- Ximena Fernandez
Finite_Spaces is a SAGE module to work with Finite Topological Spaces. We refer to [1] for the theoretical background and proofs of the correctness of the algorithms.
In [2], we developed a method to study Q**-transformations of group presentations based on finite spaces and a refinement of discrete Morse theory. It was applied to prove that potential counterexamples to the Andrews-Curtis conjecture do satisfy the conjecture. The computational experiments can be reproduced at Potential_counterexamples_AC.ipynb
Discrete Morse Theory for 3-deformations of 2-complexes in based in the notion of internal collapse showed at the following animation.
[1] Fernández, X. L. Combinatorial methods and algorithms in low dimensional topology and the Andrews-Curtis conjecture. PhD thesis, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 2017.
[2] Fernández, X. L. 3-deformations of 2-complexes and Morse Theory, arXiv:1912.00115, 2019.