24 research outputs found
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
Graphs on surfaces and Khovanov homology
Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented
surfaces. A quasi-tree of a ribbon graph is a spanning subgraph with one face,
which is described by an ordered chord diagram. We show that for any link
diagram , there is an associated ribbon graph whose quasi-trees correspond
bijectively to spanning trees of the graph obtained by checkerboard coloring
. This correspondence preserves the bigrading used for the spanning tree
model of Khovanov homology, whose Euler characteristic is the Jones polynomial
of . Thus, Khovanov homology can be expressed in terms of ribbon graphs,
with generators given by ordered chord diagrams.Comment: 8 pages, 5 figure
Indecomposable knots and concordance
R. C. Kirby and W. B. R. Lickorish have proved (cf. (4)) that any classical knot is concordant to an indecomposable knot. In the present note we show that this statement is also true for higher dimensional knots: more precisely, for any higher-dimensional knot K there exist infinitely many non-isotopic indecomposable simple knots which are concordant to K. This, together with the result of Kirby and Lickorish, gives a complete solution of problem 13 of (1
A reduced set of moves on one-vertex ribbon graphs coming from links
Every link in R^3 can be represented by a one-vertex ribbon graph. We prove a
Markov type theorem on this subset of link diagrams.Comment: 14 pages, 15 figure
A Reduced Set of Moves on One-Vertex Ribbon Graphs Coming from Links
Every link in R3 can be represented by a one-vertex ribbon graph. We prove a Markov type theorem on this subset of link diagrams
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