Graphs on surfaces: dualities, polynomials, and knots

Abstract

Graphs on Surfaces: Dualities, Polynomials, and Knots offers an accessibleand comprehensive treatment of recent developments on generalizedduals of graphs on surfaces, and their applications. The authorsillustrate the interdependency between duality, medial graphs andknots; how this interdependency is reflected in algebraic invariantsof graphs and knots; and how it can be exploited to solve problemsin graph and knot theory. Taking a constructive approach, the authorsemphasize how generalized duals and related ideas arise by localizingclassical constructions, such as geometric duals and Tait graphs,and then removing artificial restrictions in these constructionsto obtain full extensions of them to embedded graphs. The authorsdemonstrate the benefits of these generalizations to embedded graphsin chapters describing their applications to graph polynomials andknots.Graphs on Surfaces: Dualities, Polynomials, and Knots also providesa self-contained introduction to graphs on surfaces, generalizedduals, topological graph polynomials, and knot polynomials that isaccessible both to graph theorists and to knot theorists. Directedat those with some familiarity with basic graph theory and knot theory,this book is appropriate for graduate students and researchers ineither area. Because the area is advancing so rapidly, the authorsgive a comprehensive overview of the topic and include a robust bibliography,aiming to provide the reader with the necessary foundations to stayabreast of the field. The reader will come away from the text convincedof advantages of considering these higher genus analogues of constructionsof plane and abstract graphs, and with a good understanding of howthey arise.