Introduction: Topological Mesh Modeling 
Topological Mesh Modeling is an umbrella term that covers all our work based on extensions the theory of graph rotation systems.
It includes (1) Orientable 2manifol mesh modeling using graph rotation systems and its computer graphics applications,
(2) Knot modeling with immersions of nonorientable manifold meshes and (3) Topological constructions that is based on geometric and physical
constraints with graph rotation systems. We recently started to work on
immersions of 3manifolds as a representation to develop shape modeling systems. Click links below to go to related papers and manuscripts.
 Orientable Mesh Modeling: We have
provided a solid foundation for orientable 2manifold mesh modeling using graph rotation systems.
Based on this theory, we have developed TopMod ,
which is is an orientable 2manifold mesh modeling system. TopMod provides
a wide vriety of High Genus Modeling tools, Remeshings & Subdivisions, and
Extrusions & Replacements.
Using TopMod, one can find a wide variety of ways to create high genus shapes;
almost all subdivision algorithms, wide variety of ways to remeshing shapes and
new extrusions. These tools are also useful for Architectural applications, Design and Sculpting
and Sketch Based Modeling.
 Knots Modeling: We have developed provided
a solid foundation for knot, link and cyclic woven object modeling using extended graph rotation systems.
If we twist an arbitrary subset of edges of a mesh on an
orientable surface, we can obtain nonorientable surfaces. The resulting extended graph rotation system can be used
to induce a cyclic weaving on the original surface, that corresponds a 3space immedding of a nonorientable surface.
 Topological Constructions: Discrete GaussianBonnet theorem and Gaussian curvatures related
mesh topologic concepts to geometry. Using this relationship, we have developed methods to phsyically construct shapes.
 Immersions of 3Manifolds: Using an extension of graph rotation systems it is possible
to represent 3space immersions of 3manifolds by employing a topological graph theory concept called 3D thickening.

