Related papers: Studying surfaces via closed braids

We consider links that are alternating on surfaces embedded in a compact 3-manifold. We show that under mild restrictions, the complement of the link decomposes into simpler pieces, generalising the polyhedral decomposition of alternating…

In this paper, we describe the relation between the study of closed connected surfaces embedded in $S^3$ and the theory of handlebody-knots. By Fox's theorem, a pair of handlebody-knots is associated to a closed connected surface embedded…

Inspired by the work of Bemrose et al. \cite{Be16}, we delve into the study of weaving frames in Krein spaces. This paper presents a comprehensive exploration of various properties and characterizations of Krein space weaving frames. In…

Let $S$ and $X$ be two connected topological surfaces without boundary, and assume that $S$ is either of infinite type or has negative Euler characteristic. In this paper, we prove that if $p:S\rightarrow X$ is a fully ramified branched…

We use machine learning to classify examples of braids (or flat braids) as trivial or non-trivial. Our ML takes form of supervised learning using neural networks (multilayer perceptrons). When they achieve good results in classification, we…

This paper studies properly embedded surfaces in the 4-ball that are exotically knotted (i.e., topologically but not smoothly isotopic), and leverages this local phenomenon to study surfaces in larger 4-manifolds. The main results provide a…

A multibranched surface is a 2-dimensional polyhedron without vertices. We introduce moves for multibranched surfaces embedded in a 3-manifold, which connect any two multibranched surfaces with the same regular neighborhoods in finitely…

Many biological examples of fluid-structure interaction, including the transit of red blood cells through the narrow slits in the spleen and the intracellular trafficking of vesicles into dendritic spines, involve the near-contact of…

We prove that, under mild restrictions, the space of codimension-one foliations of degree one on a smooth projective complete intersection has two irreducible components of logarithmic type. We also prove that the same conclusion holds for…

A special class of braids, called woven, is introduced and it is shown that every conjugation class of the braid group contains woven braids. In consequence, links can be presented as plats or closures of woven braids. Restricting on knots,…

A combinatorial condition is obtained for when immersed or embedded incompressible surfaces in compact 3-manifolds with tori boundary components remain incompressible after Dehn surgery. A combinatorial characterisation of hierarchies is…

It is known that the linking form on the 2-cover of slice knots has a metabolizer. We show that several weaker conditions, or some other conditions related to sliceness, do not imply the existence of a metabolizer. We then show how the…

$*$-structures on quantum and braided spaces of the type defined via an R-matrix are studied. These include $q$-Minkowski and $q$-Euclidean spaces as additive braided groups. The duality between the $*$-braided groups of vectors and…

We study the boundary and lens rigidity problems on domains without assuming the convexity of the boundary. We show that such rigidities hold when the domain is a simply connected compact Riemannian surface without conjugate points. For the…

Ozsv\'ath and Szab\'o conjectured that knot Floer homology detects fibred links. We will verify this conjecture for closed 3-braids, by classifying fibred closed 3-braids. In particular, given a nontrivial closed 3-braid, either it is…

This is a case study of the algebraic boundary of convex hulls of varieties. We focus on surfaces in fourspace to showcase new geometric phenomena that neither curves nor hypersurfaces do. Our method is a detailed analysis of a general…

These notes are based on the lectures given by the author during Winter Braids IX in Reims in March 2019. We discuss slice knots and why they are interesting, as well as some ways to decide if a given knot is or is not slice. We describe…

We introduce the concept of pseudo-trisections of smooth oriented compact 4-manifolds with boundary. The main feature of pseudo-trisections is that they have lower complexity than relative trisections for given 4-manifolds. We prove…

Braided sets which are also spaces with dilations are presented and explored in this paper, in the general frame of emergent algebras arxiv:0907.1520. Examples of such spaces are the sub-riemannian symmetric spaces. Keywords: braided sets,…

We show that a closed piecewise-linear hypersurface immersed in $R^n$ ($n\ge 3$) is the boundary of a convex body if and only if every point in the interior of each $(n-3)$-face has a neighborhood that lies on the boundary of some convex…