Knot Floer homology and the four-ball genus

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We use the knot filtration on the Heegaard Floer complex CF to define an integer invariant τ(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to ℤ. As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlike the signature, τ gives sharp bounds on the four-ball genera of torus knots. As another illustration, we calculate the invariant for several ten-crossing knots.

Original languageEnglish (US)
Pages (from-to)615-639
Number of pages25
JournalGeometry and Topology
StatePublished - 2003

All Science Journal Classification (ASJC) codes

  • Geometry and Topology


  • 4-ball genus
  • Floer homology
  • Knot concordance
  • Signature


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