Related papers: Hard Unknots and Collapsing Tangles
Quantum entanglement is a particularly useful characterization of topological orders which lack conventional order parameters. In this work, we study the entanglement in topologically ordered states between two arbitrary spatial regions,…
We study properties of random subcomplexes of partitions returned by (a suitable form of) the Strong Hypergraph Regularity Lemma, which we call regular slices. We argue that these subcomplexes capture many important structural properties of…
We prove that any link admitting a diagram with a single negative crossing is strongly quasipositive. This answers a question of Stoimenow's in the (strong) positive. As a second main result, we give simple and complete characterizations of…
We show that the problem of constructing a real rational knot of a reasonably low degree can be reduced to an algebraic problem involving the pure braid group: expressing an associated element of the pure braid group in terms of the…
The state-sum invariants for knots and knotted surfaces defined from quandle cocycles are described using the Kronecker product between cycles represented by colored knot diagrams and a cocycle of a finite quandle used to color the diagram.…
In the present paper a criteria for a rectangular diagram to admit a simplification is given in terms of Legendrian knots. It is shown that there are two types of simplifications which are mutually independent in a sense. A new proof of 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…
A $1$-factorization of the complete multigraph $\lambda K_{2n}$ is said to be indecomposable if it cannot be represented as the union of $1$-factorizations of $\lambda_0 K_{2n}$ and $(\lambda-\lambda_0) K_{2n}$, where $\lambda_0<\lambda$.…
We consider homologically essential simple closed curves on Seifert surfaces of genus one knots in $S^3$, and in particular those that are unknotted or slice in $S^3$. We completely characterize all such curves for most twist knots: they…
We show that there are links whose individual components are concordant to the unknot, but which are not concordant to any link with unknotted components. We give examples in the topological category, and examples in the smooth category…
We develop a topological model of knots and links arising from a single (or multiple processive) round(s) of recombination starting with an unknot, unlink, or (2,m)-torus knot or link substrate. We show that all knotted or linked products…
Recent advances in Quantum Topology assign $q$-series to knots in at least three different ways. The $q$-series are given by generalized Nahm sums (i.e., special $q$-hypergeometric sums) and have unknown modular and asymptotic properties.…
This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants. For a knot K in the 3-sphere let \pi_K be…
A rectangular drawing of a planar graph $G$ is a planar drawing of $G$ in which vertices are mapped to grid points, edges are mapped to horizontal and vertical straight-line segments, and faces are drawn as rectangles. Sometimes this latter…
Ever since entanglement was identified as a computational and cryptographic resource, researchers have sought efficient ways to tell whether a given density matrix represents an unentangled, or separable, state. This paper gives the first…
Recently Iltgen, Lewark and Marino introduced the concept of a proper rational tangle replacement and the corresponding notion of the proper rational unknotting number. In this note we derive a version of the Montesinos trick for proper…
In Classical Knot Theory and in the new Theory of Quantum Invariants substantial effort was directed toward the search for unknotting moves on links. We solve, in this note, several classical problems concerning unknotting moves. Our…
We study congruences involving truncated hypergeometric series of the form_rF_{r-1}(1/2,...,1/2;1,...,1;\lambda)_{(mp^s-1)/2} = \sum_{k=0}^{(mp^s-1)/2} ((1/2)_k/k!)^r \lambda^k where p is a prime and m, s, r are positive integers. These…
Zeckendorf's theorem states every positive integer has a unique decomposition as a sum of non-adjacent Fibonacci numbers. This result has been generalized to many sequences $\{a_n\}$ arising from an integer positive linear recurrence, each…
We study the structure of the augmented fundamental quandle of a knot whose complement contains an incompressible torus. We obtain the relationship between the fundamental quandle of a satellite knot and the fundamental quandles/groups of…