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The diagram below indicates 2 ways to use Fenchel-Nielsen coordinates to parameterize the Teichmüller space of conformal structures on the 4-holed sphere with totally-geodesic boundary, corresponding to the two pants decompositions, as indicated by this diagram:

Here $a,b,c,d,x,y$ are lengths of the indicated geodesics in a hyperbolic instance of the conformal structure, and $\alpha,\beta$ are the twisting parameters for the gluing.

What is the coordinate transformation implied by these decompositions? That is, what is the formula for $y$ and $\beta$ as a function of $a,b,c,d,x,\alpha$?

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Just to be precise, there are infinitely many ways to give FN coordinates for the four-holed torus, one for every essential non-peripheral simple closed curve. –  Sam Nead Mar 2 at 14:32
    
Indeed, thanks for that point, I will edit the question to clarify this. –  Jamie Vicary Mar 2 at 14:36

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See the paper "Effects of a change of pants decompositions on their Fenchel-Nielsen coordinates" by Takayuki Okai, published in Kobe J. Math.

You can also find a version where some of the boundary components are cusps in the paper "The behaviour of Fenchel-Nielsen distance under a change of pants decomposition" by Alessandrini, Liu, Papadopoulos, and Su, published in Comm. Anal. Geom. and also available on the ArXiv.


Added by @JamieVicary: here are the equations as given in the Alessandrini, Liu and Papadopoulos paper in the notation of this question, with $a'=\text{cosh}(a/2)$, $b'=\text{cosh}(b/2)$, $c'=\text{cosh}(c/2)$, $d'=\text{cosh}(d/2)$, $x'=\text{cosh}(x/2)$, $y'=\text{cosh}(y/2)$, $\alpha' = \text{cosh}(\alpha)$ and $\beta' = \text{cosh}(\beta)$:

$$\begin{align} x'&=y' ^{-2} ( a'c'+d'b' + y'(a'd' + c'b') + \beta'(y'^2 + 2 a'b'y' + a'^2 + b'^2 - 1] ^{1/2}(y'^2 + 2 c'd'y' + c'^2 + d'^2)^{1/2}) \\ \alpha'&=(a'^2 + c'^2 + 2 a' c' x' + x'^2-1) ^{-1/2} ( d'^2 + b'^2 + 2 d' b' x' + x'^2 - 1) ^{-1/2} ((x'^2 - 1)y' - a'b' - c'd' - x'(a'd' + c'b')) \end{align}$$

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Hasn't this all been done earlier by Penner, with his Lambda lengths? –  Igor Rivin Mar 2 at 17:23

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