The Four Bars Problem

A four-bar linkage is a mechanism consisting of four rigid bars which are joined by their endpoints in a polygonal chain and which can rotate freely at the joints (or vertices). We assume that the linkage lies in the 2-dimensional plane so that one of the bars is held horizontally fixed. In this paper we consider the problem of reconfiguring a four-bar linkage using an operation called a \emph{pop}. Given a polygonal cycle, a pop reflects a vertex across the line defined by its two adjacent vertices along the polygonal chain. Our main result shows that for certain conditions on the lengths of the bars of the four-bar linkage, the neighborhood of any configuration that can be reached by smooth motion can also be reached by pops. The proof relies on the fact that pops are described by a map on the circle with an irrational number of rotation.Comment: 18 page

An Ergodic Algorithm for Generating Random Knots with a Thickness Constraint

The first algorithm for sampling the space of thick equilateral knots, as a function of thickness, will be described. This algorithm is based on previous algorithms of applying random reflections.To prove the usefulness of the algorithm, we create a method for turning any knot into the regular planar polygon using only thickness non-decreasing moves. This ensures that the algorithm has a positive probability of connecting any two knots with the required thickness constraint and so is ergodic. This ergodic sampling allows us to analyze the effects of thickness on properties of the geometric knot such as radius of gyration and knotting.The data from this algorithm will show that the radius of gyration increases strongly with thickness, in that the growth exponent for radius of gyration increases with thickness. It also shows how knotting is decreased by the addition of a thickness constraint

Geometric-based Optimization Algorithms for Cable Routing and Branching in Cluttered Environments

The need for designing lighter and more compact systems often leaves limited space for planning routes for the connectors that enable interactions among the system’s components. Finding optimal routes for these connectors in a densely populated environment left behind at the detail design stage has been a challenging problem for decades. A variety of deterministic as well as heuristic methods has been developed to address different instances of this problem. While the focus of the deterministic methods is primarily on the optimality of the final solution, the heuristics offer acceptable solutions, especially for such problems, in a reasonable amount of time without guaranteeing to find optimal solutions. This study is an attempt to furthering the efforts in deterministic optimization methods to tackle the routing problem in two and three dimensions by focusing on the optimality of final solutions. The objective of this research is twofold. First, a mathematical framework is proposed for the optimization of the layout of wiring connectors in planar cluttered environments. The problem looks at finding the optimal tree network that spans multiple components to be connected with the aim of minimizing the overall length of the connectors while maximizing their common length (for maintainability and traceability of connectors). The optimization problem is formulated as a bi-objective problem and two solution methods are proposed: (1) to solve for the optimal locations of a known number of breakouts (where the connectors branch out) using mixed-binary optimization and visibility notion and (2) to find the minimum length tree that spans multiple components of the system and generates the optimal layout using the previously-developed convex hull based routing. The computational performance of these methods in solving a variety of problems is further evaluated. Second, the problem of finding the shortest route connecting two given nodes in a 3D cluttered environment is considered and addressed through deterministically generating a graphical representation of the collision-free space and searching for the shortest path on the found graph. The method is tested on sample workspaces with scattered convex polyhedra and its computational performance is evaluated. The work demonstrates the NP-hardness aspect of the problem which becomes quickly intractable as added components or increase in facets are considered

New Directions in Geometric and Applied Knot Theory

The aim of this book is to present recent results in both theoretical and applied knot theory—which are at the same time stimulating for leading researchers in the field as well as accessible to non-experts. The book comprises recent research results while covering a wide range of different sub-disciplines, such as the young field of geometric knot theory, combinatorial knot theory, as well as applications in microbiology and theoretical physics

Elliptic dimers on minimal graphs and genus 1 Harnack curves

This paper provides a comprehensive study of the dimer model on infinite minimal graphs with Fock's elliptic weights [arXiv:1503.00289]. Specific instances of such models were studied in [arXiv:052711, arXiv:1612.09082, arXiv1801.00207]; we now handle the general genus 1 case, thus proving a non-trivial extension of the genus 0 results of [arXiv:math-ph/0202018, arXiv:math/0311062] on isoradial critical models. We give an explicit local expression for a two-parameter family of inverses of the Kasteleyn operator with no periodicity assumption on the underlying graph. When the minimal graph satisfies a natural condition, we construct a family of dimer Gibbs measures from these inverses, and describe the phase diagram of the model by deriving asymptotics of correlations in each phase. In the $\mathbb{Z}^2$-periodic case, this gives an alternative description of the full set of ergodic Gibbs measures constructed in [arXiv:math-ph/0311005] by Kenyon, Okounkov and Sheffield. We also establish a correspondence between elliptic dimer models on periodic minimal graphs and Harnack curves of genus 1. Finally, we show that a bipartite dimer model is invariant under the shrinking/expanding of 2-valent vertices and spider moves if and only if the associated Kasteleyn coefficients are antisymmetric and satisfy Fay's trisecant identity.Comment: 71 pages, 16 figure