Affiliation:
1. Kansas State University, USA
Abstract
Our goal is to systematically compute the [Formula: see text]-genus of as many prime knots up to 8-crossings as possible. We obtain upper bounds on the [Formula: see text]-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 [Formula: see text]. There are 27 prime knots and distinct mirrors up to 7-crossings. We now know the [Formula: see text]-genus of all of these knots. There are 64 prime knots and distinct mirrors up to 8-crossings. We now know the [Formula: see text]-genus of all but 6 of these knots, where the [Formula: see text]-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 [Formula: see text]-genus of each knot differs from that of its mirror.
Publisher
World Scientific Pub Co Pte Lt
Subject
Algebra and Number Theory
Cited by
6 articles.
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