We prove that the extrinsic Hausdorff dimension is always greater than or equal to the intrinsic Hausdorff dimension in models of triangulated random surfaces with action which is quadratic in the separation of vertices. We furthermore derive a few naive scaling relations which relate the intrinsic Hausdorff dimension to other critical exponents. These relations suggest that the intrinsic Hausd...

A Cn-move is a local move on links defined by Habiro and Goussarov, which can be regarded as a ‘higher order crossing change’. We use Milnor invariants with repeating indices to provide several classification results for links up to Cn-moves, under certain restrictions. Namely, we give a classification up to C4-moves of 2-component links, 3-component Brunnian links and n-component C3-trivial li...

The notion of compatible apparition points is introduced for non-Hausdorff manifolds, and properties of these points are studied. It is well known that the Hausdorff property is independent of the other conditions given in the standard definition of a topological manifold. In much of literature, a topological manifold of dimension n is a Hausdorff topological space which has a countable base of...

We study the Hausdorff Voronoi diagram of a set S of polygonal objects in the plane, a generalization of Voronoi diagrams based on the maximum distance of a point from a polygon, and show that it is equivalent to the Voronoi diagram of S under the Hausdorff distance function. We investigate the structural and combinatorial properties of the Hausdorff Voronoi diagram and give a divide and conque...

Andreas Zastrow conjectured, and Cannon-Conner-Zastrow proved, (see [3,pp. 44-45]) that filling one hole in the Sierpinski curve with a disk results in a planar Peano continuum that is not homotopy equivalent to a 1-dimensional set. Zastrow's example is the motivation for this paper, where we characterize those planar Peano continua that are homotopy equivalent to 1-dimensional sets. While many...

In this short note we show that the space of all ∂̄-free maps of any complex manifold Vc into Cn is always homotopically equivalent to the space of all sections of the corresponding bundle in the space of jets of smooth maps Vc → Cn. In particular, the space of all linear ordinary differential equations with complex-valued coefficients on an elliptic curve with the identical monodromy (defined a...

It has been known for over 30 years that every closed connected orientable 3manifold is obtained by surgery on a link in S [8]. However, a classification of such 3-manifolds in terms of this surgery construction has remained elusive. This is due primarily to the lack of uniqueness of the surgery description. In [5], Kirby gave us a calculus of surgery diagrams. However, the lack of a ‘canonical...

Inspired by classical puzzles in geometry that ask about probabilities of geometric phenomena, we give an explicit formula for the probability a random triangle on flat torus is homotopically trivial. Our main tool this computation involves reducing problem to new invariant measurable sets plane unchanged under area-preserving affine transformations. result show minimized at all rectangular tor...