Cubical geometry in the polygonalisation complex

Dipaths and dihomotopies in a cubical complex

In the geometric realization of a cubical complex without degeneracies, a 2-set, dipaths and dihomotopies may not be combinatorial, i.e., not geometric realizations of combinatorial dipaths and equivalences. When we want to use geometric/topological tools to classify dipaths on the 1-skeleton, combinatorial dipaths, up to dihomotopy, and in particular up to combinatorial dihomotopy, we need tha...

Vassiliev invariants and the cubical knot complex Ilya Kofman

We construct a cubical CW-complex CK(M3) whose rational cohomology algebra contains Vassiliev invariants of knots in the 3-manifold M3. We construct CK(R3) by attaching cells to CK(R3) for every degenerate 1-singular and 2-singular knot, and we show that π1(CK(R 3)) = 1 and π2(CK(R 3)) = Z. We give conditions for Vassiliev invariants to be nontrivial in cohomology. In particular, for R3 we show...

The complex geometry of

Light scattering by cubical particle in the WKB approximation

In this work, we determined the analytical expressions of the form factor of a cubical particle in the WKB approximation. We adapted some variables (size parameter, refractive index, the scattering angle) and found the form factor in the approximation of Rayleigh-Gans-Debye (RGD), Anomalous Diffraction (AD), and determined the efficiency factor of the extinction. Finally, to illustrate our form...

Toward Polygonalisation of Thick Discrete Arcs

All the polygonalisation algorithms we can find in the literature proceed on 4-connected or 8-connected discrete arcs. In this article, we aim to polygonalise “thick arcs”. A first step consists in giving a definition of such arcs based on morphological properties. In a second step, we propose two methods in order to polygonalise such arcs. The first one is based on a squelettisation of the arc...