7,651 research outputs found
BIPARTITE GRAPHS AND QUASIPOSITIVE SURFACES
We present a simple combinatorial model for quasipositive surfaces and positive braids, based on embedded bipartite graphs. As a first application, we extend the well-known duality on standard diagrams of torus links to twisted torus links. We then introduce a combinatorial notion of adjacency for bipartite graph links and discuss its potential relation with the adjacency problem for plane curve singularitie
Arrow ribbon graphs
We introduce an additional structure on ribbon graphs, arrow structure. We
extend the Bollob\'as-Riordan polynomial to ribbon graph with this structure.
The extended polynomial satisfies the contraction-deletion relations and
naturally behaves with respect to the partial duality of ribbon graphs. We
construct an arrow ribbon graph from a virtual link whose extended
Bollob\'as-Riordan polynomial specializes to the arrow polynomial of the
virtual link recently introduced by H.Dye and L.Kauffman. This result
generalizes the classical Thistlethwaite theorem to the arrow polynomial of
virtual links.Comment: to appear in Journal of Knot Theory and Its Ramification
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
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
The multivariate signed Bollobas-Riordan polynomial
We generalise the signed Bollobas-Riordan polynomial of S. Chmutov and I. Pak
[Moscow Math. J. 7 (2007), no. 3, 409-418] to a multivariate signed polynomial
Z and study its properties. We prove the invariance of Z under the recently
defined partial duality of S. Chmutov [J. Combinatorial Theory, Ser. B, 99 (3):
617-638, 2009] and show that the duality transformation of the multivariate
Tutte polynomial is a direct consequence of it.Comment: 17 pages, 2 figures. Published version: a section added about the
quasi-tree expansion of the multivariate Bollobas-Riordan polynomia
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