Normal forms of twisted wire knots

This article is a continuation of the study of new types of knot energy undertaken in [1, 2] (but is formally independent of those articles); it describes some experiments with mechanical models of knots (that we call twisted wire knots), contains rigorous definitions of their mathematical counterparts, formulations of a series of problems and conjectures. Different energy functionals for various classes of knot types and the corresponding normal forms are discussed and compared.     

Tolerance Spaces Revisited I: Almost Solutions

Mathematical Notes - The paper deals with sequences of positive numbers (dn) such that, multiplying the Fourier coefficients (Cn(f)) of functions from given function classes by these numbers, one...     

Tolerance space theory and some applications

The paper deals with the notion of tolerance space (introduced by E. C. Zeeman, but discerned earlier by H. Poincaré), which formalizes the idea of resemblance. The category of tolerance spaces is described, their homology and homotopy theories developed. Applications include almost-fixed point theorems, almost-solution existence theorems for difference schemes and the three-channel principle (a general theorem on multichannel data transmission).     

Bringing Closed Polygonal Curves in the Plane to Normal Form via Local Moves

We define normal forms of regular closed polygonal curves in R², prove that any such curve can be taken to normal form by a regular homotopy, construct two different algorithms (implemented in computer animations) designed to take a given curve to normal form via local moves, present experimental results confirming that this almost always happens, and explain the biological motivation behind the algorithms, as well as their biological interpretation.     

Mechanical normal forms of knots and flat knots

We describe a series of physical experiments with knotted resilient wires and observe that ambient isotopy classes of knots with a small number of crossing have unique normal forms (corresponding to minima of the energy of the wire). We then describe local moves that the wire knots can perform (Reidemeister moves, Markov moves, the Whitney trick). Further, we consider flat knots (i.e. wire knots squeezed between parallel planes) and prove theorems about their mathematical models.     

Elementary Theory of the Alexander–Conway Polynomial

Mathematical Notes -     

Theory of Thickened Knots

Mathematical Notes -     

Configuration spaces of planar mechanical linkages with one degree of freedom

We study, from the kinematic point of view, planar mechanical linkages known as quadrangles (or sometimes as three-bar mechanisms). We discuss geometric structures on their moduli (configuration) spaces and show that they can be classified by means of invariants of Vassiliev type in the space of all moduli spaces.     

Energies of knot diagrams

We introduce and begin the study of new knot energies defined on knot diagrams. Physically, they model the internal energy of thin metallic solid tori squeezed between two parallel planes. Thus the knots considered can perform the second and third Reidemeister moves, but not the first one. The energy functionals considered are the sum of two terms, the uniformization term (which tends to make the curvature of the knot uniform) and the resistance term (which, in particular, forbids crossing changes). We define an infinite family of uniformization functionals, depending on an arbitrary smooth function f and study the simplest nontrivial case f(x) = x ², obtaining neat normal forms (corresponding to minima of the functional) by making use of the Gauss representation of immersed curves, of the phase space of the pendulum, and of elliptic functions.     

Normal Forms of Unknotted Ribbons and DNA

We study a certain class of unknotted smooth embeddings of ribbons (i.e., surfaces diffeomorphic to S¹×[?1,1]) in Euclidean space R³ (unknotted means that the midline of the ribbon is the unknot). Studying them from the mathematical point of view, we classify them. Regarding them as ideal physical objects with certain properties, we study their behavior under natural conditions. Finally, we discuss the eventual relationship of our models with DNA, RNA, and other long molecules appearing in biophysics.