On slicing invariants of knots

Owens, B. (2010) On slicing invariants of knots. Transactions of the American Mathematical Society, 362, pp. 3095-3106. (doi: 10.1090/S0002-9947-09-04904-6)

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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Owens, Professor Brendan
• Authors: Owens, B.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Transactions of the American Mathematical Society
• Publisher: American Mathematical Society
• ISSN: 0002-9947
• Published Online: 13 August 2009