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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.

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© Robert FERRÉOL  2018