Lagrangian skeleta and plane curve singularities

Abstract

We construct closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities. This yields closed Lagrangian skeleta for Weinstein pairs $$(\mathbb {C}^2,\Lambda )$$ ( C 2 , Λ ) and Weinstein 4-manifolds $$W(\Lambda )$$ W ( Λ ) associated to max-tb Legendrian representatives of algebraic links $$\Lambda \subseteq (\mathbb {S}^3,\xi _\text {st})$$ Λ ⊆ ( S 3 , ξ st ) . We provide computations of Legendrian and Weinstein invariants, and discuss the contact topological nature of the Fomin–Pylyavskyy–Shustin–Thurston cluster algebra associated to a singularity. Finally, we present a conjectural ADE-classification for Lagrangian fillings of certain Legendrian links and list some related problems.

On Newton polytopes of Lagrangian augmentations

“…In the examples above, only finitely many nonorientable Lagrangian fillings can be distinguished. In line with [3], one would naturally conjecture that only finitely many such actually exist. In order to prove Theorem 1.3, we need to construct a different example.…”

“…Recently, [5] constructed Legendrian links in the contact Darboux 3-ball (ℝ 3 , 𝜉 𝑠𝑡𝑑 ) with infinitely many orientable exact Lagrangian fillings, distinct up to compactly supported Hamiltonian isotopy. Further examples of such Legendrians links in (ℝ 3 , 𝜉 𝑠𝑡𝑑 ) were found in [6,11,12,22] using methods from cluster algebras and the microlocal theory of sheaves.…”

“…Then, Φ(𝑁(𝛽 𝑛 )) is the convex hull of the points (1, 0, 0), (𝑛, 0, 0), and (0, 0, 2𝑛) in ℝ 3 ⟨𝑥,𝑦,𝑧⟩ , and it is contained in the ⟨𝑥, 𝑧⟩ plane. Note that Φ(𝑁(𝛽 𝑛 )) has 2𝑛 boundary lattice points and area 𝑛 2 − 𝑛; by Pick's Theorem, Φ(𝑁(𝛽 𝑛 )) has 𝑛 2 − 𝑛 − 1 interior lattice points, and therefore, has 𝑛 2 + 𝑛 − 1 lattice points in total.…”

“…Then, Φ(𝑁(𝛽 𝑛 ) is again the convex hull of the points (1, 0, 0), (𝑛, 0, 0), and (0, 0, 2𝑛) in ℝ 3 ⟨𝑥,𝑦,𝑧⟩ . As above, 𝑖(𝑁(𝛽 𝑛 )) is not unimodular equivalent to 𝑖(𝑁(𝛽 𝑚 (𝑥, 𝑦, 𝑧))) in ℝ 𝑑 if 𝑛 ≠ 𝑚.…”

Braid loops with infinite monodromy on the Legendrian contact DGA

“…Two of these knots, $$10_{139}$$ and $$m(10_{152})$$, are positive braid closures and indeed their Legendrian representatives are rainbow closures of positive braids. We remark that the only other knots with crossing number $$\leqslant 10$$ that are positive braid closures are the torus knots $$T(2,3)$$, $$T(2,5)$$, $$T(2,7)$$, $$T(3,4)$$, $$T(2,9)$$, and $$T(3,5)$$; it is conjectured that the (max‐tb) Legendrian representatives of each of these knots has finitely many fillings [8, Conjecture 5.1].…”