Lagrangian Approximation of Totally Real Concordances

TL;DR

The paper proves a low-dimensional h-principle for totally real concordances: after stabilising Legendrian boundaries with sufficiently many positive and negative moves, a totally real two-dimensional concordance in a four-dimensional symplectisation can be C^0-approximated by a genuine Lagrangian concordance. This flexibility enables the construction of knotted Lagrangian cylinders in arbitrary four-dimensional symplectisations and knotted Lagrangian tori in symplectisations of overtwisted contact 3-manifolds. It also shows smooth unknottedness in the unstabilised standard-unknot setting, underscoring the necessity of both-sign stabilisations for knotted phenomena. The methods combine h-principle flexibility for totally real submanifolds with a precise control of Legendrian stabilisations, fundamental-group computations of complements, and a careful Lagrangian extension via a standard neighbourhood and cobordism technique. Collectively, the results expand the landscape of knotted Lagrangians in low-dimensional symplectic and contact topology and provide a toolkit for generating infinite families of knotted Lagrangian structures.

Abstract

We show that a two-dimensional totally real concordance can be approximated by a Lagrangian concordance whose Legendrian boundary has been stabilised both positively and negatively sufficiently many times. The main applications that we provide are constructions of knotted Lagrangian concordances in arbitrary four-dimensional symplectiations, as well as of knotted Lagrangian tori in symplectisations of overtwisted contact three-manifolds.

Figures (5)

• Figure 1: Left: An unknotted arc. Right: A knotted arc that coincides with $\{p_{\theta}=p_x=0\}$ near the boundary and which is nowhere tangent to $\partial_{p_{\theta}}$.
• Figure 2: A long properly embedded knotted arc $A \subset \mathbb{R}^3$ and a knot $K_A \subset \mathbb{R}^3$ obtained by closing up the arc; the fundamental groups of the complements in $\mathbb{R}^3$ of these two submanifolds are isomorphic.
• Figure 3: Left: A convex annulus with Legendrian boundary and six dividing curves, each with boundary intersecting both components of the boundary of the annulus. Right: An extension of the annulus along one of the boundary component $\Lambda$ (shown in blue on the left), exhibiting a positive and a negative stabilisation of that Legendrian. The destabilised Legendrian is $\Lambda'$ shown in blue on the right.
• Figure 4: Top: the front projection of a negative and positive stabilisation, shown on the left and right, respectively. Bottom: the corresponding Lagrangian projection. The stabilisation of either sign contributes with the self-linking number $-1$ for the framing given by the Reeb vector field (i.e. the black board framing with respect to the Lagrangian projection), as can be computed e.g. using the lower knot diagram.
• Figure 5: Two canceling coil springs. The zig-zags on the left are positive stabilisations (they increase the rotation number of the Lagrangian projection shown below) while the zig-zags on the right are negative stabilisations (they decrease the rotation number). Here each loop is $W$-fold covered with $W=7$.

Theorems & Definitions (32)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.5
• Theorem 1.6
• Lemma 3.1
• proof
• Lemma 3.2
• Corollary 3.3
• Corollary 3.4