Symmetric ribbon numbers of low-complexity knots

Abstract

Every knot $K\subset S^3$ that admits a symmetric union presentation bounds an immersed ribbon disk in $S^3$, while the converse is an open problem due to Christoph Lamm. The symmetric ribbon number $r_s(K)$ of $K$ is the minimum number of ribbon singularities in any symmetric ribbon disk bounded by $K$. In this paper, we undertake a systematic investigation of symmetric ribbon numbers of knots with at most 12 crossings. Along the way, we exhibit novel lower bounds for $r_s(K)$ arising from knot determinants, Alexander polynomials, Jones polynomials, and Kauffman polynomials.

31 pages, 32 figures, 5 tables, comments welcome!