Operator Formalism for Topology-Conserving Crossing Dynamics in Planar Knot Diagrams

Use of Crossing-State Equivalence Classes for Rapid Relabeling of Knot-Diagrams Representing 21/2D Scenes

In our previous research, we have demonstrated a sophisticated computer-assisted drawing program called Druid, which permits easy construction of 21/2D scenes. A 21/2D scene is a representation of surfaces that is fundamentally two-dimensional, but which also represents the relative depths of those surfaces in the third dimension. This paper improves Druid’s efficiency by exploitating a topolog...

Formalism for interactions: Feynman diagrams

Computing Chebyshev knot diagrams

A Chebyshev curve C(a, b, c, φ) has a parametrization of the form x(t) = Ta(t); y(t) = Tb(t); z(t) = Tc(t + φ), where a, b, c are integers, Tn(t) is the Chebyshev polynomial of degree n and φ ∈ R. When C(a, b, c, φ) is nonsingular, it defines a polynomial knot. We determine all possible knot diagrams when φ varies. Let a, b, c be integers, a is odd, (a, b) = 1, we show that one can list all pos...

Invariants of Knot Diagrams

We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams.

vertex centered crossing number for maximal planar graph

the crossing number of a graph  is the minimum number of edge crossings over all possible drawings of  in a plane. the crossing number is an important measure of the non-planarity of a graph, with applications in discrete and computational geometry and vlsi circuit design. in this paper we introduce vertex centered crossing number and study the same for maximal planar graph.