arXiv:math/0506523v5; 30 pages, 15 figures. V5: minor revision. Notes for V5: Fixed a typo in the statement of the Brunnian properties, and clarified a counter-intuitive convention that knots are considered ‘split’ for the purposes of a theorem of Kanenobu; Journal Ref: L’enseignement Mathe’matique (2) 52 (2006), 319–359