Related papers: Nullification of knots and links
Knots naturally appear in continuous dynamical systems as flow periodic trajectories. However, discrete dynamical systems are also closely connected with the theory of knots and links. For example, for Pixton diffeomorphisms, the…
We prove that for knots, the evaluation of the Jones polynomial at the sixth root of unity, as well as the evaluation of the $Q$-polynomial at the reciprocal of the golden ratio, are uniquely determined by the oriented homeomorphism type of…
The art of tying knots is exploited in nature and occurs in multiple applications ranging from being an essential part of scouting programs to engineering molecular knots. Biomolecular knots, such as knotted proteins, bear various cellular…
Algorithm of construction of all knots, links with given number of crosses on diagram of knot, link is offered. This algorithm is based on simple proposition, that there is a representation of knot (link) as closure of braid with n threads…
We study symmetric crossing change operations for strongly invertible knots. Our main theorem is that the most natural notion of equivariant unknotting number is not additive under connected sum, in contrast with the longstanding conjecture…
Knotted and tangled structures frequently appear in physical fields, but so do mechanisms for untying them. To understand how this untying works, we simulate the behavior of 1,458 superfluid vortex knots of varying complexity and scale in…
Networks play a prominent role in the study of complex systems of interacting entities in biology, sociology, and economics. Despite this diversity, we demonstrate here that a statistical model decomposing networks into matching and…
We introduce a new numerical knot invariant, termed the \textit{segment number}, which is derived from partitioned knot diagrams subject to specific over/under-crossing constraints. We prove that a knot is non-trivial if and only if its…
A classical knot is described by a one-stroke trajectory with entanglements of a string. The replica method appears as a powerful tool in statistical mechanics for a polymer or self-avoiding walk. We consider this replica N to 0 limit in…
In this paper we study the theory of {\it pseudo knots}, which are knots with some missing crossing information, and we introduce and study the theory of {\it pseudo tied links} and the theory of {\it pseudo knotoids}. In particular, we…
All knots are fused isotopic to the unknot via a process known as virtualization. We extend and adapt this process to show that, up to fused isotopy, classical links are classified by their linking numbers.
This paper discusses some geometric ideas associated with knots in real projective 3-space $\mathbb{R}P^3$. These ideas are borrowed from classical knot theory. Since knots in $\mathbb{R}P^3$ are classified into three disjoint classes, -…
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…
Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into $R^2$, such that no more than two points project to the same point in $R^2$. These diagrams are drawings of 4-regular plane multigraphs. Knots are…
Topological entanglements are abundant, and often detrimental, in polymeric systems in biology and materials science. Here we theoretically investigate the topological simplification of knots by diffusing slip-links (SLs), which may…
Recent progress in string theory has led to a reformulation of quantum-group polynomial invariants for knots and links into new polynomial invariants whose coefficients can be understood in topological terms. We describe in detail how to…
This paper contains linear systems of equations which can distinguish knots without knot invariants. Let $M_n$ be the topological moduli space of all n-component string links and such that a fixed projection into the plane is an immersion.…
Grid diagrams with their relatively simple mathematical formalism provide a convenient way to generate and model projections of various knots. It has been an open question whether these 2D diagrams can be used to model a complex 3D process…
A reliable inference of networks from data is of key interest in the Neurosciences. Several methods have been suggested in the literature to reliably determine links in a network. To decide about the presence of links, these techniques rely…
We will strengthen the known upper and lower bounds on the delta-crossing number of knots in therms of the triple-crossing number. The latter bound turns out to be strong enough to obtain (unknown values of) triple-crossing numbers for a…