SurfaceAutlso

This application integrates the core computational functions of surface mapping groups, supporting Dehn twist generation, Nielsen-Thurston classification, and pseudo-Anosov mapping analysis. Users can input mapping-related terms through an intuitive graphical interface to perform real-time calculations of Alexander polynomials and Jones polynomials, and verify Thistlethwaite's theorem. The application employs double-sheet surface visualization technology to dynamically display the leaf-like structures of stable and unstable flows, coupled with a stretching factor adjustment function, helping users deeply understand the mathematical essence of pseudo-Anosov mappings. The geometric implementation module supports switching between hyperbolic, Euclidean, and spherical geometries, coupled with the visual presentation of parameters such as geodesics, area, and curvature, making abstract geometric concepts clear and tangible. The surface type selector provides functions for genus and marked point configuration, automatically calculating the Euler characteristic and fundamental group representation.