The Search for Nonsymmetric Ribbon Knots

Abstract

We present the results of Axel Seeliger’s tabulation of symmetric union presentations for ribbon knots with crossing numbers 11 and 12 and exhibit possible examples for ribbon knots which are not representable as symmetric unions. In addition, we give a complete atlas of band diagrams for prime ribbon knots with 11 and 12 crossings.

keywords:

• symmetric union presentations of ribbon knots
• atlas of ribbon knots

2000 Mathematics Subject Classification.:

• 57M25
• 57M27

View correction statement:

Correction

Correction Statement

This article was originally published with errors, which have now been corrected in the online version. Please see Correction. https://doi.org/10.1080/10586458.2023.2223097.

Acknowledgments

I thank Lukas Lewark and Johannes Renkl for commenting on earlier versions of this article and the referee for additional valuable help. The support of Axel Seeliger, who has left knot theory research to work in industry, is gratefully acknowledged.

Notes

1 In the context of two-dimensional knots the notion of “symmetric ribbon spheres” was used by Takeshi Yajima [Yajima 64]. He proved that ribbon 2-knots can be deformed to a symmetric position in ${\mathbb{R}}^4$ (symmetric to the plane $x_4=0$). For comparison with classical knots, we summarize the situation for 2-knots: all 2-knots are slice. There are slice 2-knots which are not ribbon. All ribbon 2-knots are symmetric (in the above sense).