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We define a variation of Khovanov homology formally similar to totally twisted Heegaard–Floer homology. Over a certain field, this version of Khovanov homology has a completely explicit description in terms of the spanning trees of a link projection. We prove that this new theory is a link invariant and describe some of its properties. Finally, we provide the results of some computer computations of the invariant.
A concrete model for a –dimensional gauge theory under special holonomy is proposed, within the paradigm of Donaldson and Thomas, over the asymptotically cylindrical –manifolds provided by Kovalev’s solution to a noncompact version of the Calabi conjecture.
One obtains a solution to the –instanton equation from the associated Hermitian Yang–Mills problem, to which the methods of Simpson et al are applied, subject to a crucial asymptotic stability assumption over the “boundary at infinity”.
We determine the –graded Eilenberg–MacLane cohomology (with mod coefficients) of the real, infinite Grassmannians in the case . Possible connections to motivic characteristic classes of quadratic bundles are briefly discussed.
We give new tightness criteria for positive surgeries along knots in the –sphere, generalising results of Lisca and Stipsicz, and Sahamie. The main tools will be Honda, Kazez and Matić’s, and Ozsváth and Szabó’s Floer-theoretic contact invariants. We compute Ozsváth–Szabó contact invariant of positive contact surgeries along Legendrian knots in the –sphere in terms of the classical invariants of the knot. We also combine a Legendrian cabling construction with contact surgeries to get results about rational contact surgeries.
We show that the figure-eight knot complement admits a uniformizable spherical structure, ie it occurs as the manifold at infinity of a complex hyperbolic orbifold. The uniformization is unique provided we require the peripheral subgroups to have unipotent holonomy.
For any symmetric collection of natural numbers, we construct a smooth complex projective variety whose weight- Hodge structure has Hodge numbers ; if is even, then we have to impose that is bigger than some quadratic bound in . Combining these results for different weights, we solve the construction problem for the truncated Hodge diamond under two additional assumptions. Our results lead to a complete classification of all nontrivial dominations among Hodge numbers of Kähler manifolds.
We address the question of which convex shapes, when packed as densely as possible under certain restrictions, fill the least space and leave the most empty space. In each different dimension and under each different set of restrictions, this question is expected to have a different answer or perhaps no answer at all. As the problem of identifying global minima in most cases appears to be beyond current reach, in this paper we focus on local minima. We review some known results and prove these new results: in two dimensions, the regular heptagon is a local minimum of the double-lattice packing density, and in three dimensions, the directional derivative (in the sense of Minkowski addition) of the double-lattice packing density at the point in the space of shapes corresponding to the ball is in every direction positive.
We build homogeneous quasimorphisms on the universal cover of the contactomorphism group for certain prequantizations of monotone symplectic toric manifolds. This is done using Givental’s nonlinear Maslov index and a contact reduction technique for quasimorphisms. We show how these quasimorphisms lead to a hierarchy of rigid subsets of contact manifolds. We also show that the nonlinear Maslov index has a vanishing property, which plays a key role in our proofs. Finally we present applications to orderability of contact manifolds and Sandon-type metrics on contactomorphism groups.
Let be a translation surface. We show that certain deformations of supported on the set of all cylinders in a given direction remain in the –orbit closure of . Applications are given concerning complete periodicity, affine field of definition and the number of parallel cylinders which may be found on a translation surface in a given orbit closure.
We investigate constraints on embeddings of a nonorientable surface in a –manifold with the homology of , where is a rational homology –sphere. The constraints take the form of inequalities involving the genus and normal Euler class of the surface, and either the Ozsváth–Szabó –invariants or Atiyah–Singer –invariants of . One consequence is that the minimal genus of a smoothly embedded surface in is the same as the minimal genus of a surface in . We also consider embeddings of nonorientable surfaces in closed –manifolds.
Proposition 5.6 and Corollary 5.7 of our paper “Monopole Floer homology and Legendrian knots”, which use the invariants of that paper to construct nonloose Legendrian knots in overtwisted contact manifolds, are incorrect. In this erratum we explain the problem with the proof of Proposition 5.6 and why it cannot be true.
We construct hyperbolic integer homology –spheres where the injectivity radius is arbitrarily large for nearly all points of the manifold. As a consequence, there exists a sequence of closed hyperbolic –manifolds that Benjamini–Schramm converge to whose normalized Ray–Singer analytic torsions do not converge to the –analytic torsion of . This contrasts with the work of Abert et al who showed that Benjamini–Schramm convergence forces convergence of normalized Betti numbers. Our results shed light on a conjecture of Bergeron and Venkatesh on the growth of torsion in the homology of arithmetic hyperbolic –manifolds, and we give experimental results which support this and related conjectures.