13,615 research outputs found
The Intersection Graph Conjecture for Loop Diagrams
Vassiliev invariants can be studied by studying the spaces of chord diagrams
associated with singular knots. To these chord diagrams are associated the
intersection graphs of the chords. We extend results of Chmutov, Duzhin and
Lando to show that these graphs determine the chord diagram if the graph has at
most one loop. We also compute the size of the subalgebra generated by these
"loop diagrams."Comment: 23 pages, many figures. arXiv admin note: Figures 1, 2, 5 and 11
included in sources but in format not supported by arXi
Forbidden induced subgraph characterization of circle graphs within split graphs
A graph is circle if its vertices are in correspondence with a family of
chords in a circle in such a way that every two distinct vertices are adjacent
if and only if the corresponding chords have nonempty intersection. Even though
there are diverse characterizations of circle graphs, a structural
characterization by minimal forbidden induced subgraphs for the entire class of
circle graphs is not known, not even restricted to split graphs (which are the
graphs whose vertex set can be partitioned into a clique and a stable set). In
this work, we give a characterization by minimal forbidden induced subgraphs of
circle graphs, restricted to split graphs.Comment: 59 pages, 15 figure
Automorphism Groups of Geometrically Represented Graphs
Interval graphs are intersection graphs of closed intervals and circle graphs are intersection graphs of chords of a circle. We study automorphism groups of these graphs. We show that interval graphs have the same automorphism groups as trees, and circle graphs have the same
as pseudoforests, which are graphs with at most one cycle in every connected component.
Our technique determines automorphism groups for classes with a
strong structure of all geometric representations, and it can be applied to other graph classes. Our results imply polynomial-time algorithms for computing automorphism groups in term of group products
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 Kontsevich integral for knotted trivalent graphs
We construct an extension of the Kontsevich integral of knots to knotted
trivalent graphs, which commutes with orientation switches, edge deletions,
edge unzips, and connected sums. In 1997 Murakami and Ohtsuki [MO] first
constructed such an extension, building on Drinfel'd's theory of associators.
We construct a step by step definition, using elementary Kontsevich integral
methods, to get a one-parameter family of corrections that all yield invariants
well behaved under the graph operations above.Comment: Journal version, 47 page
Ribbon graphs and bialgebra of Lagrangian subspaces
To each ribbon graph we assign a so-called L-space, which is a Lagrangian
subspace in an even-dimensional vector space with the standard symplectic form.
This invariant generalizes the notion of the intersection matrix of a chord
diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual)
and Vassiliev moves on ribbon graphs are reinterpreted nicely in the language
of L-spaces, becoming changes of bases in this vector space. Finally, we define
a bialgebra structure on the span of L-spaces, which is analogous to the
4-bialgebra structure on chord diagrams.Comment: 21 pages, 13 figures. v2: major revision, Sec 2 and 3 completely
rewritten; v3: minor corrections. Final version, to appear in Journal of Knot
Theory and its Ramification
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