Related papers: Ropes, fractions, and moduli spaces
In the paper Blocked-braid Groups, submitted to Applied Categorical Structures, the present authors together with Davide Maglia introduced the blocked-braid groups BB_n on n strands, and proved that a blocked torsion has order either 2 or…
Inspired by artistic practices such as beadwork and himmeli, we study the problem of threading a single string through a set of tubes, so that pulling the string forms a desired graph. More precisely, given a connected graph (where edges…
Curve stitching is a classic educational activity where one constructs elegant curves from a family of straight lines. We perform curve stitching around a circle to make a modular stitch graph. Take $m$ points equally spaced around a…
A special class of braids, called woven, is introduced and it is shown that every conjugation class of the braid group contains woven braids. In consequence, links can be presented as plats or closures of woven braids. Restricting on knots,…
This is the writeup of a lecture given at the May Wisconsin workshop on mathematical aspects of orbifold string theory. In the first part of this lecture, we review recent work on discrete torsion, and outline how it is currently understood…
We show how the theory of tangles is equivalent to that of well-connected tangles. These are drawn on a surface with boundary, and equivalent via Reidemeister moves of a restricted kind. This reworking of the graphical foundations for link…
It is well known that knots are countable in ordinary knot theory. Recently, knots {\it with intersections} have raised a certain interest, and have been found to have physical applications. We point out that such knots --equivalence…
The paper provides bounds for the ropelength of a link in terms of the crossing numbers of its split components. As in earlier papers, the bounds grow with the square of the crossing number; however, the constant involved is a substantial…
Entangled networks are ubiquitous in tissues, polymers, and fabrics. However, their mechanics remain insufficiently understood due to the complexity of the topological constraints at the network level. Here, we develop a mathematical…
In the present paper we give a new method for converting virtual knots and links to virtual braids. Indeed the braiding method given in this paper is quite general, and applies to all the categories in which braiding can be accomplished. We…
Contraction theory is a mathematical framework for studying the convergence, robustness, and modularity properties of dynamical systems and algorithms. In this opinion paper, we provide five main opinions on the virtues of contraction…
We present a simple method to efficiently compute a lower limit of the topological entropy and its spatial distribution for two-dimensional mappings. These mappings could represent either two-dimensional time-periodic fluid flows or…
This article is devoted to the investigation of wrap groups of connected fiber bundles over the fields of real $\bf R$, complex $\bf C$ numbers, the quaternion skew field $\bf H$ and the octonion algebra $\bf O$. Cohomologies of wrap groups…
This is a new and short proof of the main theorem of classical structure tree theory. Namely, we show the existence of certain automorphism-invariant tree-decompositions of graphs based on the principle of removing finitely many edges. This…
The space of elliptic modular forms of fixed weight and level can be identfied with a space of intertwining operators, from a holomorphic discrete series representation of SL2(R) to a space of automorphic forms. Moreover, multiplying…
Pursuing conjectures of John Roe, we use the stable Higson corona of foliated cones to construct a new $K$-theory model for the leaf space of a foliation. This new $K$-theory model is -- in contrast to Alain Connes' $K$-theory model -- a…
We describe moduli spaces of logarithmic rank $2$ connections on elliptic curves with $n \geq 1$ poles and generic residues. In particular, we generalize a previous work by the first and second named authors. Our main approach is to analyze…
In this paper we extend our correlation functions to the open/closed case. This gives rise to actions of an open/closed version of the Sullivan PROP as well as an action of the relevant moduli space. There are several unexpected structures…
Since the braid group was discovered by E. Artin, the question of its conjugacy problem has been solved by Garside and Birman, Ko and Lee. However, the solutions given thus far are difficult to compute with a computer, since the number of…
The Kontsevich integral $Z$ associates to each braid $b$ (or more generally knot $k$) invariants $Z_i(b)$ lying in finite dimensional vector spaces, for $i = 0, 1, 2, ...$. These values are not yet known, except in special cases. The…