Registered users receive a variety of benefits including the ability to customize email alerts, create favorite journals list, and save searches.
Please note that a Project Euclid web account does not automatically grant access to full-text content. An institutional or society member subscription is required to view non-Open Access content.
Contact firstname.lastname@example.org with any questions.
In a –manifold , let be a knot and be an annulus which meets transversely. We define the notion of the pair being caught by a surface in the exterior of the link . For a caught pair , we consider the knot gotten by twisting times along and give a lower bound on the bridge number of with respect to Heegaard splittings of ; as a function of , the genus of the splitting, and the catching surface . As a result, the bridge number of tends to infinity with . In application, we look at a family of knots found by Teragaito that live in a small Seifert fiber space and where each admits a Dehn surgery giving . We show that the bridge number of with respect to any genus- Heegaard splitting of tends to infinity with . This contrasts with other work of the authors as well as with the conjectured picture for knots in lens spaces that admit Dehn surgeries giving .
We define a homology for closed braids by applying Khovanov and Rozansky’s matrix factorization construction with potential . Up to a grading shift, is the HOMFLYPT homology defined by Khovanov and Rozansky. We demonstrate that for , is a –graded –module that is invariant under transverse Markov moves, but not under negative stabilization/destabilization. Thus, for , this homology is an invariant for transverse links in the standard contact , but not for smooth links. We also discuss the decategorification of and the relation between and the Khovanov–Rozansky homology.
Let be a group and let be a field of characteristic zero. We prove that if the Farrell–Jones conjecture for the –theory of is satisfied for every smooth –algebra , then it is also satisfied for every commutative –algebra .
We consider various constructions of monotone Lagrangian submanifolds of , , and quadric hypersurfaces of . In and we show that several different known constructions of exotic monotone tori yield results that are Hamiltonian isotopic to each other, in particular answering a question of Wu by showing that the monotone fiber of a toric degeneration model of is Hamiltonian isotopic to the Chekanov torus. Generalizing our constructions to higher dimensions leads us to consider monotone Lagrangian submanifolds (typically not tori) of quadrics and of which can be understood either in terms of the geodesic flow on or in terms of the Biran circle bundle construction. Unlike previously known monotone Lagrangian submanifolds of closed simply connected symplectic manifolds, many of our higher-dimensional Lagrangian submanifolds are provably displaceable.
We study the classifying space of a twisted loop group , where is a compact Lie group and is an automorphism of of finite order modulo inner automorphisms. Equivalently, we study the –twisted adjoint action of on itself. We derive a formula for the cohomology ring and explicitly carry out the calculation for all automorphisms of simple Lie groups. More generally, we derive a formula for the equivariant cohomology of compact Lie group actions with constant rank stabilizers.
We give a combinatorial proof of the quasi-invertibility of in bordered Heegaard Floer homology, which implies a Koszul self-duality on the dg-algebra , for each pointed matched circle . We do this by giving an explicit description of a rank 1 model for , the quasi-inverse of . To obtain this description we apply homological perturbation theory to a larger, previously known model of .
We show that the strong asymptotic class of Weil–Petersson geodesic rays with narrow end invariant and bounded annular coefficients is determined by the forward ending laminations of the geodesic rays. This generalizes the recurrent ending lamination theorem of Brock, Masur and Minsky. As an application we provide a symbolic condition for divergence of Weil–Petersson geodesic rays in the moduli space.
Let be a finite group. We define a suitable model-categorical framework for –equivariant homotopy theory, which we call –model categories. We show that the diagrams in a –model category which are equipped with a certain equivariant structure admit a model structure. This model category of equivariant diagrams supports a well-behaved theory of equivariant homotopy limits and colimits. We then apply this theory to study equivariant excision of homotopy functors.
Let be the fundamental group of the complement of the torus knot of type . It has a presentation . We find a geometric description of the character variety of characters of representations of into , and .
We use categorical skew Howe duality to find recursion rules that compute categorified invariants of rational tangles colored by exterior powers of the standard representation. Further, we offer a geometric interpretation of these rules which suggests a connection to Floer theory. Along the way we make progress towards two conjectures about the colored HOMFLY homology of rational links and discuss consequences for the corresponding decategorified invariants.
We consider the moduli spaces of a closed linkage with links and prescribed lengths in –dimensional Euclidean space. For these spaces are no longer manifolds generically, but they have the structure of a pseudomanifold.
We use intersection homology to assign a ring to these spaces that can be used to distinguish the homeomorphism types of for a large class of length vectors. These rings behave rather differently depending on whether is even or odd, with the even case having been treated in an earlier paper. The main difference in the odd case comes from an extra generator in the ring, which can be thought of as an Euler class of a stratified bundle.
We use the technique of quantum skew Howe duality to investigate the monoidal category generated by exterior powers of the standard representation of . This produces a complete diagrammatic description of the category in terms of trivalent graphs, with the usual MOY relations plus one additional family of relations. The technique also gives a useful connection between a system of symmetries on and the braiding on the category of –representations which can be used to construct the Alexander polynomial and coloured variants.
The definition of a category of spaces given in An étaléspace construction forstacks is problematic in that the main examples do not satisfy the axioms listed there. In this erratum, we give an alternative definition which encompasses our main examples and for which the results and proofs of this paper still hold.
We prove that the pure braid group of a nonorientable surface (closed or with boundary, but different from ) is residually –finite. Consequently, this group is residually nilpotent. The key ingredient in the closed case is the notion of –almost direct product, which is a generalization of the notion of almost direct product. We also prove some results on lower central series and augmentation ideals of –almost direct products.
We show that the exterior powers of the matrix valued random walk invariant of string links, introduced by Lin, Tian, and Wang, are isomorphic to the graded components of the tangle functor associated to the Alexander polynomial by Ohtsuki divided by the zero graded invariant of the functor. Several resulting properties of these representations of the string link monoids are discussed.
The main goal of this note is to suggest an algebraic approach to the quasi-isometric classification of partially commutative groups (alias right-angled Artin groups). More precisely, we conjecture that if the partially commutative groups and are quasi-isometric, then is a (nice) subgroup of and vice-versa. We show that the conjecture holds for all known cases of quasi-isometric classification of partially commutative groups, namely for the classes of –trees and atomic graphs. As in the classical Mostow rigidity theory for irreducible lattices, we relate the quasi-isometric rigidity of the class of atomic partially commutative groups with the algebraic rigidity, that is, with the co-Hopfian property of their –completions.