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Alternating plane graphs
A plane graph is called alternating if all adjacent vertices have different degrees, and all neighboring faces as well. Alternating plane graphs were introduced in 2008. This paper presents the previous research on alternating plane graphs.
There are two smallest alternating plane graphs, having 17 vertices and 17 faces each. There is no alternating plane graph with 18 vertices, but alternating plane graphs exist for all cardinalities from 19 on. From a small set of initial building blocks, alternating plane graphs can be constructed for all large cardinalities. Many of the small alternating plane graphs have been found with extensive computer help.
Theoretical results on alternating plane graphs are included where all degrees have to be from the set {3,4,5}. In addition, several classes of “weak alternating plane graphs” (with vertices of degree 2) are presented
Resonance graphs of plane bipartite graphs as daisy cubes
We characterize all plane bipartite graphs whose resonance graphs are daisy
cubes and therefore generalize related results on resonance graphs of benzenoid
graphs, catacondensed even ring systems, as well as 2-connected outerplane
bipartite graphs. Firstly, we prove that if is a plane elementary bipartite
graph other than , then the resonance graph is a daisy cube if and
only if the Fries number of equals the number of finite faces of , which
in turn is equivalent to being homeomorphically peripheral color
alternating. Next, we extend the above characterization from plane elementary
bipartite graphs to all plane bipartite graphs and show that the resonance
graph of a plane bipartite graph is a daisy cube if and only if is
weakly elementary bipartite and every elementary component of other than
is homeomorphically peripheral color alternating. Along the way, we prove
that a Cartesian product graph is a daisy cube if and only if all of its
nontrivial factors are daisy cubes
Construction of harmonic diffeomorphisms and minimal graphs
We study complete minimal graphs in HxR, which take asymptotic boundary
values plus and minus infinity on alternating sides of an ideal inscribed
polygon Γ in H. We give necessary and sufficient conditions on the
"lenghts" of the sides of the polygon (and all inscribed polygons in Γ)
that ensure the existence of such a graph. We then apply this to construct
entire minimal graphs in HxR that are conformally the complex plane C. The
vertical projection of such a graph yields a harmonic diffeomorphism from C
onto H, disproving a conjecture of Rick Schoen
Unsigned state models for the Jones polynomial
It is well a known and fundamental result that the Jones polynomial can be
expressed as Potts and vertex partition functions of signed plane graphs. Here
we consider constructions of the Jones polynomial as state models of unsigned
graphs and show that the Jones polynomial of any link can be expressed as a
vertex model of an unsigned embedded graph.
In the process of deriving this result, we show that for every diagram of a
link in the 3-sphere there exists a diagram of an alternating link in a
thickened surface (and an alternating virtual link) with the same Kauffman
bracket. We also recover two recent results in the literature relating the
Jones and Bollobas-Riordan polynomials and show they arise from two different
interpretations of the same embedded graph.Comment: Minor corrections. To appear in Annals of Combinatoric
On Finite Order Invariants of Triple Points Free Plane Curves
We describe some regular techniques of calculating finite degree invariants
of triple points free smooth plane curves . They are a direct
analog of similar techniques for knot invariants and are based on the calculus
of {\em triangular diagrams} and {\em connected hypergraphs} in the same way as
the calculation of knot invariants is based on the study of chord diagrams and
connected graphs.
E.g., the simplest such invariant is of degree 4 and corresponds to the
diagram consisting of two triangles with alternating vertices in a circle in
the same way as the simplest knot invariant (of degree 2) corresponds to the
2-chord diagram . Also, following V.I.Arnold and other authors we
consider invariants of {\em immersed} triple points free curves and describe
similar techniques also for this problem, and, more generally, for the
calculation of homology groups of the space of immersed plane curves without
points of multiplicity for any $k \ge 3.
Decomposition theorem on matchable distributive lattices
A distributive lattice structure has been established on the
set of perfect matchings of a plane bipartite graph . We call a lattice {\em
matchable distributive lattice} (simply MDL) if it is isomorphic to such a
distributive lattice. It is natural to ask which lattices are MDLs. We show
that if a plane bipartite graph is elementary, then is
irreducible. Based on this result, a decomposition theorem on MDLs is obtained:
a finite distributive lattice is an MDL if and only if each factor
in any cartesian product decomposition of is an MDL. Two types of
MDLs are presented: and , where
denotes the cartesian product between -element
chain and -element chain, and is a poset implied by any
orientation of a tree.Comment: 19 pages, 7 figure
The Jones polynomial and graphs on surfaces
The Jones polynomial of an alternating link is a certain specialization of
the Tutte polynomial of the (planar) checkerboard graph associated to an
alternating projection of the link. The Bollobas-Riordan-Tutte polynomial
generalizes the Tutte polynomial of planar graphs to graphs that are embedded
in closed oriented surfaces of higher genus.
In this paper we show that the Jones polynomial of any link can be obtained
from the Bollobas-Riordan-Tutte polynomial of a certain oriented ribbon graph
associated to a link projection. We give some applications of this approach.Comment: 19 pages, 9 figures, minor change
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