Related papers: On detecting the trivial rational $3$-tangle

We study random knots, which we define as a triple of random periodic functions (where a random function is a random trigonometric series, \[f(\theta) = \sum_{k=1}^\infty a_k \cos (k \theta) +b_k (\sin k \theta),\] with $a_k, b_k$ are…

The $k$-truss, introduced by Cohen (2005), is a graph where every edge is incident to at least $k$ triangles. This is a relaxation of the clique. It has proved to be a useful tool in identifying cohesive subnetworks in a variety of…

The ability to compare complex systems can provide new insight into the fundamental nature of the processes captured in ways that are otherwise inaccessible to observation. Here, we introduce the $n$-tangle method to directly compare two…

There is a natural way to associate with a transformation of an isotopy class of rational tangles to another, an element of the modular group. The correspondence between the isotopy classes of rational tangles and rational numbers follows,…

In this paper we study the operation of cutting off edges of a simple $3$-polytope $P$ along the graph $\Gamma$. We give the criterion when the resulting polytope is simple and when it is flag. As a corollary we prove the analog of…

We study tangle replacement in the context of spatial graphs. The main results show that, for certain spatial handcuff graphs, there is a one-to-one correspondence between the neighborhood equivalence classes of the spatial graphs obtained…

There is a map, defined and studied by Jones, from Thompson's group $F$ to knots. Jones proved that every knot is in the image of this map -- that is, that every knot can be seen as the "knot closure" of a Thompson group element. We…

We discuss the problem of classifying birational extremal contractions of smooth threefolds where the canonical bundle is trivial along the curves contracted, in the case when a divisor is contracted to a point. We prove the analytic…

Let $F$ be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle $(B,T)$. Then $F$ separates the strings of $T$ in $B$ and the boundary slope of $F$ is…

We study algebraic tangles as fundamental components in knot theory, developing a systematic approach to classify and tabulate prime tangles using a novel canonical representation. The canonical representation enables us to distinguish…

In this paper we classify, up to rigid isotopy, non-singular real rational curves of degrees less than or equal to 6 in a quadric homeomorphic to the 3-sphere. We also study their connections with rigid isotopy classes of real rational…

In this paper, we show the trivializing number of all minimal diagrams of positive 2-bridge knots and study the relation between the trivializing number and the unknotting number for a part of these knots.

This paper is concerned with configurations of points in a plane lattice which determine angles that are rational multiples of $\pi$. We shall study how many such angles may appear in a given lattice and in which positions, allowing the…

Periodic networks serve as models for the structural organisation of biological and chemical crystalline systems. Single or multiple networks can have different configurations in space, where entanglement may arise due to the way the…

The notion of a pseudoknot is defined as an equivalence class of knot diagrams that may be missing some crossing information. We provide here a topological invariant schema for pseudoknots and their relatives, 4-valent rigid vertex spatial…

We use the Chebyshev knot diagram model of Koseleff and Pecker in order to introduce a random knot diagram model by assigning the crossings to be positive or negative uniformly at random. We give a formula for the probability of choosing a…

We present a practical algorithm to test whether a 3-manifold given by a triangulation or an ideal triangulation contains a closed essential surface. This property has important theoretical and algorithmic consequences. As a testament to…

The Meridional Rank Conjecture asks whether the bridge number of a knot in $S^3$ is equal to the minimal number of meridians needed to generate the fundamental group of its complement. In this paper we investigate the analogous conjecture…

We initiate the study of combinatorial algorithms for Triangle Detection in $H$-free graphs. The goal is to decide if a graph that forbids a fixed pattern $H$ as a subgraph contains a triangle, using only "combinatorial" methods that…

We give three algorithms to determine the crosscap number of a knot in the 3-sphere using $0$-efficient triangulations and normal surface theory. Our algorithms are shown to be correct for a larger class of complements of knots in closed…