Configuration space integrals have been used in recent years for studying the cohomology of spaces of (string) knots and links in R n for n > 3 since they provide a map from a certain differential graded algebra of diagrams to the deRham complex of differential forms on the spaces of knots and links. We refine this construction so that it now applies to the space of homotopy string links -the space of smooth maps of some number of copies of R in R n with fixed behavior outside a compact set and such that the images of the copies of R are disjoint -even for n = 3. We further study the case n = 3 in degree zero and show that our integrals represent a universal finite type invariant of the space of classical homotopy string links. As a consequence, we deduce that Milnor invariants of string links can be written in terms of configuration space integrals.
We present a novel graph embedding space (i.e., a set of measures on graphs) for performing statistical analyses of networks. Key improvements over existing approaches include discovery of "motif-hubs" (multiple overlapping significant subgraphs), computational efficiency relative to subgraph census, and flexibility (the method is easily generalizable to weighted and signed graphs). The embedding space is based on scalars, functionals of the adjacency matrix representing the network. Scalars are global, involving all nodes; although they can be related to subgraph enumeration, there is not a one-to-one mapping between scalars and subgraphs. Improvements in network randomization and significance testing-we learn the distribution rather than assuming gaussianity-are also presented. The resulting algorithm establishes a systematic approach to the identification of the most significant scalars and suggests machine-learning techniques for network classification.PACS numbers: 87.10.+e, 89.75.Fb, 87.16.Ac, 87.23.Kg Background: Recent studies of real-world biological, social, and technological networks have catalyzed an explosion of research from a broad range of disciplines. Much of the effort in this emerging field has focused on characterizing the structure of networks using various statistical properties that are local (analysis relies on subset of nodes) or global (relying on all nodes) in scope. The former analysis includes subgraph census (comparing frequency of subgraph occurrences in a given graph with those over a distribution of graphs [2,7]), while examples of the latter include path lengths and degree distributions (see citations in [1]).To study local structure statistics, sociologists developed the k-subgraph census, an enumeration of all possible subgraphs of k nodes appearing in networks. For example, sociologists used the 3-subgraph census, compared with 3-subgraph distributions in randomized graphs, to quantify network transitivity [3,6,8] (in the context of a social network, high transitivity means that many of your friends are friends with each other). Applying such techniques first to the E. coli genetic network [7] and later to various biological and physical networks [9], Milo et al showed that different networks have different "most significant" subgraphs.Major limitations of these subgraph approaches include computational cost and generalizability. The number of isomorphism classes of digraphs grows rapidly with graph size [6,12] and subgraph isomorphism is an NP-complete 2 problem [13] [40]. These computational limitations bias results, since structures with more than three or four nodes would not be counted. Moreover, it is not readily obvious how to extend subgraph census to weighted and/or signed graphs. This is particularly relevant for genetic regulatory networks in which the interactions can be described quantitatively via binding affinities and qualitatively as activating or repressing, or similarly neuronal networks, in which the interactions are often weighted by the number of syna...
We show that the map on components from the space of classical long knots to the nth stage of its Goodwillie-Weiss embedding calculus tower is a map of monoids whose target is an abelian group, and which is invariant under clasper surgery. We deduce that this map on components is a finite type-(n − 1) knot invariant. We compute the E 2 -page in total degree zero for the spectral sequence converging to the components of this tower, identifying it as Z-modules of primitive chord diagrams, providing evidence for the conjecture that the tower is a universal finite-type invariant over the integers. Key to these results is the development of a group structure on the tower compatible with connect-sum of knots, which in contrast with the corresponding results for the (weaker) homology tower requires novel techniques involving operad actions, evaluation maps, and cosimplicial and subcubical diagrams.
Bott and Taubes constructed knot invariants by integrating differential forms along the fiber of a bundle over the space of knots, generalizing the Gauss linking integral. Their techniques were later used to construct real cohomology classes in spaces of knots and links in higher-dimensional Euclidean spaces. In previous work, we constructed cohomology classes in knot spaces with arbitrary coefficients by integrating via a Pontrjagin-Thom construction. We carry out a similar construction over the space of string links, but with a refinement in which configuration spaces are glued together according to the combinatorics of weight systems. This gluing is somewhat similar to work of Kuperberg and Thurston. We use a formula of Mellor for weight systems of Milnor invariants, and we thus recover the Milnor triple linking number for string links, which is in some sense the simplest interesting example of a class obtained by this gluing refinement of our previous methods. Along the way, we find a description of this triple linking number as a degree of a map from the 6-sphere to a quotient of the product of three 2-spheres. triple linking number for string links. This invariant is already known to be integer-valued, but generalizations of this work should have more novel implications.We conjecture that this gluing refinement of our "homotopy-theoretic Bott-Taubes integrals" produces integral multiples of all the Bott-Taubes/Vassiliev-type cohomology classes of Cattaneo, Cotta-Ramusino and Longoni [CCRL02], showing that these classes are rational. This work is currently in progress. In this paper, we will focus only on the specific but interesting example of the Milnor triple linking number, rather than all finite-type invariants or all the cohomology classes of Cattaneo et al.As our title suggests, this gluing is inspired by and similar to a construction of Kuperberg and Thurston [KT] (see also the paper of Lescop [Les04]). This gluing idea was also present in the work of Bott and Taubes [BT94, Equation 1.18], as once pointed out to the author by Habegger. The difference between our construction and that of Kuperberg and Thurston is that we do a gluing which is specific to the weight system for the invariant. Thus, in our work in progress, we generalize this to arbitrary cohomology classes by constructing one glued-up space for each weight system (or graph cocycle), rather than a glued-up space that accounts for all weight systems. This slightly different approach is necessary to produce bundles whose fibers are nice enough to admit neat embeddings and Pontrjagin-Thom constructions.1.1. The Gauss linking integral. Before stating our main results, we describe the analogue of our constructions in the simpler case of the pairwise linking number. We first discuss in detail this pairwise linking number in terms of a Pontrjagin-Thom construction and the space of links in the case of closed links, where this invariant is completely straightforward.For any space X, denote the configuration space of q points in X by C q (X) := {(...
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