Separability and the genus of a partial dual

Partial duality generalizes the fundamental concept of the geometric dual of an embedded graph. A partial dual is obtained by forming the geometric dual with respect to only a subset of edges. While geometric duality preserves the genus of an embedded graph, partial duality does not. Here we are interested in the problem of determining which edge sets of an embedded graph give rise to a partial dual of a given genus. This problem turns out to be intimately connected to the separability of the embedded graph. We determine how separability is related to the genus of a partial dual. We use this to characterize partial duals of graphs embedded in the plane, and in the real projective plane, in terms of a particular type of separation of an embedded graph. These characterizations are then used to determine a local move relating all partially dual graphs in the plane and in the real projective plane

Partial duals of plane graphs, separability and the graphs of knots

There is a well-known way to describe a link diagram as a (signed) plane graph, called its Tait graph. This concept was recently extended, providing a way to associate a set of embedded graphs (or ribbon graphs) to a link diagram. While every plane graph arises as a Tait graph of a unique link diagram, not every embedded graph represents a link diagram. Furthermore, although a Tait graph describes a unique link diagram, the same embedded graph can represent many different link diagrams. One is then led to ask which embedded graphs represent link diagrams, and how link diagrams presented by the same embedded graphs are related to one another. Here we answer these questions by characterizing the class of embedded graphs that represent link diagrams, and then using this characterization to find a move that relates all of the link diagrams that are presented by the same set of embedded graphs.Comment: v2: major change

3nj Morphogenesis and Semiclassical Disentangling

Recoupling coefficients (3nj symbols) are unitary transformations between binary coupled eigenstates of N=(n+1) mutually commuting SU(2) angular momentum operators. They have been used in a variety of applications in spectroscopy, quantum chemistry and nuclear physics and quite recently also in quantum gravity and quantum computing. These coefficients, naturally associated to cubic Yutsis graphs, share a number of intriguing combinatorial, algebraic, and analytical features that make them fashinating objects to be studied on their own. In this paper we develop a bottom--up, systematic procedure for the generation of 3nj from 3(n-1)j diagrams by resorting to diagrammatical and algebraic methods. We provide also a novel approach to the problem of classifying various regimes of semiclassical expansions of 3nj coefficients (asymptotic disentangling of 3nj diagrams) for n > 2 by means of combinatorial, analytical and numerical tools

New directions in Nielsen-Reidemeister theory

The purpose of this expository paper is to present new directions in the classical Nielsen-Reidemeister fixed point theory. We describe twisted Burnside-Frobenius theorem, groups with $R_\infty$ \emph{property} and a connection between Nielsen fixed point theory and symplectic Floer homology.Comment: 50 pages, surve