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Martin Scharlemann

  1. Berge's distance 3 pairs of genus 2 Heegaard splittings.

    Mon, 03/01/2010 - 16:01 — SomNambul
    Authors: Martin Scharlemann
    Subjects: Geometric Topology
    Abstract

    Following an example discovered by John Berge, we show that there is a
    4-component link L \subset (S^1 x S^2)#(S^1 x S^2) so that, generically, the
    result of Dehn surgery on L is a 3-manifold with two inequivalent genus 2
    Heegaard splittings, and each of these Heegaard splittings is of Hempel
    distance 3.

  2. Multiple genus 2 Heegaard splittings: a missed case.

    Wed, 10/21/2009 - 14:15 — SomNambul
    Authors: Martin Scharlemann, John Berge
    Subjects: Geometric Topology
    Abstract

    A gap in a paper of Rubinstein-Scharlemann is explored: new examples are
    found of closed orientable 3-manifolds with possibly multiple genus 2 Heegaard
    splittings. Properties common to all the examples in the original paper are not
    universally shared by the new examples: some of the new examples have Hempel
    distance 3, and it is not clear that a single stabilization always makes the
    multiple splittings isotopic.

  3. Multiple genus 2 Heegaard splittings: a missed case.

    Wed, 10/21/2009 - 14:15 — SomNambul
    Authors: Martin Scharlemann, John Berge
    Subjects: Geometric Topology
    Abstract

    A gap in a paper of Rubinstein-Scharlemann is explored: new examples are
    found of closed orientable 3-manifolds with possibly multiple genus 2 Heegaard
    splittings. Properties common to all the examples in the original paper are not
    universally shared by the new examples: some of the new examples have Hempel
    distance 3, and it is not clear that a single stabilization always makes the
    multiple splittings isotopic.

  4. Fibered knots and Property 2R, II.

    Thu, 08/20/2009 - 23:06 — SomNambul
    Authors: Robert E. Gompf, Martin Scharlemann
    Subjects: Geometric Topology
    Abstract

    A knot K in the 3-sphere is said to have Property nR if, whenever K is a
    component of an n-component link L and some integral surgery on L produces the
    connected sum of n copies of S^1 x S^2, there is a sequence of handle slides on
    L that converts L into a 0-framed unlink. The Generalized Property R Conjecture
    is that all knots have Property nR for all n.

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