Kindred diagrams

Abstract

By a diagram we mean a topological space obtained by gluing to a standard circle a finite number of pairwise non-intersecting closed rectangles along their lateral sides, the glued rectangles are pairwise disjoint. Diagrams are not new objects; they were used in many areas of low-dimensional topology. Our main goal is to develop the theory of diagrams to a level sufficient for application in one more branch — in the theory of tangles. We provide the diagrams with simple additional structures — the smoothness of the circles and rectangles that are pairwise consistent with each other, the orientation of the circle, a point on the circle; we introduce new equivalence relation (that is, as far as the author knows, not previously encountered in the scientific literature) — kindred relation; we define a surjective mapping of the set of classes of kindred diagrams onto the set of classes of diffeomorphic smooth compact connected two-dimensional manifolds with boundary, and note that in the simplest cases this surjection is also a bijection. The application of the constructed theory to the tangle theory requires additional preparation and therefore is not included in this article; the author intends to devote a separate publication to this application.