282 research outputs found
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
Quasi-tree expansion for the Bollob\'as-Riordan-Tutte polynomial
Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented
surfaces. The Bollob\'as-Riordan-Tutte polynomial is a three-variable
polynomial that extends the Tutte polynomial to oriented ribbon graphs. A
quasi-tree of a ribbon graph is a spanning subgraph with one face, which is
described by an ordered chord diagram. We generalize the spanning tree
expansion of the Tutte polynomial to a quasi-tree expansion of the
Bollob\'as-Riordan-Tutte polynomial.Comment: This version to be published in the Bulletin of the London
Mathematical Society. 17 pages, 4 figure
Chord Diagrams and Gauss Codes for Graphs
Chord diagrams on circles and their intersection graphs (also known as circle
graphs) have been intensively studied, and have many applications to the study
of knots and knot invariants, among others. However, chord diagrams on more
general graphs have not been studied, and are potentially equally valuable in
the study of spatial graphs. We will define chord diagrams for planar
embeddings of planar graphs and their intersection graphs, and prove some basic
results. Then, as an application, we will introduce Gauss codes for immersions
of graphs in the plane and give algorithms to determine whether a particular
crossing sequence is realizable as the Gauss code of an immersed graph.Comment: 20 pages, many figures. This version has been substantially
rewritten, and the results are stronge
On the refined counting of graphs on surfaces
Ribbon graphs embedded on a Riemann surface provide a useful way to describe
the double line Feynman diagrams of large N computations and a variety of other
QFT correlator and scattering amplitude calculations, e.g in MHV rules for
scattering amplitudes, as well as in ordinary QED. Their counting is a special
case of the counting of bi-partite embedded graphs. We review and extend
relevant mathematical literature and present results on the counting of some
infinite classes of bi-partite graphs. Permutation groups and representations
as well as double cosets and quotients of graphs are useful mathematical tools.
The counting results are refined according to data of physical relevance, such
as the structure of the vertices, faces and genus of the embedded graph. These
counting problems can be expressed in terms of observables in three-dimensional
topological field theory with S_d gauge group which gives them a topological
membrane interpretation.Comment: 57 pages, 12 figures; v2: Typos corrected; references adde
Combinatorial Formulae for Vassiliev Invariants from Chern-Simons Gauge Theory
We analyse the perturbative series expansion of the vacuum expectation value
of a Wilson loop in Chern-Simons gauge theory in the temporal gauge. From the
analysis emerges the notion of the kernel of a Vassiliev invariant. The kernel
of a Vassiliev invariant of order n is not a knot invariant, since it depends
on the regular knot projection chosen, but it differs from a Vassiliev
invariant by terms that vanish on knots with n singular crossings. We
conjecture that Vassiliev invariants can be reconstructed from their kernels.
We present the general form of the kernel of a Vassiliev invariant and we
describe the reconstruction of the full primitive Vassiliev invariants at
orders two, three and four. At orders two and three we recover known
combinatorial expressions for these invariants. At order four we present new
combinatorial expressions for the two primitive Vassiliev invariants present at
this order.Comment: 73 pages, latex, epsf, 18 figures, 2 table
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