Related papers: Closed surfaces and character varieties
We give a brief introduction to some of the recent works on finding geometric structures on triangulated surfaces using variational principles.
This is a report on our ongoing research on a combinatorial approach to knot recognition, using coloring of knots by certain algebraic objects called quandles. The aim of the paper is to summarize the mathematical theory of knot coloring in…
We consider closed and orientable immersed hypersurfaces of translational manifolds. Given a vector field on such a hypersurface, we define a perturbation of its Gauss map, which allows us to obtain topological invariants for the immersion…
Given a unit vector field on a closed Euclidean hypersurface, we define a map from the hypersurface to a sphere in the Euclidean space. This application allows us to exhibit a list of topological invariants which combines the second…
We study the existence of topologically closed complex curves normalized by bordered Riemann surfaces in complex spaces. Our main result is that such curves abound in any noncompact complex space admitting an exhaustion function whose Levi…
Quandle cocycle invariants form a powerful and well developed tool in knot theory. This paper treats their variations - namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret the former as particular…
We describe the various types of singularities that can arise for second order rational mappings and we discuss the historical and present-day, practical, role the singularity confinement property plays as an integrability detector. In…
This elementary article introduces easy-to-manage invariants of genus one knots in homology 3-spheres. To prove their invariance, we investigate properties of an invariant of 3-dimensional genus two homology handlebodies called the…
We show that Thurston's skinning maps of Teichmuller space have finite fibers. The proof centers around a study of two subvarieties of the SL_2(C) character variety of a surface, one associated to complex projective structures and the other…
Classical shape descriptors such as Heat Kernel Signature (HKS), Wave Kernel Signature (WKS), and Signature of Histograms of OrienTations (SHOT), while widely used in shape analysis, exhibit sensitivity to mesh connectivity, sampling…
This paper studies properly embedded surfaces in the 4-ball that are exotically knotted (i.e., topologically but not smoothly isotopic), and leverages this local phenomenon to study surfaces in larger 4-manifolds. The main results provide a…
We study invariant Seifert surfaces for strongly invertible knots, and prove that the gap between the equivariant genus (the minimum of the genera of invariant Seifert surfaces) of a strongly invertible knot and the (usual) genus of the…
We characterize contractible curves on proper normal algebraic surfaces in terms of complementary Weil divisors. Using this we generalize the classical criteria of Castelnuovo and Artin. As application we derive a finiteness result on…
We extend the configurations discussed in Burghelea's book and Burghelea-Haller's paper on topology of angle-valued maps, equivalently the closed, open and closed-open bar codes from real- or angle-valued maps, to topological closed one…
In this note we give an algebraic and topological interpretation of essential coordinate components of characteristic varieties and illustrate their importance with an example.
We show that for each pair of positive integers g and n, there are infinitely many tunnel number one knots, whose exteriors contain an essential meridional surface of genus g, and with 2n boundary components. We also show that for each…
The conditions for a cuspidal edge, swallowtail and other fundamental singularities are given in the context of Lie sphere geometry. We then use these conditions to study the Lie sphere transformations of a surface.
Effective bounds for the finite number of surjective holomorphic maps between canonically polarized compact complex manifolds of any dimension with fixed domain are proven. Both the case of a fixed target and the case of varying targets are…
We define singular points of the first kind and singular points of the second kind as singular points of mappings between surfaces. Typical examples of these singular points are fold singular points and cusp singular points, respectively.…
Through numerical examples we show that the complex scaling method is suited to explore the pole structure in multichannel scattering problems. All poles lying on the multisheeted Riemann energy surface, including shadow poles, can be…