Picture theory : algorithms and software
This thesis is concerned with developing and implementing algorithms based upon the geometry of pictures. Spherical pictures have been used in many areas of combinatorial group theory, and particularly, they have shown to be a useful method when studying the second homotopy module, 1T2, of a presentation (,,,, and ). Computational programs that implement picture theoretical and design algorithms could advance the areas in which picture theory can be used, due to the much faster time taken to derive results than that of manual calculations. A variety of algorithms are presented. A data structure has been devised to represent spherical pictures. A method is given that verifies that a given data structure represents a picture, or set of pictures, over a group presentation. This method includes a new planarity testing algorithm, which can be performed on any graph. A computational algorithm has been implemented that determines if a given presentation defines a group extension. This work is based upon the algorithm of Baik et al.  which has been developed using the theory of pictures. A 3-presentation for a group G is given by < P, s >, where P is a presentation for G and s is a set of generators for 1T2. The set s can be described in a number of ways. An algorithm is given that produces a generating set of spherical pictures for 1T2 when s is given in the form of identity sequences. Conversely, if s is given in terms of spherical pictures, then the corresponding identity sequences that describe 1T2 can be determined. The above algorithms are contained in the Spherical PIcture Editor (SPICE). SPICE is a software package that enables a user to manually draw pictures over group presentations and, for these pictures, call the algorithms described above. It also contains a library of generating pictures for the non abelian groups of order at most 30. Furthermore, a method has been implemented that automatically draws a spherical picture from a corresponding identity sequence. Again, this new graph drawing technique can be performed on any arbitrary graph.