LINEAR CELTIC KNOT

 Fantasist page.

Using the following three basic pieces:   A, and its left/right flip, that will provide a crossing, B, and its left/right flip, that will provide two crossings, M, that will provide three crossings,
we build structures of the type AMnA, with 3n+2 crossings, of the type BMnA, with 3n+3 crossings, and of the type BMnB, with 3n+4 crossings.
This way, we get a sequence of knots or links with one example for any number of crossings.

First examples:

 TYPE AA: 1+1 = 2 crossings, Hopf link TYPE BA: 2+1 = 3 crossings, trefoil knot TYPE BB: 2+2 = 4 crossings, Solomon's knot TYPE AMA: 1+3+1 = 5 crossings, Whitehead link TYPE BMA: 2+3+1 = 6 crossings, knot 6.1.2 TYPE BMB: 2+3+2 = 7 crossings, knot 7.1.4 TYPE AM2A: 1+6+1 = 8 crossings, link 8.2.7 TYPE BM2A: 2+6+1 = 9 crossings, knot 9.1.20 TYPE BM2B: 2+6+2 = 10 crossings, link L10a101 Small theorem: the BMnA always are knots, the AMnA always have two blades, and the BMnB, which are rectangular billiard knots of the type (n+2, 2), are knots for odd values of n, and have two blades otherwise. This Wehrmacht officer's epaulette (see here) is a BM3A; knot K12a541

© Robert FERRÉOL  2018