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Maciej Borodzik

  1. On the signatures of torus knots.

    Чт, 02/25/2010 - 16:01 — SomNambul
    Authors: Maciej Borodzik, Krzysztof Oleszkiewicz
    Subjects: Geometric Topology
    Abstract

    We study properties of the signature function of the torus knot $T_{p,q}$.
    First we provide a very elementary proof of the formula for the integral of the
    signatures over the circle. We obtain also a closed formula for the
    Tristram--Levine signature of a torus knot in terms of Dedekind sums.

  2. An efficient method of finding a Puiseux expansion of a parametric singularity.

    Ср, 11/11/2009 - 16:01 — SomNambul
    Authors: Maciej Borodzik
    Subjects: Algebraic Geometry
    Abstract

    We provide a very effective and explicit algorithm of finding a Puiseux
    expansion of a cuspidal singularity of a plane curve, when this singularity is
    given in a parametric form.

  3. Deformations of singularities of plane curves. Topological approach.

    Сб, 09/26/2009 - 04:10 — SomNambul
    Authors: Maciej Borodzik
    Subjects: Algebraic Geometry
    Abstract

    We use a knot invariant, namely the Tristram--Levine signature to study
    deformations of singular points of plane curves. We find a bound on the sum of
    M numbers over all singularities of a generic fiber in terms of the M number of
    the singularity at the central fiber and some topological data.

  4. Puiseux coefficients and parametric deformation of plane curve singularities.

    Сб, 09/26/2009 - 04:10 — SomNambul
    Authors: Maciej Borodzik
    Subjects: Algebraic Geometry
    Abstract

    We study deformations of plane curve singularities from an analytic point of
    view and obtain some new concrete results. We show some rather unexpected
    properties of Puiseux coefficients treated as functions on a suitably defined
    parameter space. The methods used in paper are very elementary.

  5. Number of singular points of a genus $g$ curve with one point at infinity.

    Втр, 09/01/2009 - 12:47 — SomNambul
    Authors: Maciej Borodzik
    Subjects: Algebraic Geometry
    Abstract

    We bound the maximal number N of singular points of a plane algebraic curve C
    that has precisely one place at infinity with one branch in terms of its first
    Betti number $b_1(C)$. Asymptotically we prove that $N<\sim{17/11}b_1(C)$ for
    large $b_1$. In particular, in the case of curves with one place at infinity,
    we confirm the Zaidenberg and Lin conjecture stating that $N\le 2b_1+1$.

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