2019/12/04 by Pichelmeyer, Jacob
#FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1912.01787
Our goal is to systematically compute the ℂP2-genus of all prime knots up to 8-crossings. We obtain upper bounds on the ℂP2-genus via coherent band surgery. We obtain lower bounds by obstructing homological degrees of potential slice discs. The obstructions are pulled from a variety of sources in low-dimensional topology and adapted to ℂP2. There are 27 prime knots and distinct mirrors up to 7-crossings. We now know the ℂP2-genus of all but 2 of these knots. There are 64 prime knots and distinct mirrors up to 8-crossings. We now know the ℂP2-genus of all but 9 of these knots. Where the ℂP2-genus was not determined explicitly, it was narrowed down to 2 possibilities. As a consequence of this work, we show an infinite family of knots such that the ℂP2-genus of each knot differs from that of it's mirror.