2013/11/26 by Brendan Owens, Owens, Brendan, Saso Strle +1 · 2 citations
Mathematics · #57M25 #57M27 (Primary) #57Q60 (Secondary) #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:57M25 #msc:57M27 #msc:57Q60
paper · pdf · doi:10.48550/arxiv.1311.6702
24 pages, 5 figures. V2: added section on Gordian distances between torus knots. V3: Improved exposition incorporating referees' suggestions. Accepted for publication in Comm. Anal. Geom
arxiv created 2015/11/04 · arxiv updated 2015/11/05
We study three knot invariants related to smoothly immersed disks in the four-ball. These are the four-ball crossing number, which is the minimal number of normal double points of such a disk bounded by a given knot; the slicing number, which is the minimal number of crossing changes to a slice knot; and the concordance unknotting number, which is the minimal unknotting number in a smooth concordance class. Using Heegaard Floer homology we obtain bounds that can be used to determine two of these invariants for all prime knots with crossing number ten or less, and to determine the concordance unknotting number for all but thirteen of these knots. As a further application we obtain some new bounds on Gordian distance between torus knots. We also give a strengthened version of Ozsvath and Szabo's obstruction to unknotting number one.