The 6j symbols

Allow me to speak very loosely here. I’m still learning this algebra.

From one perspective, the 6j symbols describe how to translate from one splitting of a morphism to another. You may care to view a morphism as a directed graph. Here we have two morphisms viewed as trees; one directed upwards and one directed downwards. Though they branch differently, their ends correspond as shown below.

6jsymbols-cobordism edges

Though the two trees branch differently, there is a cobordism between them. As we slide the oddball branch from one position to the other, we sweep out 6 faces. This forms a pleasant sort of 2-complex with only one triple point.

6jsymbols-cobordism faces

Let us deform this cobordism version of the 2-complex to something more symmetric.

6jsymbols-equivalence

Here’s a larger picture of the end result.

6jsymbols-cone

This 2-complex is the cone of the closed graph below. And the 6j symbol can be seen as the trace of the corresponding morphism.

6jsymbols-cone edges

Here it is again in a different context.

6jsymbols-tetrahedron

Lol and behold: the spine of a tetrahedron!

6jsymbols-tetrahedron edges
And again with the edges dual to the faces.

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~ by Ken Baker on December 12, 2007.

7 Responses to “The 6j symbols”

  1. very interesting.
    i’m adding in RSS Reader

  2. Brilliant. You’ve read the descriptions of all of these things on Baez’ TWF and his QG seminar, right? The pictures really bring all that out.

  3. Thanks John, I hadn’t seen his descriptions.

    For everyone else, here’s Baez’s Quantum Geometry seminar and This Week’s Finds.

  4. Multivalency says : I absolutely agree with this !

  5. Intractability says : I absolutely agree with this !

  6. Miniskirted says : I absolutely agree with this !

  7. [...] Trivalent Cobordism Here’s a quickie — just checking out some Section Plane functionality of SketchUp. You can build a 2-complex and see how its slices evolve. This 2-complex is the same as the spine of a tetrahedron. [...]

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