## On slicing invariants of knots

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- by Brendan Owens PDF
- 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.## References

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## Additional Information

**Brendan Owens**- Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
- Address at time of publication: Department of Mathematics, University of Glasgow, Glasgow, G12 8QW, United Kingdom
- Received by editor(s): April 11, 2008
- Published electronically: August 13, 2009
- Additional Notes: The author was supported in part by NSF grant DMS-0604876.
- © Copyright 2009 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**362**(2010), 3095-3106 - MSC (2000): Primary 57M25
- DOI: https://doi.org/10.1090/S0002-9947-09-04904-6
- MathSciNet review: 2592947