Monsters in the hollow: counting Naiki braid patterns using de Bruijn's Monster theorem

ABSTRACT

The Japanese braids known as Naiki, which are distinguished by their hollow interior, have a simple structure shared by many other fiber arts and crafts. The way in which this structure forms a cylindrical braid imposes a particular set of symmetries on the final product. This paper uses enumerative combinatorics, including de Bruijn's Monster Theorem, to count the number of two-color Naiki braids under equivalence by this natural set of symmetries.

GRAPHICAL ABSTRACT

KEYWORDS:

• Braiding
• Japanese crafts
• fiber arts
• weaving
• counting
• patterns
• enumerative combinatorics
• generating functions
• Monster theorem
• symmetry

Acknowledgments

The author would like to thank Rosalie Neilson for introducing him to the Naiki technique and suggesting the project. Thanks also to the editors and to the anonymous reviewers for many helpful suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Traditionally called Burnside's Lemma but not due to Burnside, see Neumann (1979) for more.