247 research outputs found
A Penrose polynomial for embedded graphs
We extend the Penrose polynomial, originally defined only for plane graphs,
to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial
of embedded graphs leads to new identities and relations for the Penrose
polynomial which can not be realized within the class of plane graphs. In
particular, by exploiting connections with the transition polynomial and the
ribbon group action, we find a deletion-contraction-type relation for the
Penrose polynomial. We relate the Penrose polynomial of an orientable
checkerboard colourable graph to the circuit partition polynomial of its medial
graph and use this to find new combinatorial interpretations of the Penrose
polynomial. We also show that the Penrose polynomial of a plane graph G can be
expressed as a sum of chromatic polynomials of twisted duals of G. This allows
us to obtain a new reformulation of the Four Colour Theorem
Knot invariants and the Bollobas-Riordan polynomial of embedded graphs
For a graph G embedded in an orientable surface \Sigma, we consider
associated links L(G) in the thickened surface \Sigma \times I. We relate the
HOMFLY polynomial of L(G) to the recently defined Bollobas-Riordan polynomial
of a ribbon graph. This generalizes celebrated results of Jaeger and Traldi. We
use knot theory to prove results about graph polynomials and, after discussing
questions of equivalence of the polynomials, we go on to use our formulae to
prove a duality relation for the Bollobas-Riordan polynomial. We then consider
the specialization to the Jones polynomial and recent results of Chmutov and
Pak to relate the Bollobas-Riordan polynomials of an embedded graph and its
tensor product with a cycle.Comment: v2: minor corrections, to appear in European Journal of Combinatoric
Simple realizability of complete abstract topological graphs simplified
An abstract topological graph (briefly an AT-graph) is a pair
where is a graph and is a set of pairs of its edges. The AT-graph is simply
realizable if can be drawn in the plane so that each pair of edges from
crosses exactly once and no other pair crosses. We show that
simply realizable complete AT-graphs are characterized by a finite set of
forbidden AT-subgraphs, each with at most six vertices. This implies a
straightforward polynomial algorithm for testing simple realizability of
complete AT-graphs, which simplifies a previous algorithm by the author. We
also show an analogous result for independent -realizability,
where only the parity of the number of crossings for each pair of independent
edges is specified.Comment: 26 pages, 17 figures; major revision; original Section 5 removed and
will be included in another pape
Knot Graphs
We consider the equivalence classes of graphs induced by the unsigned
versions of the Reidemeister moves on knot diagrams.
Any graph which is
reducible by some finite sequence of these moves, to a graph with no
edges is called a knot graph. We show that the class of knot graphs
strictly contains the set of delta-wye graphs. We prove that the
dimension of the intersection of the cycle and cocycle spaces is an
effective numerical invariant of these classes
- …