On slicing invariants of knots

Owens, Brendan

doi:10.1090/S0002-9947-09-04904-6

On slicing invariants of knots
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Trans. Amer. Math. Soc. 362 (2010), 3095-3106 Request permission

Abstract:

The slicing number of a knot, $u_s(K)$, is the minimum number of crossing changes required to convert $K$ to a slice knot. This invariant is bounded above by the unknotting number and below by the slice genus $g_s(K)$. We show that for many knots, previous bounds on the unknotting number obtained by Ozsváth and Szabó and by the author in fact give bounds on the slicing number. Livingston defined another invariant $U_s(K)$, which takes into account signs of crossings changed to get a slice knot and which is bounded above by the slicing number and below by the slice genus. We exhibit an infinite family of knots $K_n$ with slice genus $n$ and Livingston invariant greater than $n$. Our bounds are based on restrictions (using Donaldson’s diagonalisation theorem or Heegaard Floer homology) on the intersection forms of four-manifolds bounded by the double branched cover of a knot.

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