occasional meanderings in physics' brave new world
- Name: Kea
- Location: New Zealand
Marni D. Sheppeard
Wednesday, March 21, 2007
A paper by D. Thurston introduces the idea of a knotted trivalent (ribbon) graph (KTG). It turns out that these gadgets may be generated from three simple graphs, namely the unknotted tetrahedron and two unknotted Mobius strips, one with a left twist and one with a right twist. The moves allowed on single graphs are the bubble move and the unzip move There is also a connected sum operation, which splices two graphs together along an edge. Any knot may be represented by a string of KTG operations. Bar-Natan's important paper on non-associative tangles includes a pentagon relation, which Thurston encodes via KTG moves. The sequence of three moves begins with three unknotted tetrahedra, which are connected via sum and then unzipped to obtain the final triangular prism This is of course reminiscent of the gluing of three sides of the parity cube on which the categorified Mac Lane pentagon appears.