相关论文: Ropelength of tight polygonal knots
Primitive path analysis algorithms are now routinely employed to analyze entanglements in computer simulations of polymeric systems, but different analysis protocols result in different estimates of the entanglement length, N_e. Here we…
The space writhe of a knot is a property of its three-dimensional embedding that contains information about its underlying topology, but the correspondence between space writhe and other topological invariants is not fully understood. We…
Global textures are interesting because they are promising candidates for seeds in the structure formation of the universe. The important configurations are those who will be able to collapse. The type of configurations that I will consider…
Complex systems, ranging from soft materials to wireless communication, are often organised as random geometric networks in which nodes and edges evenly fill up the volume of some space. Studying such networks is difficult because they…
We present a computer simulation study of the compact self-avoiding loops as regards their length and topological state. We use a Hamiltonian closed path on the cubic-shaped segment of a 3D cubic lattice as a model of a compact polymer. The…
Computational topology is a vibrant contemporary subfield and this article integrates knot theory and mathematical visualization. Previous work on computer graphics developed a sequence of smooth knots that were shown to converge point wise…
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…
The values of writhe of the most tight conformations, found by the SONO algorithm, of all alternating prime knots with up to 10 crossings are analysed. The distribution of the writhe values is shown to be concentrated around the equally…
Robust topology optimization (RTO), as a class of topology optimization problems, identifies a design with the best average performance while reducing the response sensitivity to input uncertainties, e.g. load uncertainty. Solving RTO is…
We explore several problems related to ruled polygons. Given a ruling of a polygon $P$, we consider the Reeb graph of $P$ induced by the ruling. We define the Reeb complexity of $P$, which roughly equates to the minimum number of points…
We introduce two algorithms for computing tight guarantees on the probabilistic robustness of Bayesian Neural Networks (BNNs). Computing robustness guarantees for BNNs is a significantly more challenging task than verifying the robustness…
We present a fast algorithm for global rigid symmetry detection with approximation guarantees. The algorithm is guaranteed to find the best approximate symmetry of a given shape, to within a user-specified threshold, with very high…
In this paper, we determine geometric information on slope lengths of a large class of knots in the 3-sphere, based only on diagrammatical properties of the knots. In particular, we show such knots have meridian length strictly less than 4,…
We study near-alternating links whose diagrams satisfy conditions generalized from the notion of semi-adequate links. We extend many of the results known for adequate knots relating their colored Jones polynomials to the topology of…
A ribbon is, intuitively, a smooth mapping of an annulus $S^1 \times I$ in 3-space having constant width $\varepsilon$. This can be formalized as a triple $(x,\varepsilon, \mathbf{u})$ where $x$ is smooth curve in 3-space and $\mathbf{u}$…
The problem of modeling ropes arises in many applications, including providing haptic feedback to surgeons who are using surgical robots to realign the distal and proximal ends of split bones. Here, we consider a simplified, 2D variant of…
We introduce a notion of intrinsic linking and knotting for virtual spatial graphs. Our theory gives two filtrations of the set of all graphs, allowing us to measure, in a sense, how intrinsically linked or knotted a graph is; we show that…
Knots are deeply entangled with every branch of science. One of the biggest open challenges in knot theory is to formalise a knot invariant that can unambiguously and efficiently distinguish any two knotted curves. Additionally, the…
We define cylinder knots as billiard knots in a cylinder. We present a necessary condition for cylinder knots: after dividing cylinder knots by possible rotational symmetries we obtain ribbon knots. We obtain an upper bound for the number…
Using a lattice model of polymers in a tube, we define one way to characterise different configurations of a given knot as either "local" or "non-local" and, for several ring polymer models, we provide both theoretical and numerical…