861 research outputs found

    Cyclic operads and homology of graph complexes

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    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

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    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 LL, there is an associated ribbon graph whose quasi-trees correspond bijectively to spanning trees of the graph obtained by checkerboard coloring LL. This correspondence preserves the bigrading used for the spanning tree model of Khovanov homology, whose Euler characteristic is the Jones polynomial of LL. 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

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    In this paper, we study the homology of the cyclic coloring complex of three different types of kk-uniform hypergraphs. For the case of a complete kk-uniform hypergraph, we show that the dimension of the (n−k−1)st(n-k-1)^{st} homology group is given by a binomial coefficient. Further, we discuss a complex whose rr-faces consist of all ordered set partitions [B1,...,Br+2][B_1, ..., B_{r+2}] where none of the BiB_i contain a hyperedge of the complete kk-uniform hypergraph HH and where 1∈B11 \in B_1. 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 H}H \}. 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

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    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

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    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

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    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

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    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

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    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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