Numerical work with sagemath 24: 6j Symbols and non-eucledean Tetrahedra

This week I have begun to look at Hyperbolic Tetrahedra and their geometry. In the paper ‘6j Symbols for Uq and non-eucledean Tetrahedra‘, Taylor and Woodward  relate the semiclassical asymptotics of the 6j symbols for the  quantized enveloping algebra Uq(sl2) to the geometry of spherical and  hyperbolic tetrahedra.

The quantum 6j symbol is a function of a 6-tuple jab, 1 ≤ a ≤ b ≤ 4. The 6j symbols

6jsym6j for q = 1 were introduced as a tool in atomic spectroscopy
by Racah, and then studied mathematically by Wigner. 6j symbols
for Uq(sl2) were introduced by Kirillov and Reshetikhin, who used them to generalize the Jones knot invariant. Turaev and Viro used them to define three manifold invariants or  quantum gravity with a cosmological constant.

6jsymfig1

 

I have started doing some preliminary work with sagemath on 3j symbols, 6j symbols, the quantum integer and on the gram matrix.

graph 1 program6jvsj graph1

graph 2 program

quantumnvsn graph2

 

graph 3 program

3jvsj3 graph3

graph 4 program

ampltudevsj graph4

 

gram matrix

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