Non-contractible loops of Legendrian tori from families of knots

TL;DR

The paper addresses whether loops of knots give nontrivial loops of their unit conormal Legendrian tori in $U^*\mathbb{R}^3$ by tracing monodromy on Legendrian contact homology in degree $0$. It develops a framework linking knot cord algebras, string homology, and Legendrian contact homology through parametrized moduli spaces and exact Lagrangian cobordisms, proving monodromy can be detected topologically. The main result produces an infinite family of non-contractible loops of Legendrian tori that are smoothly contractible, by exhibiting nontrivial automorphisms on cord algebras via explicit knot-loop constructions (blue-box loops). This work thus connects knot-theoretic invariants to Legendrian topology, providing tools to distinguish Legendrian loops that are invisible to smooth isotopy. The findings offer new insights into the interplay between contact topology and knot theory, with potential extensions to more general cord algebras and higher-dimensional conormal constructions.

Abstract

In the unit cotangent bundle of $\mathbb{R}^3$, we consider loops of Legendrian tori arising as families of the unit conormal bundles of smooth knots in $\mathbb{R}^3$. In this paper, using the cord algebra of knots, we give a topological method to compute the monodromy on the Legendrian contact homology in degree $0$ induced by those loops. As an application, we obtain an infinite family of non-contractible loops of Legendrian tori which are contractible in the space of smoothly embedded tori.

Figures (10)

• Figure 3.1: The image of $u$ in $T^*\mathbb{R}^3$. $a$ is a Reeb chord of $\Lambda$. The black boundaries are contained $L_b$ and the red boundaries are contained in $\mathbb{R}^3$. The switching points $p_1,\dots ,p_{2l}$ are mapped to $L_b\cap \mathbb{R}^3 =K_b$.
• Figure 3.2: $L_b$ contains a cylindrical region identified with $[0,\infty)\times \Lambda$ via $E\circ \tau_{r'}$. $L^s_b$ is an exact Lagrangian filling of $\Lambda$ obtained by replacing the cylindrical region with $M_s\cap ([0,\infty)\times U^*\mathbb{R}^3)$. In the right picture, $P^0_a\coloneqq (2r_1,a(0))$ and $P^1_a\coloneqq (2r_1,a(T_a))$.
• Figure 3.3: The leftmost picture represents a sequence $(b_n,u_n,\kappa_n)_{n=1,2,\dots}$ in $\mathcal{M}_{\mathscr{L},l}(a)$ when $l=2$. Taking a subsequence if necessary, such a sequence can converge to a $J'_{\rho}$-holomorphic curve $u$ with a marked point $z$ on the boundary such that $u(z)\in K_b$ for $b=\lim_{n\to \infty}b_n$. (We may think of it as $J'_{\rho}$-holomorphic curve with a boundary node at $z$ such that a constant disk at $u(z)$ is attached to $u$ at the node.) A priori, the sequence may converge to a $J'_{\rho}$-holomorphic building as in the rightmost picture, but this cannot happen since $\bar{\mathcal{M}}_{\Lambda_{K_0},J}(a;a_1,\dots,a_m)=\emptyset$ when $|a|=0$.
• Figure 4.1: The relations generating the ideal $\mathcal{I}$. '$\simeq$' means a homotopy equivalence.
• Figure 5.1: When $b=0$, $L_0 = L_{K_0}$ (the left picture). When $b\in [0,1]$, $L_b$ is an exact Lagrangian filling of $\Lambda_{K_0}$ obtained by concatenating the conormal bundle $L_{K_{t(b)}}$ with $\widetilde{L}_{t(b)}$ via $E$ (the middle picture). When $b\geq 1$, $L_b$ is given by a concatenation of $L_{K_1}$ and $\widetilde{L}_1$, with the shaded cylindrical region stretched as $b\to \infty$ (the right picture).

Theorems & Definitions (66)

• Definition 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Remark 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• Lemma 2.4