Thus a true octonion contains three trefoil knots, whereas a split octonion may be specified by mixing a pair of quaternion trefoil lines.
14 years ago
Sunday, February 6, 2011
Theory Update 59
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.
Thus a true octonion contains three trefoil knots, whereas a split octonion may be specified by mixing a pair of quaternion trefoil lines.
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This sounds like an important conclusion and seems to me to relate to that difference between bosons and fermions.
ReplyDeleteIndeed, ThePeSla, the particle zoo becomes far, far clearer in this language. There is a transformation between fermions and bosons, but it is not the same thing as the (now dead) Susy so beloved of stringers, because it occurs discretely in the octonion algebras of braids.
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