Knotted surfaces as vanishing sets of polynomials

Knotted toroidal sets, attractors and incompressible surfaces

In this paper we give a complete characterization of those knotted toroidal sets that can be realized as attractors for both discrete and continuous dynamical systems globally defined in $\mathbb{R}^3$. We also see that the techniques used to solve this problem can be used to give sufficient conditions to ensure that a wide class of subcompacta of $\mathbb{R}^3$ that are attractors for homeomorphisms must also be attractors for flows. In addition we study certain attractor-repeller decompositions of $\mathbb{S}^3$ which arise naturally when considering toroidal sets.

4-流形中的光滑非平凡且拓扑平凡的零同调曲面

Smoothly knotted and topologically unknotted nullhomologous surfaces in 4-manifolds

本文利用不等价光滑结构，在4-流形中构造无穷多光滑互异但拓扑同伦的零同调曲面，并指出其拓扑边界为柄体。

扭曲Yang-Baxter集、上同调理论及其在纽结中的应用

Twisted Yang-Baxter sets, cohomology theory, and application to knots

该研究引入扭曲Yang-Baxter集并构建拓展的上同调理论，结合扭曲双广群上循环与Alexander编号构造纽结及曲面的状态和不变量，成功区分2-扭转纺三叶结与其反向构型，验证了方法有效性。

纽结环面的分类

Classification of knotted tori

本文针对光滑嵌入 $S^p\times S^q$ 到 $\mathbb R^m$ 的同痕类集合，在维数条件 $m\ge2p+q+3$ 下引入阿贝尔群结构并描述其至扩张问题，利用 Skopenkov 正合序列给出更明确的分类推论。

Universal components of random nodal sets

We give, as $L$ grows to infinity, an explicit lower bound of order $L^{n/m}$ for the expected Betti numbers of the vanishing locus of a random linear combination of eigenvectors of $P$ with eigenvalues below $L$. Here, $P$ denotes an elliptic self-adjoint pseudo-differential operator of order $m\textgreater{}0$, bounded from below and acting on the sections of a Riemannian line bundle over a smooth closed $n$-dimensional manifold $M$ equipped with some Lebesgue measure. In fact, for every closed hypersurface $Σ$ of $\mathbb R^n$, we prove that there exists a positive constant $p\_Σ$ depending…