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On torsion in (bi)linearized Legendrian contact homology in dimension 3

Frédéric Bourgeois, Salammbo Connolly

TL;DR

This work characterizes torsion phenomena in linearized Legendrian contact homology over $\mathbb{Z}$ and completely determines the geography of bilinearized LCH for Legendrian knots in $\mathbb{R}^3$. It proves that the linearized invariant splits as $LCH^{\varepsilon}(\Lambda) \simeq \mathbb{Z}[-1] \oplus F \oplus \overline{F} \oplus T \oplus \overline{T}[1]$, with torsion controlled by a pair of dual modules $T$ and $\overline{T}$, and that the free part is constrained by duality via $r_k = r_{-k}$ for $|k|>1$ and $r_1 = 1 + r_{-1}$. For bilinearized LCH, the invariant takes the form $LCH^{\varepsilon_1,\varepsilon_2}(\Lambda) \simeq \mathbb{Z}[0] \oplus F \oplus T$, with $F$ of even rank, and the authors construct explicit knots realizing arbitrary prescribed free and torsion data, using a toolkit of connected sums, copies of the Legendrian unknot, and unclasp moves. The results yield a complete (geographic) picture of what torsion can occur and how to realize it in dimension 3, offering a concrete set of building blocks to realize any allowable pair of invariants. This advances understanding of Legendrian invariants with integral coefficients and provides new methods to manipulate and realize prescribed torsion in both linearized and bilinearized frameworks.

Abstract

Linearized Legendrian contact homology (LCH) and bilinearized LCH are important homological invariants for Legendrian submanifolds in contact geometry. For legendrian knots in $\mathbb{R}^3$, very little was previously known about the possibility of having torsion in these invariants when they are defined over integer coefficients. In this paper, we give properties of torsion that can appear in linearized LCH with integer coefficients, and also give the full geography of bilinearized LCH with integer coefficients.

On torsion in (bi)linearized Legendrian contact homology in dimension 3

TL;DR

This work characterizes torsion phenomena in linearized Legendrian contact homology over and completely determines the geography of bilinearized LCH for Legendrian knots in . It proves that the linearized invariant splits as , with torsion controlled by a pair of dual modules and , and that the free part is constrained by duality via for and . For bilinearized LCH, the invariant takes the form , with of even rank, and the authors construct explicit knots realizing arbitrary prescribed free and torsion data, using a toolkit of connected sums, copies of the Legendrian unknot, and unclasp moves. The results yield a complete (geographic) picture of what torsion can occur and how to realize it in dimension 3, offering a concrete set of building blocks to realize any allowable pair of invariants. This advances understanding of Legendrian invariants with integral coefficients and provides new methods to manipulate and realize prescribed torsion in both linearized and bilinearized frameworks.

Abstract

Linearized Legendrian contact homology (LCH) and bilinearized LCH are important homological invariants for Legendrian submanifolds in contact geometry. For legendrian knots in , very little was previously known about the possibility of having torsion in these invariants when they are defined over integer coefficients. In this paper, we give properties of torsion that can appear in linearized LCH with integer coefficients, and also give the full geography of bilinearized LCH with integer coefficients.
Paper Structure (21 sections, 8 theorems, 69 equations, 16 figures)

This paper contains 21 sections, 8 theorems, 69 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Linearized and bilinearized Legendrian contact homology
  4. Chekanov-Eliashberg DGA
  5. Linearized LCH
  6. Bilinearized LCH
  7. Duality
  8. Connected sums
  9. Torsion in linearized Legendrian contact homology
  10. Constraints on the linearized LCH module
  11. Construction of torsion in linearized LCH
  12. Construction of the free part of linearized LCH
  13. Proof of Theorem \ref{['thm:linLCH']}
  14. Torsion in bilinearized Legendrian contact homology
  15. Bilinearized Legendrian contact homology of the trefoil knot
  16. ...and 6 more sections

Key Result

Theorem 1.1

For any Legendrian knot $\Lambda$ in $(\mathbb{R}^3, \xi_{\rm std})$ and any $\mathbb{Z}$-valued augmentation $\varepsilon$ of its Chekanov-Eliashberg DGA, their linearized Legendrian contact homology has the form where $F$ is a finitely generated graded free $\mathbb{Z}$-module and $T$ is a finitely generated graded torsion $\mathbb{Z}$-module. Conversely, for any finitely generated graded free

Figures (16)

  • Figure 1: Connected sum of two Legendrian knots.
  • Figure 2: Front projection of the trefoil knot.
  • Figure 3: Link between the trefoil knot and a legendrian unknot.
  • Figure 4: Legendrian Hopf link before connected sum.
  • Figure 5: 6-copy of the Legendrian unknot.
  • ...and 11 more figures

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Proposition 2.3
  • proof
  • Example 4.1
  • Lemma 4.2
  • ...and 6 more