arXiv subjects

Yukihiro Okamoto

Publications and source records attributed to Yukihiro Okamoto.

6 recordsLinked to original sources

arXiv2026

Topological constraints on clean Lagrangian intersections from $\mathbb{Q}$-valued augmentations

Let $K$ be a knot in $\mathbb{R}^3$ which has the $(2,q)$-torus knot for $q\neq \pm 1$ or the figure-eight knot as a component of connected sum. For its conormal bundle $L_K$ in $T^*\mathbb{R}^3$, we show that there is no compactly supported Hamiltonian diffeomorphism $φ$ on $T^*\mathbb{R}^3$ such that $φ(L_K)$ intersects the zero section $\mathbb{R}^3$ cleanly along the unknot in $\mathbb{R}^3$. Using symplectic field theory, the proof is reduced to studying the augmentation variety $V_{\mathbf{k}}(K)$ of $K$ over a filed $\mathbf{k}$. The key point of this paper is finding an algebraic constraint on $V_{\mathbf{k}}(K)$ which is valid only when $\mathbf{k}$ is not algebraically closed, and the proof is completed by some arithmetic argument with $\mathbf{k}=\mathbb{Q}$.

math.SG↗

arXiv2026

Topological constraints on clean Lagrangian intersections via microlocal sheaf theory

Fix a knot $K_0$ in $\mathbb{R}^3$ and consider a Lagrangian submanifold $L$ of $T^*\mathbb{R}^3$ that is isotopic to the conormal bundle of $K_0$ by a compactly supported Hamiltonian isotopy and intersects the zero section $\mathbb{R}^3$ cleanly along a knot. In this paper, using microlocal sheaf theory and some results in $3$-manifold theory, we prove that the knot type of $K_1 := L\cap \mathbb{R}^3$ in $\mathbb{R}^3$ is strictly constrained from the knot type of $K_0$. Specifically, we deduce the existence of a surjective group homomorphism $π_1(\mathbb{R}^3\setminus K_0) \to π_1(\mathbb{R}^3\setminus K_1)$ preserving the longitude and meridian with respect to the Seifert framing. Moreover, combining with a previous work by the second author, we obtain a rigidity result which was only known for the unknot: If $K_0$ is the $(2,q)$-torus knot or the figure-eight knot, $K_1$ must have the same knot type as $K_0$.

math.SG↗

arXiv2026

Non-contractible loops of Legendrian tori from families of knots

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.

math.SG↗

arXiv2025

Legendrian non-isotopic unit conormal bundles in high dimensions

For any compact connected submanifold $K$ of $\mathbb{R}^n$, let $Λ_K$ denote its unit conormal bundle, which is a Legendrian submanifold of the unit cotangent bundle of $\mathbb{R}^n$. In this paper, we give examples of pairs $(K_0,K_1)$ of compact connected submanifolds of $\mathbb{R}^n$ such that $Λ_{K_0}$ is not Legendrian isotopic to $Λ_{K_1}$, although they cannot be distinguished by classical invariants. Here, $K_1$ is the image of an embedding $ι_f \colon K_0 \to \mathbb{R}^n$ which is regular homotopic to the inclusion map of $K_0$ and the codimension in $\mathbb{R}^n$ is greater than or equal to $4$. As non-classical invariants, we define the strip Legendrian contact homology and a coproduct on it under certain conditions on Legendrian submanifolds. Then, we give a purely topological description of these invariants for $Λ_K$ when the codimension of $K$ is greater than or equal to $4$. The main examples $Λ_{K_0}$ and $Λ_{K_1}$ are distinguished by the coproduct, which is computed by using an idea of string topology.

math.SG↗

arXiv2025

On knot types of clean Lagrangian intersections in $T^*\mathbb{R}^3$

Let $K_0$ and $K$ be knots in $\mathbb{R}^3$. Suppose that by a compactly supported Hamiltonian isotopy on $T^*\mathbb{R}^3$, the conormal bundle of $K_0$ is isotopic to a Lagrangian submanifold which intersects the zero section cleanly along $K$. In this paper, we prove some constraints on the pair of knot types of $K_0$ and $K$. One example is that if $K_0$ is the unknot, then $K$ is also the unknot. We also consider some cases where $K_0$ and $K$ have specific knot types, such as torus knots and connected sums of trefoil knots. The key step is finding a DGA map between the Chekanov-Eliashberg DGAs of the unit conormal bundles of knots. The main results are deduced from a relation between the augmentation varieties of $K_0$ and $K$ determined by these DGAs.

math.SG↗

arXiv2023

Toward a topological description of Legendrian contact homology of unit conormal bundles

For a smooth compact submanifold $K$ of a Riemannian manifold $Q$, its unit conormal bundle $Λ_K$ is a Legendrian submanifold of the unit cotangent bundle of $Q$ with a canonical contact structure. Using pseudo-holomorphic curve techniques, the Legendrian contact homology of $Λ_K$ is defined when, for instance, $Q=\mathbb{R}^n$. In this paper, aiming at giving another description of this homology, we define a graded $\mathbb{R}$-algebra for any pair $(Q,K)$ with orientations from a perspective of string topology and prove its invariance under smooth isotopies of $K$. The author conjectures that it is isomorphic to the Legendrian contact homology of $Λ_K$ with coefficients in $\mathbb{R}$ in all degrees. This is a reformulation of a homology group, called string homology, introduced by Cieliebak, Ekholm, Latschev and Ng when the codimension of $K$ is $2$, though the coefficient is reduced from original $\mathbb{Z}[π_1(Λ_K)]$ to $\mathbb{R}$. We compute our invariant (i) in all degrees for specific examples, and (ii) in the $0$-th degree when the normal bundle of $K$ is a trivial $2$-plane bundle.

math.SG↗