Related papers: A Non-classification Result for Wild Knots
We construct a map from knots to (abstract) 2-knots which can be extended to higher dimensions; this map is the natural "knot" counterpart for "braid" theory of groups $G_{n}^{k}$.
We usually construct mathematical objects that are accessible, on which we can put our hands, but a huge part of the mathematical existing is actually wild. Here we explore part of the wild world: its inhabitants are knots that are…
Identifying defective items in larger sets is a main problem with many applications in real life situations. We consider the problem of localizing defective nodes in networks through an approach based on boolean network tomography (BNT),…
We present several results on counting untyped lambda terms, i.e., on telling how many terms belong to such or such class, according to the size of the terms and/or to the number of free variables.
Structured prediction is a powerful framework for coping with joint prediction of interacting outputs. A central difficulty in using this framework is that often the correct label dependence structure is unknown. At the same time, we would…
We prove the Categorified Wrapping Number Conjecture for large classes of annular links, including alternating annular links and tangle closures exhibiting plumbed link phenomena. We do so by characterizing when a resolution is sufficient…
A textile structure is a periodic arrangement of threads in the thickened plane. A topological classification of textile structures is harder than for classical knots and links that are non-periodic and restricted to a bounded region. The…
A relation modification problem gets a logical structure and a natural number k as input and asks whether k modifications of the structure suffice to make it satisfy a predefined property. We provide a complete classification of the…
We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, {\sc…
We show that the problem `whether a finite set of regular-linear axioms defines a rigid theory' is undecidable.
One important issue implied by the finite nature of real-world networks regards the identification of their more external (border) and internal nodes. The present work proposes a formal and objective definition of these properties, founded…
Non-classical virtual knots may have non-isomorphic upper and lower quandles. We exploit this property to define the quandle difference invariant, which can detect non-classicality by comparing the numbers of homomorphisms into a finite…
We consider countable linear orders and study the quasi-order of convex embeddability and its induced equivalence relation. We obtain both combinatorial and descriptive set-theoretic results, and further extend our research to the case of…
Physical knot classification is a challenging fine-grained recognition task in which the intended discriminative cue is rope crossing structure; however, high closed-set accuracy may still arise from low-level appearance shortcuts rather…
In this paper, we give a simple description of the deformations of a map between two smooth curves with partially prescribed branching, in the cases that both curves are fixed, and that the source is allowed to vary. Both descriptions work…
We analyze the connections between the mathematical theory of knots and quantum physics by addressing a number of algorithmic questions related to both knots and braid groups. Knots can be distinguished by means of `knot invariants', among…
This manuscript introduces a new framework for the study of knots by exploring the neighborhood of knot embeddings in the space of simple open and closed curves in 3-space. The latter gives rise to a knotoid spectrum, which determines the…
We address the important question of the extent to which random variables and vectors with truncated power tails retain the characteristic features of random variables and vectors with power tails. We define two truncation regimes, soft…
We examine computer experiments that can be performed to understand the dynamics of knots under self-repulsion. In the course of specific computer exploration we use the knot theory of rational knots and rational tangles to produce classes…
Entanglement detection in high dimensional systems is a NP-hard problem since it is lacking an efficient way. Given a bipartite quantum state of interest free entanglement can be detected efficiently by the PPT-criterion (Peres-Horodecki…