The equation of the Kenyon-Smillie $(2,3,4)$-Teichm\"uller curve

TL;DR

This paper computes the algebraic equation of the universal family over the Kenyon-Smillie (2,3,4)-Teichmüller curve, providing geometric insights and confirming its Teichmüller curve status through cohomology analysis.

Contribution

It explicitly determines the algebraic equation of the universal family and offers a geometric description of the torsion map, with an independent proof of the curve's Teichmüller nature.

Findings

Explicit algebraic equation of the universal family

Geometric description of the torsion map

Confirmation of Teichmüller curve via Picard-Fuchs equation

Abstract

We compute the algebraic equation of the universal family over the Kenyon-Smillie $(2,3,4)$-Teichm\"uller curve and give a nice geometric description of the torsion map. Moreover, we re-prove independently that the found algebraic equation describes a Teichm\"uller curve by computing the Picard-Fuchs equation associated to the absolute cohomology bundle.