Circular knotworks consisting of pattern no. 295: a mathematical approach

Abstract

In Easter Ross in Scotland, there are a number of cross-slabs with various carvings of knotworks. This article deals with a specific circular knotwork on two of the stones originally found there, the Hilton of Cadboll and the Nigg stone. The knotworks are referred to as No. 791 and No. 792 respectively by J. Romilly Allen and Joseph Anderson in their book The Early Christian Monuments of Scotland. Both stones show recurring loops, referred to as pattern No. 295 by Allen and Anderson. This article will consider circular knotworks that can be created with pattern No. 295. We present a mathematical model for this circular knotwork and variations of it. Three parameters are used to characterize the knotwork: the number of circular layers, the number of loops in the innermost layer and finally, a parameter that specifies how the loops are interwoven in the innermost layer. We derive equations to calculate the number of strands for different parameters of the knotwork. The relationships derived involve the greatest common divisor function.

Keywords:

• Pictish sculpture
• Hilton of Cadboll
• knotworks
• greatest common divisor

AMS Subject Classifications:

• 57N16
• 57M25

Acknowledgements

We thank Dr Simon Morgan for fruitful discussions and Dr Colin Byfleet for helpful suggestions of improvements. We also thank the referees and the editor of the Journal of Mathematics and the Arts for their help in improving the presentation of our results.

Notes

1. The editors have noted that the layer index can also be calculated non-recursively by letting ϱ(u) = a + 1 for u = 1, … , 2 ^l  − 1 where a is the exponent of the highest power of 2 that divides u.