2008/02/15 by Brendan Owens, Owens, Brendan
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #math.GT #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.0802.2109
14 pages, 2 figures
arxiv created 2008/02/15 · openalex publication_date 2008/02/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The slicing number of a knot, us(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 gs(K). We show that for many knots, previous bounds on unknotting number obtained by Ozsvath and Szabo and by the author in fact give bounds on the slicing number. Livingston defined another invariant Us(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 Kn 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.