a corollary, homotopic augmentations determine the same graded normal ruling of $L$ and have isomorphic linearized contact homology groups. A second corollary states that the count of equivalence classes of Morse complex sequences of a Legendrian knot is a Legendrian isotopy invariant.

#### Article information

Source
Algebr. Geom. Topol., Volume 15, Number 6 (2015), 3323-3353.

Dates
Revised: 10 April 2015
Accepted: 15 April 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841069

Digital Object Identifier
doi:10.2140/agt.2015.15.3323

Mathematical Reviews number (MathSciNet)
MR3450763

Zentralblatt MATH identifier
1334.57025

#### Citation

Henry, Michael B; Rutherford, Dan. Equivalence classes of augmentations and Morse complex sequences of Legendrian knots. Algebr. Geom. Topol. 15 (2015), no. 6, 3323--3353. doi:10.2140/agt.2015.15.3323. https://projecteuclid.org/euclid.agt/1510841069

#### References

• Y Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002) 441–483
• Y,V Chekanov, Invariants of Legendrian knots, from: “Proceedings of the International Congress of Mathematicians, II”, (T Li, editor), Higher Ed. Press, Beijing (2002) 385–394
• Y,V Chekanov, P,E Pushkar', Combinatorics of fronts of Legendrian links, and Arnol'd's $4$–conjectures, Uspekhi Mat. Nauk 60 (2005) 99–154 In Russian; translated in Russian Math. Surveys 60 (2005) 95–149
• T Ekholm, K Honda, T Kálmán, Legendrian knots and exact Lagrangian cobordisms, preprint
• Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. (2000) 560–673
• Y Félix, S Halperin, J-C Thomas, Differential graded algebras in topology, from: “Handbook of algebraic topology”, (I,M James, editor), North-Holland, Amsterdam (1995) 829–865
• D Fuchs, Chekanov–Eliashberg invariant of Legendrian knots: Existence of augmentations, J. Geom. Phys. 47 (2003) 43–65
• D Fuchs, T Ishkhanov, Invariants of Legendrian knots and decompositions of front diagrams, Mosc. Math. J. 4 (2004) 707–717, 783
• D Fuchs, D Rutherford, Generating families and Legendrian contact homology in the standard contact space, J. Topol. 4 (2011) 190–226
• M,B Henry, Connections between Floer-type invariants and Morse-type invariants of Legendrian knots, Pacific J. Math. 249 (2011) 77–133
• M,B Henry, D Rutherford, A combinatorial DGA for Legendrian knots from generating families, Commun. Contemp. Math. 15 (2013)
• M,B Henry, D Rutherford, Ruling polynomials and augmentations over finite fields, J. Topol. 8 (2015) 1–37
• J Jordan, L Traynor, Generating family invariants for Legendrian links of unknots, Algebr. Geom. Topol. 6 (2006) 895–933
• T Kálmán, Contact homology and one parameter families of Legendrian knots, Geom. Topol. 9 (2005) 2013–2078
• L,L Ng, Computable Legendrian invariants, Topology 42 (2003) 55–82
• L,L Ng, J,M Sabloff, The correspondence between augmentations and rulings for Legendrian knots, Pacific J. Math. 224 (2006) 141–150
• J,M Sabloff, Augmentations and rulings of Legendrian knots, Int. Math. Res. Not. 2005 (2005) 1157–1180
• J,M Sabloff, L Traynor, Obstructions to Lagrangian cobordisms between Legendrians via generating families, Algebr. Geom. Topol. 13 (2013) 2733–2797
• L Traynor, Legendrian circular helix links, Math. Proc. Cambridge Philos. Soc. 122 (1997) 301–314
• L Traynor, Generating function polynomials for Legendrian links, Geom. Topol. 5 (2001) 719–760