Legendrian and Transverse Invariants: Chekanov, EHK, and Transverse Invariants¶
Job: develop the Legendrian knots / classical invariants (tb, rot, Maslov, Bennequin), the Chekanov DGA (Reeb chords / holomorphic disks / linearized), the EFH loose vs rigid (dimensions \(\ge5\)), and the transverse knots / self-linking / HF invariants — with the Chekanov pair (same classical, different DGA) as the worked example.
Core¶
Legendrian knots and the classical invariants¶
A Legendrian knot \(L\subset(Y^3,\xi)\) (\(TL\subset\xi\), the maximal integral submanifold of the contact distribution). The classical invariants:
| Classical invariant | Description |
|---|---|
| Topological type | the smooth knot type of \(L\) in \(Y\) |
| Thurston–Bennequin \(\mathrm{tb}(L)\) | the framing of \(L\) relative to \(\xi\) (the contact framing) |
| Rotation number \(\mathrm{rot}(L)\) | the rotation of \(TL\) in \(\xi\) (the Maslov class) |
| Bennequin inequality | $\mathrm{tb}(L)+ |
The Chekanov contact homology and the Legendrian DGA¶
The Chekanov differential graded algebra (the Legendrian contact homology, the DGA generated by the Reeb chords of \(L\) with the differential counting the holomorphic disks in the symplectization, the \(A_\infty\) / the DGA).
| Chekanov DGA | Description |
|---|---|
| DGA | generated by the Reeb chords of \(L\), the differential counts holomorphic disks in the symplectization |
| Chekanov invariants | the DGA the non-classical invariant (Legendrian not distinguished by the classical / the Chekanov distinguishes) |
| Linearized contact homology | the linearization of the DGA (the augmentation, the linearized) |
The Eliashberg–Fraser–Hofer (EFH) and the Legendrian in high dimensions¶
Theorem (EFH). The loose Legendrian in dimensions \(\ge5\) the classification: the two loose Legendrians with the same classical invariants are isotopic, the \(h\)-principle for the loose.
| EFH / loose / rigid | Description |
|---|---|
| EFH | the loose Legendrian in dim \(\ge5\) the \(h\)-principle (same classical \(\Rightarrow\) isotopic) |
| Loose Legendrian | the loose chart, the flexible / no-rigidity (Borman–Eliashberg–Murphy) |
| Rigid Legendrian | the tight / non-loose, the Chekanov / the DGA, the rigidity |
| Dimensions | the loose the \(\ge5\), the 3D the tight / rigid |
Transverse knots and the transverse invariants¶
| Transverse / self-linking / HF | Description |
|---|---|
| Transverse knot \(K\) | \(TK\not\subset\xi\) (everywhere transverse to \(\xi\)) |
| Self-linking \(\mathrm{sl}(K)\) | the framing relative to \(\xi\) (the transverse invariant) |
| Transverse / Legendrian | the transverse the push-off of the Legendrian (\(\mathrm{sl}=\mathrm{tb}-\mathrm{rot}\)) |
| HF / knot Floer | the \(\widehat{\mathrm{HFK}}\) the transverse invariant (Ozsváth–Szabó–Thurston, the LOSS) |
See it¶
Chekanov pair: two Legendrian knots with the same classical invariants but different Chekanov DGA. The Chekanov pair \((L_0,L_1)\) (the two Legendrian representatives of the \(m(5_2)\) knot, the same topological type \(5_2\) and the same \(\mathrm{tb}\) and \(\mathrm{rot}\), but the Chekanov DGA the different — the \(L_0\) and \(L_1\) the non-isotopic Legendrian, the Chekanov the first non-classical invariant). The DGA: the Reeb chords (the crossings of the Lagrangian projection), the differential (the holomorphic disks, the count), the \(L_0\) and the \(L_1\) the different DGA (the algebra the non-isomorphic, the Poincaré polynomial of the linearized the different). The key point: the Chekanov DGA the non-classical Legendrian invariant, distinguishing the Legendrian knots with the same classical invariants — the contact / holomorphic-curve invariant beyond the classical.
Check¶
Show that the Bennequin inequality \(\mathrm{tb}(L)+|\mathrm{rot}(L)|\le-\chi(\Sigma)\) holds in the tight contact structure but fails in the overtwisted.
The Bennequin inequality: for a Legendrian knot \(L\) in a tight contact 3-manifold \((Y,\xi)\), bounding a surface \(\Sigma\), \(\mathrm{tb}(L)+|\mathrm{rot}(L)|\le-\chi(\Sigma)\) (the upper bound on the tb + rot from the Euler characteristic of the spanning surface). The tight: the inequality the consequence of the tightness (the contact structure the no-overtwisted-disk, the Bennequin the original for the standard \(S^3\), the Eliashberg the general tight). The overtwisted: the inequality fails — the overtwisted disk allows the Legendrian knots with the \(\mathrm{tb}\) exceeding the \(-\chi\) (the overtwisted the \(h\)-principle, the no-bound, the Bennequin the failure). So the tight the Bennequin (the upper bound), the overtwisted the failure (the no-bound).
Explain why the Chekanov DGA is a non-classical invariant, and what the Chekanov pair demonstrates.
The classical invariants (the topological type, the tb, the rot) do not distinguish all Legendrian knots — there exist Legendrian knots with the same classical invariants but different Legendrian isotopy types. The Chekanov DGA (the Legendrian contact homology): the DGA generated by the Reeb chords (the crossings of the Lagrangian projection), the differential counting the holomorphic disks in the symplectization. The DGA the non-classical: it depends on the contact / holomorphic-curve data (the Reeb chords, the holomorphic disks), not just the topological / the classical. The Chekanov pair: the two Legendrian representatives of \(m(5_2)\) with the same topological type, tb, and rot, but the different DGA (the algebra non-isomorphic, the linearized the different Poincaré polynomial) — the Chekanov the first non-classical invariant, distinguishing the Legendrian knots the classical cannot.
Pitfalls¶
- The Chekanov DGA the non-classical (the Legendrian not distinguished by the classical / the Chekanov the non-classical) — the DGA the holomorphic-curve invariant, not the topological / classical.
- The EFH the loose in dimensions \(\ge5\) (the 3D the tight / rigid, the loose the higher-D) — the loose the \(h\)-principle the \(\ge5\), the 3D the rigid / the Chekanov.
- The Bennequin inequality the tight (the upper bound), the overtwisted the failure — the tight the rigid / the holomorphic-curve (the Bennequin), the overtwisted the flexible / the \(h\)-principle.
- The transverse / Legendrian correspondence the \(\mathrm{sl}=\mathrm{tb}-\mathrm{rot}\) (the push-off) — the transverse the Legendrian push-off, the self-linking the tb - rot.
- The HF / knot Floer transverse invariant (the Ozsváth–Szabó–Thurston / the LOSS) the Heegaard Floer / the knot Floer, not the Chekanov DGA — the two the different invariants (the DGA the contact homology, the HF the gauge-theoretic).
Refs¶
- Chekanov, "Differential algebra of Legendrian links" (Invent. 2002); Eliashberg–Fraser–Hofer, "On the h-principle for loose Legendrians" (2005)
- Etnyre, "Legendrian and transversal knots" (Handbook of Knot Theory, 2005); Ng, "Computable Legendrian invariants"
- Ozsváth–Szabó–Thurston, "Legendrian knots, transverse knots and combinatorial Floer homology" (2008); LOSS (Lisca–Ozsváth–Stipsicz–Szabó)
- Borman–Eliashberg–Murphy, "Existence and classification of overtwisted contact structures" (2015)
Next¶
CG index · Foundations · Contact Homology & SFT · h-Principle & Flexibility · ECH & Weinstein · Symplectic / Floer · SCV / CR