Exotically knotted disks and complex curves

exotic 结合的 2 球面及其补集的基本群

Exotically knotted 2-spheres and the fundamental groups of their complements

该研究展示了一种构造包含特定 2 链的 4 维 manifold 的方法，其 2 链群可为任意有限生成群，并证明这些链的组成部分具有空同源性质。

来自族版本Donaldson对角化的奇异纽结闭曲面

Exotically knotted closed surfaces from Donaldson's diagonalization for families

本研究利用族版本Donaldson对角化定理，提出无需显式用4-流形不变量的奇异曲面检测法，构造出首个补空间微分同胚的不可定向奇异纽结曲面例子，并得到多类奇异嵌入的相关结果。

Knotted surfaces in 4-manifolds and stabilizations

In this paper, we study stable equivalence of exotically knotted surfaces in 4-manifolds, surfaces that are topologically isotopic but not smoothly isotopic. We prove that any pair of embedded surfaces in the same homology class become smoothly isotopic after stabilizing them by handle additions in the ambient 4-manifold, which can moreover assumed to be attached in a standard way (locally and unknottedly) in many favorable situations. In particular, any exotically knotted pair of surfaces with cyclic fundamental group complements become smoothly isotopic after a same number of standard stabil…

一次稳定化对于闭纽结曲面是不够的

One stabilization is not enough for closed knotted surfaces

本文证明存在包含成对异构纽结2-球面的光滑4-流形，其异构性在一次稳定化后仍保持，表明“一次即足够”定理不适用于同调类为特征类的闭曲面。

Khovanov homology and exotic surfaces in the 4-ball

We show that the cobordism maps on Khovanov homology can distinguish smooth surfaces in the 4-ball that are exotically knotted (i.e., isotopic through ambient homeomorphisms but not ambient diffeomorphisms). We develop new techniques for distinguishing cobordism maps on Khovanov homology, drawing on knot symmetries and braid factorizations. We also show that Plamenevskaya's transverse invariant in Khovanov homology is preserved by maps induced by positive ascending cobordisms.

Bridge trisections of knotted surfaces in 4--manifolds

We prove that every smoothly embedded surface in a 4--manifold can be isotoped to be in bridge position with respect to a given trisection of the ambient 4--manifold; that is, after isotopy, the surface meets components of the trisection in trivial disks or arcs. Such a decomposition, which we call a \emph{generalized bridge trisection}, extends the authors' definition of bridge trisections for surfaces in $S^4$. Using this new construction, we give diagrammatic representations called \emph{shadow diagrams} for knotted surfaces in 4--manifolds. We also provide a low-complexity classification f…