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 being pairwise disjoint. Diagrams are not new objects; they have been 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 yet another branch: the theory of tangles. We provide diagrams with simple additional structures: the smoothness of the circles and rectangles that are pairwise consistent with each other, the orientation of the circle, and a point on the circle. We introduce a new equivalence relation (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 a 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.