Given a line of three words, one cannot necessarily impose the trefoil quandle rules. However, under the product $(123)(456) = ((14)(25)(36))$ the Pauli quandle rules will apply to any cyclic triple of words. That is, we can use Pauli type quandles to fill in lines in a Fano plane. Now on the tetractys there are three sets of natural trefoil triples. One is given by the triple of lines:
Thus a true octonion contains three trefoil knots, whereas a split octonion may be specified by mixing a pair of quaternion trefoil lines.
6 years ago