occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

Tuesday, March 24, 2009

M Theory Lesson 266

Thinking once again about knotted and coloured string networks, it seems that trivalent vertices on three colours are a lot like a (Tutte) dual representation of a Pauli quandle for a trefoil, where each arc of the knot is a different colour.

These networks also allow crossings, such that the undercrossing preserves the colour. As a quandle rule, such colourings correspond to the action \$b \circ a = a\$. An example of a quandle that obeys this simple law is one generated by a single invertible operator \$M\$, such that \$M^{n} = 1\$. The quandle operation (as is usual for a group) is conjugation, and this acts trivially on \$M^{k}\$ because we have only powers of \$M\$ to play with. Without the trivalent vertices, this quandle generates \$n\$ separate link components, because the arcs at a crossing never mix.