M Theory Lesson 172
In a post about TWF 261, Baez asks about the trefoil and a knot quandle. The knot quandle looks like
$i \cdot j = k$
$j \cdot k = i$
$k \cdot i = j$
which M theorists will recognise as a three dimensional cyclic rule similar to the logic of mass operators. The braid group $B_3$ (recall that this is the fundamental group of the complement of the trefoil) is associated to this quandle, and this is our favourite group covering the modular group.
Michael Batanin pointed out that Loday had put the trefoil on the Stasheff associahedron. In M Theory we like to put knot crossings on the squares of this $3D$ polytope, because the polytope can be turned into the pair of pants with marked trivalent vertices which we put onto the Riemann sphere (which has a lot to do with the modular group) and recall that the squares end up on the real axis, where we might eventually want branch cuts that can accommodate knot crossings, just as in the Ghrist ribbon templates.
Clearly there is something very fundamental about knots here that we do not really understand. Recall that we also wanted knots on the squares (rather circle boundaries) of the pants so that we could use planar diagrams orthogonally to the Chern-Simons type knots contained in the tubes, and where we could rotate sources and targets on the circles before gluing.
$i \cdot j = k$
$j \cdot k = i$
$k \cdot i = j$
which M theorists will recognise as a three dimensional cyclic rule similar to the logic of mass operators. The braid group $B_3$ (recall that this is the fundamental group of the complement of the trefoil) is associated to this quandle, and this is our favourite group covering the modular group.
Michael Batanin pointed out that Loday had put the trefoil on the Stasheff associahedron. In M Theory we like to put knot crossings on the squares of this $3D$ polytope, because the polytope can be turned into the pair of pants with marked trivalent vertices which we put onto the Riemann sphere (which has a lot to do with the modular group) and recall that the squares end up on the real axis, where we might eventually want branch cuts that can accommodate knot crossings, just as in the Ghrist ribbon templates.
Clearly there is something very fundamental about knots here that we do not really understand. Recall that we also wanted knots on the squares (rather circle boundaries) of the pants so that we could use planar diagrams orthogonally to the Chern-Simons type knots contained in the tubes, and where we could rotate sources and targets on the circles before gluing.
posted by Kea | 7:43 PM
1 Comments:
Hi Kea,
1 - In TWF, Baez has an image of a trefoil on torus [just above section 5)].
This reminds me of a Tokamak used for fusion power in plasma physics. This is also like a Smale Solenoid.
2 - Look at what 63 year old AN Trahtman did in symbolic dynamics The road coloring problem
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