Dehn filling of the "magic" 3-manifold

We classify all the non-hyperbolic Dehn fillings of the complement of the chain-link with 3 components, conjectured to be the smallest hyperbolic 3-manifold with 3 cusps. We deduce the classification of all non-hyperbolic Dehn fillings of infinitely many 1-cusped and 2-cusped hyperbolic manifolds, including most of those with smallest known volume. Among other consequences of this classification, we mention the following: - for every integer n we can prove that there are infinitely many hyperbolic knots in the 3-sphere having exceptional surgeries n, n+1, n+2, n+3, with n+1, n+2 giving small Seifert manifolds and n, n+3 giving toroidal manifolds; - we exhibit a 2-cusped hyperbolic manifold that contains a pair of inequivalent knots having homeomorphic complements; - we exhibit a chiral 3-manifold containing a pair of inequivalent hyperbolic knots with orientation-preservingly homeomorphic complements; - we give explicit lower bounds for the maximal distance between small Seifert fillings and any other kind of exceptional filling.Comment: 56 pages, 10 figures, 16 tables. Some consequences of the classification adde

Mutations and short geodesics in hyperbolic 3-manifolds

In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurabiltiy classes by analyzing their cusp shapes. The knot complements in each class differ by a topological cut-and-paste operation known as mutation. Ruberman has shown that mutations of hyperelliptic surfaces inside hyperbolic 3-manifolds preserve volume. Here, we provide geometric and topological conditions under which such mutations also preserve the initial (complex) length spectrum. This work requires us to analyze when least area surfaces could intersect short geodesics in a hyperbolic 3-manifold.Comment: This is the final (accepted) version of this pape

Complexes of not ii-connected graphs

Complexes of (not) connected graphs, hypergraphs and their homology appear in the construction of knot invariants given by V. Vassiliev. In this paper we study the complexes of not $i$-connected $k$-hypergraphs on $n$ vertices. We show that the complex of not $2$-connected graphs has the homotopy type of a wedge of $(n-2)!$ spheres of dimension $2n-5$. This answers one of the questions raised by Vassiliev in connection with knot invariants. For this case the $S_n$-action on the homology of the complex is also determined. For complexes of not $2$-connected $k$-hypergraphs we provide a formula for the generating function of the Euler characteristic, and we introduce certain lattices of graphs that encode their topology. We also present partial results for some other cases. In particular, we show that the complex of not $(n-2)$-connected graphs is Alexander dual to the complex of partial matchings of the complete graph. For not $(n-3)$-connected graphs we provide a formula for the generating function of the Euler characteristic

Topology and Order

We will discuss topologies as orders, orders on sets of topologies, and topologies on ordered sets. More specifically, we will discuss Alexandroff topologies as quasiorders, the lattice of topologies on a finite set, and partially ordered topological spaces. Some topological properties of Alexandroff spaces are characterized in terms of their order. Complementation in the lattice of topologies on a set and in the lattice of convex topologies on a partially ordered set will be discussed

Characteristic Varieties of Hypersurface Complements

We give divisibility results for the (global) characteristic varieties of hypersurface complements expressed in terms of the local characteristic varieties at points along one of the irreducible components of the hypersurface. As an application, we recast old and obtain new finiteness and divisibility results for the classical (infinite cyclic) Alexander modules of complex hypersurface complements. Moreover, for the special case of hyperplane arrangements, we translate our divisibility results for characteristic varieties in terms of the corresponding resonance varieties.Comment: v2: much of the paper has been re-written, including a more detailed introduction and updated reference

Topology of Exceptional Orbit Hypersurfaces of Prehomogeneous Spaces

We consider the topology for a class of hypersurfaces with highly nonisolated singularites which arise as exceptional orbit varieties of a special class of prehomogeneous vector spaces, which are representations of linear algebraic groups with open orbits. These hypersurface singularities include both determinantal hypersurfaces and linear free (and free*) divisors. Although these hypersurfaces have highly nonisolated singularities, we determine the topology of their Milnor fibers, complements and links. We do so by using the action of linear algebraic groups beginning with the complement, instead of using Morse type arguments on the Milnor fibers. This includes replacing the local Milnor fiber by a global Milnor fiber which has a complex geometry resulting from a transitive action of an appropriate algebraic group, yielding a compact model submanifold for the homotopy type of the Milnor fiber. The topology includes the (co)homology (in characteristic 0, and 2 torsion in one family) and homotopy groups, and we deduce the triviality of the monodromy transformations on rational (or complex) cohomology. The cohomology of the Milnor fibers and complements are isomorphic as algebras to exterior algebras or for one family, modules over exterior algebras; and cohomology of the link is, as a vector space, a truncated and shifted exterior algebra, for which the cohomology product structure is essentially trivial. We also deduce from Bott's periodicity theorem, the homotopy groups of the Milnor fibers for determinantal hypersurfaces in the stable range as the stable homotopy groups of the associated infinite dimensional symmetric spaces. Applying a Theorem of Oka we obtain a class of formal linear combinations of exceptional orbit hypersurfaces which have Milnor fibers which are homotopy equivalent to joins of the compact model submanifolds.Comment: to appear in the Journal of Topolog