相关论文: Knotted optical fields: Seifert fibrations, braide…
Knots and links are fascinating and intricate topological objects. Their influence spans from DNA and molecular chemistry to vortices in superfluid helium, defects in liquid crystals and cosmic strings in the early universe. Here, we find…
With its boundary tracing out a link or knot in 3D, the Seifert surface is a 2D surface of core importance to topological classification. We propose the first-ever experimentally realistic setup where Seifert surfaces emerge as the boundary…
In this article, we propose a new approach for describing and understanding knots and links in a 3-manifold through the use of an embedded non-orientable surface. Specifically, we define a plat-like representation based on this…
Knots and links represent a fundamental motif of non-local connectivity that permeates the physical sciences from string theory to protein folds. While spectral braiding has been explored in two-band non-Hermitian models across various…
Ozsv\'ath and Szab\'o conjectured that knot Floer homology detects fibred knots in $S^3$. We will prove this conjecture for null-homologous knots in arbitrary closed 3--manifolds. Namely, if $K$ is a knot in a closed 3--manifold $Y$, $Y-K$…
In this paper we present a detailed study of \emph{bonded knots} and their related structures, integrating recent developments into a single framework. Bonded knots are classical knots endowed with embedded bonding arcs modeling physical or…
Knots are familiar entities that appear at a captivating nexus of art, technology, mathematics, and science. As topologically stable objects within field theories, they have been speculatively proposed as explanations for diverse persistent…
Sutured manifolds defined by Gabai are useful in the geometrical study of knots and 3-dimensional manifolds. On the other hand, homology cylinders are in an important position in the recent theory of homology cobordisms of surfaces and…
This work seeks to make explicit the operational connection between the preparation of two-level quantum systems with their corresponding description (as states) in a Hilbert space. This may sound outdated, but we show there is more to this…
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…
The Skyrme-Faddeev model is a three-dimensional non-linear field theory that has topological soliton solutions, called hopfions, which are novel string-like solutions taking the form of knots and links. Solutions found thus far take the…
The deep connection among braids, knots and topological physics has provided valuable insights into studying topological states in various physical systems. However, identifying distinct braid groups and knot topology embedded in…
We consider the question of when is the closed manifold obtained by elementary surgery on an $n$-knot Seifert fibred over a 2-orbifold. After some observations on the classical case, we concentrate on the cases n=2 and 3. We have found a…
Construction of (colored) knot polynomials for double-fat graphs is further generalized to the case when "fingers" and "propagators" are substituting R-matrices in arbitrary closed braids with m-strands. Original version of arXiv:1504.00371…
Knotted fields enrich a variety of physical phenomena, ranging from fluid flows, electromagnetic fields, to textures of ordered media. Maxwell's electrostatic equations, whose vacuum solution is mathematically known as a harmonic field,…
Bialgebras and Hopf (bi)modules are typical algebraic structures with several interacting operations. Their structural and homological study is therefore quite involved. We develop the machinery of braided systems, tailored for handling…
An interesting question is whether two 3-manifolds can be distinguished by computing and comparing their collections of finite covers; more precisely, by the profinite completions of their fundamental groups. In this paper, we solve this…
Classical knot theory deals with {\em diagrams} and {\em invariants}. By means of horizontal {\em trisecants}, we construct a new theory of classical braids with invariants valued in {\em pictures}. These pictures are closely related to…
In this paper we find a family of knots with trivial Alexander polynomial, and construct two non-isotopic Seifert surfaces for each member in our family. In order to distinguish the surfaces we study the sutured Floer homology invariants of…
We give an explicit description of a fibration of the complement of the closure of a homogeneous braid, understanding how each fiber intersects every cross-section of $S^3$.