arXiv research

In the 1980's Daryl Cooper introduced the notion of a C-complex (or clasp-complex) bounded by a link and explained how to compute signatures and polynomial invariants using a C-complex. Since then this was extended by works of Cimasoni, Florens, Mellor, Melvin, Conway, Toffoli, Friedl, and others to compute other link …

Corrected a false lemma in Cimasoni's work on linking theory.

Any two knots admit orientation preserving homeomorphic Seifert surfaces, as can be seen by stabilizing. There is a generalization of a Seifert surface to the setting of links called a C-complex. In this paper, we ask when two links will admit orientation preserving homeomorphic C-complexes. In the case of 2-component …

We introduce and study the notion of relative rigidity for pairs $(X,\JJ)$ where 1) $X$ is a hyperbolic metric space and $\JJ$ a collection of quasiconvex sets 2) $X$ is a relatively hyperbolic group and $\JJ$ the collection of parabolics 3) $X$ is a higher rank symmetric space and $\JJ$ an equivariant collection of ma…

For a symplectic manifold admitting a metaplectic structure and for a Kuiper map, we construct a complex of differential operators acting on exterior differential forms with values in the dual of the Kostant's symplectic spinor bundle. Defining a Hilbert $C^*$-structure on this bundle for a suitable $C^*$-algebra, we o…

The slope invariant is shown to be unchanged by concordance of colored links.

It is well known that the Blanchfield pairing of a knot can be expressed using Seifert matrices. In this paper, we compute the Blanchfield pairing of a colored link with non-zero Alexander polynomial. More precisely, we show that the Blanchfield pairing of such a link can be written in terms of generalized Seifert matr…

We extend our earlier work in [TZ1], where an analytic approach to the Guillemin-Sternberg conjecture [GS] was developed, to cases where the Spin$^c$-complex under consideration is allowed to be further twisted by certain natural exterior power bundles. The main result is a weighted quantization formula in the presence…

We extend our earlier work in [TZ1], where an analytic approach to the Guillemin-Sternberg conjecture [GS] was developed, to cases where the Spin^c-complex under consideration is allowed to be further twisted by certain exterior power bundles of the cotangent bundle. The main result is a weighted quantization formula i…

The intersection pattern of the translates of the limit set of a quasi-convex subgroup of a hyperbolic group can be coded in a natural incidence graph, which suggests connections with the splittings of the ambient group. A similar incidence graph exists for any subgroup of a group. We show that the disconnectedness of …

Algorithm calculates Seifert matrices for colored links.

Classifies complex structures on SL(2,C), finding new non-regular ones.

We show that mapping class groups associated to all types of real algebraic curves are virtual duality groups. We also deduce some results about the orbifold homotopy groups of the moduli spaces of real algebraic curves. We achieve these results by defining a new complex associated to a not necessarily orientable surfa…

Study complex slices on real algebraic varieties and their properties.

We test the 3d-3d correspondence for theories that are labelled by Lens spaces. We find a full agreement between the index of the 3d ${\cal N}=2$ "Lens space theory" $T[L(p,1)]$ and the partition function of complex Chern-Simons theory on $L(p,1)$. In particular, for $p=1$, we show how the familiar $S^3$ partition func…

AlgebraNets use alternative algebras for neural networks, improving performance on image and language tasks.

Introduces a space of almost complex structures for complex Lie group bundles.