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Register a new userMilnor invariants of string links, trivalent trees, and configuration space integrals
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We study configuration space integral formulas for Milnors homotopy link invariants, showing that they are in correspondence with certain linear combinations of trivalent trees. Our proof is essentially a combinatorial analysis of a certain space of trivalent homotopy link diagrams which corresponds to all finite type homotopy link invariants via configuration space integrals. An important ingredient is the fact that configuration space integrals take the shuffle product of diagrams to the product of invariants. We ultimately deduce a partial recipe for writing explicit integral formulas for products of Milnor invariants from trivalent forests. We also obtain cohomology classes in spaces of link maps from the same data.
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Configuration space integrals have in recent years been used for studying the cohomology of spaces of (string) knots and links in $mathbb{R}^n$ for $n>3$ since they provide a map from a certain differential 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 $mathbb{R}$ in $mathbb{R}^n$ 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 obtain configuration space integral expressions for Milnor invariants of string links.
We use rational formality of configuration spaces and the bar construction to study the cohomology of the space of braids in dimension four or greater. We provide a diagram complex for braids and a quasi-isomorphism to the de Rham cochains on the space of braids. The quasi-isomorphism is given by a configuration space integral followed by Chens iterated integrals. This extends results of Kohno and of Cohen and Gitler on the cohomology of the space of braids to a CDGA suitable for integration. We show that this integration is compatible with Bott-Taubes configuration space integrals for long links via a map between two diagram complexes. As a corollary, we get a surjection in cohomology from the space of long links to the space of braids. We also discuss to what extent our results apply to the case of classical braids.
We consider Milnor invariants for certain covering links as a generalization of covering linkage invariants formulated by R. Hartley and K. Murasugi. A set of Milnor invariants for covering links is a cobordism invariant of a link, and that this invariant can distinguish some links for which the ordinary Milnor invariants coincide. Moreover, for a Brunnian link $L$, the first non-vanishing Milnor invariants of $L$ is modulo-$2$ congruent to a sum of Milnor invariants of covering links. As a consequence, a sum of linking numbers of iterated covering links gives the first non-vanishing Milnor invariant of $L$ modulo $2$.
In a previous paper, the authors proved that Milnor link-homotopy invariants modulo $n$ classify classical string links up to $2n$-move and link-homotopy. As analogues to the welded case, in terms of Milnor invariants, we give here two classifications of welded string links up to $2n$-move and self-crossing virtualization, and up to $V^{n}$-move and self-crossing virtualization, respectively.
For a classical link, Milnor defined a family of isotopy invariants, called Milnor $overline{mu}$-invariants. Recently, Chrisman extended Milnor $overline{mu}$-invariants to welded links by a topological approach. The aim of this paper is to show that Milnor $overline{mu}$-invariants can be extended to welded links by a combinatorial approach. The proof contains an alternative proof for the invariance of the original $overline{mu}$-invariants of classical links.
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