861 research outputs found
Cyclic operads and homology of graph complexes
We will consider P-graph complexes, where P is a cyclic operad. P-graph
complexes are natural generalizations of Kontsevich's graph complexes -- for P
= the operad for associative algebras it is the complex of ribbon graphs, for P
= the operad for commutative associative algebras, the complex of all graphs.
We construct a `universal class' in the cohomology of the graph complex with
coefficients in a theory. The Kontsevich-type invariant is then an evaluation,
on a concrete cyclic algebra, of this class. We also explain some results of M.
Penkava and A. Schwarz on the construction of an invariant from a cyclic
deformation of a cyclic algebra. Our constructions are illustrated by a `toy
model' of tree complexes.Comment: LaTeX 2.09 + article12pt,leqno style, 10 page
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
The cyclic coloring complex of a complete k-uniform hypergraph
In this paper, we study the homology of the cyclic coloring complex of three
different types of -uniform hypergraphs. For the case of a complete
-uniform hypergraph, we show that the dimension of the
homology group is given by a binomial coefficient. Further, we discuss a
complex whose -faces consist of all ordered set partitions where none of the contain a hyperedge of the complete
-uniform hypergraph and where . It is shown that the
dimensions of the homology groups of this complex are given by binomial
coefficients. As a consequence, this result gives the dimensions of the
multilinear parts of the cyclic homology groups of \C[x_1,...,x_n]/
\{x_{i_1}...x_{i_k} \mid i_{1}...i_{k} is a hyperedge of . For the other
two types of hypergraphs, star hypergraphs and diagonal hypergraphs, we show
that the dimensions of the homology groups of their cyclic coloring complexes
are given by binomial coefficients as well
Moduli spaces of colored graphs
We introduce moduli spaces of colored graphs, defined as spaces of
non-degenerate metrics on certain families of edge-colored graphs. Apart from
fixing the rank and number of legs these families are determined by various
conditions on the coloring of their graphs. The motivation for this is to study
Feynman integrals in quantum field theory using the combinatorial structure of
these moduli spaces. Here a family of graphs is specified by the allowed
Feynman diagrams in a particular quantum field theory such as (massive) scalar
fields or quantum electrodynamics. The resulting spaces are cell complexes with
a rich and interesting combinatorial structure. We treat some examples in
detail and discuss their topological properties, connectivity and homology
groups
On a theorem of Kontsevich
In two seminal papers M. Kontsevich introduced graph homology as a tool to
compute the homology of three infinite dimensional Lie algebras, associated to
the three operads `commutative,' `associative' and `Lie.' We generalize his
theorem to all cyclic operads, in the process giving a more careful treatment
of the construction than in Kontsevich's original papers. We also give a more
explicit treatment of the isomorphisms of graph homologies with the homology of
moduli space and Out(F_r) outlined by Kontsevich. In [`Infinitesimal operations
on chain complexes of graphs', Mathematische Annalen, 327 (2003) 545-573] we
defined a Lie bracket and cobracket on the commutative graph complex, which was
extended in [James Conant, `Fusion and fission in graph complexes', Pac. J. 209
(2003), 219-230] to the case of all cyclic operads. These operations form a Lie
bi-algebra on a natural subcomplex. We show that in the associative and Lie
cases the subcomplex on which the bi-algebra structure exists carries all of
the homology, and we explain why the subcomplex in the commutative case does
not.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-42.abs.htm
Polynomial 6j-Symbols and States Sums
For q a root of unity of order 2r, we give explicit formulas of a family of
3-variable Laurent polynomials J_{i,j,k} with coefficients in Z[q] that encode
the 6j-symbols associated with nilpotent representations of U_qsl_2. For a
given abelian group G, we use them to produce a state sum invariant
tau^r(M,L,h_1,h_2) of a quadruplet (compact 3-manifold M, link L inside M,
homology class h_1\in H_1(M,Z), homology class h_2\in H_2(M,G)) with values in
a ring R related to G. The formulas are established by a "skein" calculus as an
application of the theory of modified dimensions introduced in
[arXiv:0711.4229]. For an oriented 3-manifold M, the invariants are related to
TV(M,L,f\in H^1(M,C^*)) defined in [arXiv:0910.1624] from the category of
nilpotent representations of U_qsl_2. They refine them as TV(M,L,f)= Sum_h
tau^r(M,L,h,f') where f' correspond to f with the isomorphism H_2(M,C^*) ~
H^1(M,C^*).Comment: 31 pages, 14 figures. Minor changes and reference added in version
A spanning tree model for the Heegaard Floer homology of a branched double-cover
Given a diagram of a link K in S^3, we write down a Heegaard diagram for the
branched-double cover Sigma(K). The generators of the associated Heegaard Floer
chain complex correspond to Kauffman states of the link diagram. Using this
model we make some computations of the homology \hat{HF}(Sigma(K)) as a graded
group. We also conjecture the existence of a delta-grading on
\hat{HF}(Sigma(K)) analogous to the delta-grading on knot Floer and Khovanov
homology.Comment: 43 pages, 20 figure
Quantum invariants of periodic three-manifolds
Let p be an odd prime and r be relatively prime to p. Let G be a finite
p-group. Suppose an oriented 3-manifold M-tilde has a free G-action with orbit
space M. We consider certain Witten-Reshetikhin-Turaev SU(2) invariants w_r(M).
We will give a fomula for w_r(M) in terms of the defect of M-tilde --> M and
the number of elements in G. We also give a version of this result if M and
M-tilde contain framed links or colored fat graphs. We give similar formulas
for non-free actions which hold for a specified finite set of values for r.Comment: 19 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTMon2/paper9.abs.htm
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