Disk-band graphs in the theory of framed tangles

Abstract

A disk-band graph is a smooth compact two-dimensional manifold with boundary, partitioned into handles; the partition contains only index zero and index one handles and imitates the structure of the graph. (Index zero handles are analogues of the vertices of the graph, index one handles are analogues of the graph edges.) A disk-band graph is called spatial if it is a smooth submanifold of three-dimensional Euclidean space. A tangle is usually understood as a smooth compact one-dimensional submanifold of the standard three-dimensional ball that intersects the boundary of the ball orthogonally, only along its boundary, the intersection is contained in the equator. We call a tangle framed if it is equipped with a smooth field of normal straight lines. It is well known that there is a reduction of the problem of isotopic classification of spatial disk-band graphs to the problem of isotopic classification of framed tangles. This work focuses on the application of (abstract) disk-band graphs to study the set of isotopic classes of framed tangles.