Related papers: Knots with compressible thin levels
We estimate by Monte Carlo simulations the configurational entropy of $N$-steps polygons in the cubic lattice with fixed knot type. By collecting a rich statistics of configurations with very large values of $N$ we are able to analyse the…
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…
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…
We study by Monte Carlo simulations a model of knotted polymer ring adsorbing onto an impenetrable, attractive wall. The polymer is described by a self-avoiding polygon (SAP) on the cubic lattice. We find that the adsorption transition…
The slicing degree of a knot $K$ is defined as the smallest integer $k$ such that $K$ is $k$-slice in $\#^n \overline{\mathbb{CP}^2}$ for some $n$. In this paper, we establish bounds for the slicing degrees of knots using Rasmussen's…
We prove lower bounds for the entanglement of formation and the squashed entanglement for any a bipartite density matrix in terms of the conditional entropy of the bipartite state with respect to either of its partial traces, and prove that…
The problem of finding robust and effective methods for locating entanglement in embedded curves is relevant to both applications and theoretical investigations. Rather than focusing on an exact determination, we introduce the knot…
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…
The entropic pressure in the vicinity of a cubic lattice knot is examined as a model of the entropic pressure near a knotted ring polymer in a good solvent. A model for the scaling of the pressure is developed and this is tested numerically…
This paper employs various computational techniques to determine the bridge numbers of both classical and virtual knots. For classical knots, there is no ambiguity of what the bridge number means. For virtual knots, there are multiple…
Frequently, knots are enumerated by their crossing number. However, the number of knots with crossing number $c$ grows exponentially with $c$, and to date computer-assisted proofs can only classify diagrams up to around twenty crossings.…
Let M be $S^3$, $S^1\times S^2$, or a lens space L(p,q), and let k be a (1,1)-knot in M, i.e., a knot which is of 1-bridge with respect to a Heegaard torus. We show that if there is a closed meridionally incompressible surface in the…
The viscous flow of polymer chains in dense melts is dominated by topological constraints whenever the single chain contour length, N, becomes larger than the characteristic scale Ne, defining comprehensively the macroscopic rheological…
We consider diagrams of links in $S^2$ obtained by projection from $S^3$ with the Hopf map and the minimal crossing number for such diagrams. Knots admitting diagrams with at most one crossing are classified. Some properties of these knots…
It is a consequence of theorems of Gordon-Reid [Tangle decompositions of tunnel number one knots and links, J. Knot Theory and its Ramifications, 4 (1995) 389-409] and Thompson [Thin position and bridge number for knots in the 3-sphere,…
We classify all knot diagrams of genus two and three, and give applications to positive, alternating and homogeneous knots, including a classification of achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a proof…
In this paper we compute the sharp lower bounds for the crossing number of $n$-string $k$-loop essential tangles. For essential tangles with only string components, we characterise the ones with the minimum crossing number for a given…
For any given number of crossings $c$, there exists a formula to determine the number of 2-bridge knots of $c$ crossings, and indeed it is a simple matter to actually construct presentations of these knots. However, the determination of…
This paper introduces a new approach to finding knots and links with hidden symmetries using "hidden extensions", a class of hidden symmetries defined here. We exhibit a family of tangle complements in the ball whose boundaries have…
We define new invariants of knots by means of quandle colorings and longitudinal information. These invariants can be applied to a tangle embedding problem and recognizing non-classical virtual knots.