Counting generalized Jenkins-Strebel differentials

We study the combinatorial geometry of "lattice" Jenkins--Strebel differentials with simple zeroes and simple poles on $\mathbb{C}P^1$ and of the corresponding counting functions. Developing the results of M. Kontsevich we evaluate the leading term of the symmetric polynomial counting the number of such "lattice" Jenkins-Strebel differentials having all zeroes on a single singular layer. This allows us to express the number of general "lattice" Jenkins-Strebel differentials as an appropriate weighted sum over decorated trees. The problem of counting Jenkins-Strebel differentials is equivalent to the problem of counting pillowcase covers, which serve as integer points in appropriate local coordinates on strata of moduli spaces of meromorphic quadratic differentials. This allows us to relate our counting problem to calculations of volumes of these strata . A very explicit expression for the volume of any stratum of meromorphic quadratic differentials recently obtained by the authors leads to an interesting combinatorial identity for our sums over trees.Comment: to appear in Geometriae Dedicata. arXiv admin note: text overlap with arXiv:1212.166

Configuration spaces and Vassiliev classes in any dimension

The real cohomology of the space of imbeddings of S^1 into R^n, n>3, is studied by using configuration space integrals. Nontrivial classes are explicitly constructed. As a by-product, we prove the nontriviality of certain cycles of imbeddings obtained by blowing up transversal double points in immersions. These cohomology classes generalize in a nontrivial way the Vassiliev knot invariants. Other nontrivial classes are constructed by considering the restriction of classes defined on the corresponding spaces of immersions.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-39.abs.htm

Spiders for rank 2 Lie algebras

A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. We define certain combinatorial spiders by generators and relations that are isomorphic to the representation theories of the three rank two simple Lie algebras, namely A2, B2, and G2. They generalize the widely-used Temperley-Lieb spider for A1. Among other things, they yield bases for invariant spaces which are probably related to Lusztig's canonical bases, and they are useful for computing quantities such as generalized 6j-symbols and quantum link invariants.Comment: 33 pages. Has color figure

From Zwiebach invariants to Getzler relation

We introduce the notion of Zwiebach invariants that generalize Gromov-Witten invariants and homotopical algebra structures. We outline the induction procedure that induces the structure of Zwiebach on the subbicomplex, that gives the structure of Gromov-Witten invariants on subbicomplex with zero diffferentials. We propose to treat Hodge dGBV with 1/12 axiom as the simplest set of Zwiebach invariants, and explicitely prove that it induces WDVV and Getzler equations in genera 0 and 1 respectively.Comment: 35 page

Moduli of Tropical Plane Curves

We study the moduli space of metric graphs that arise from tropical plane curves. There are far fewer such graphs than tropicalizations of classical plane curves. For fixed genus $g$, our moduli space is a stacky fan whose cones are indexed by regular unimodular triangulations of Newton polygons with $g$ interior lattice points. It has dimension $2g+1$ unless $g \leq 3$ or $g = 7$. We compute these spaces explicitly for $g \leq 5$.Comment: 31 pages, 25 figure