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3 results

  • PL-Genus of surfaces in homology balls

  • Part of
  • Jennifer Hom, Matthew Stoffregen, Hugo Zhou
  • Journal:
    Forum of Mathematics, Sigma / Volume 12 / 2024
    Published online by Cambridge University Press:
    25 January 2024, e14
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  • We consider manifold-knot pairs $(Y,K)$, where Y is a homology 3-sphere that bounds a homology 4-ball. We show that the minimum genus of a PL surface $\Sigma $ in a homology ball X, such that $\partial (X, \Sigma ) = (Y, K)$ can be arbitrarily large. Equivalently, the minimum genus of a surface cobordism in a homology cobordism from $(Y, K)$ to any knot in $S^3$ can be arbitrarily large. The proof relies on Heegaard Floer homology.

  • NONSURJECTIVE SATELLITE OPERATORS AND PIECEWISE-LINEAR CONCORDANCE

  • Part of
  • ADAM SIMON LEVINE
  • Journal:
    Forum of Mathematics, Sigma / Volume 4 / 2016
    Published online by Cambridge University Press:
    23 December 2016, e34
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  • We exhibit a knot $P$ in the solid torus, representing a generator of first homology, such that for any knot $K$ in the 3-sphere, the satellite knot with pattern $P$ and companion $K$ is not smoothly slice in any homology 4-ball. As a consequence, we obtain a knot in a homology 3-sphere that does not bound a piecewise-linear disk in any homology 4-ball.

  • Concordance des nœuds de dimension 4

  • Vincent Blanloeil, Osamu Saeki
  • Journal:
    Canadian Mathematical Bulletin / Volume 50 / Issue 4 / 01 December 2007
    Published online by Cambridge University Press:
    20 November 2018, pp. 481-485
    Print publication:
    01 December 2007
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  • Nous démontrons que tous les plongements d’une variété compacte sans bord et simplement connexe de dimension quatre dans la sphère de dimension six sont concordants.