712 research outputs found
On Link Homology Theories from Extended Cobordisms
This paper is devoted to the study of algebraic structures leading to link
homology theories. The originally used structures of Frobenius algebra and/or
TQFT are modified in two directions. First, we refine 2-dimensional cobordisms
by taking into account their embedding into the three space. Secondly, we
extend the underlying cobordism category to a 2-category, where the usual
relations hold up to 2-isomorphisms. The corresponding abelian 2-functor is
called an extended quantum field theory (EQFT). We show that the Khovanov
homology, the nested Khovanov homology, extracted by Stroppel and Webster from
Seidel-Smith construction, and the odd Khovanov homology fit into this setting.
Moreover, we prove that any EQFT based on a Z_2-extension of the embedded
cobordism category which coincides with Khovanov after reducing the
coefficients modulo 2, gives rise to a link invariant homology theory
isomorphic to those of Khovanov.Comment: Lots of figure
Coxeter groups and random groups
For every dimension d, there is an infinite family of convex co-compact
reflection groups of isometries of hyperbolic d-space --- the superideal
(simplicial and cubical) reflection groups --- with the property that a random
group at any density less than a half (or in the few relators model) contains
quasiconvex subgroups commensurable with some member of the family, with
overwhelming probability.Comment: 18 pages, 14 figures; version 2 incorporates referee's correction
Embeddability of open-ended carbon nanotubes in hypercubes
AbstractA graph that can be isometrically embedded into a hypercube is called a partial cube. An open-ended carbon nanotube is a part of hexagonal tessellation of a cylinder. In this article we determine all open-ended carbon nanotubes which are partial cubes
Asymptotic Equivalence of Symplectic Capacities
A long-standing conjecture states that all normalized symplectic capacities
coincide on the class of convex subsets of . In this note we
focus on an asymptotic (in the dimension) version of this conjecture, and show
that when restricted to the class of centrally symmetric convex bodies in
, several symplectic capacities, including the
Ekeland-Hofer-Zehnder capacity, the displacement energy capacity, and the
cylindrical capacity, are all equivalent up to an absolute constant.Comment: 12 pages, no figure
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