Unimodal category and the monotonicity conjecture

Autor: Dejan Govc
Rok vydání: 2021
Předmět:
Zdroj: Journal of Applied and Computational Topology. 5:621-669
ISSN: 2367-1734
2367-1726
DOI: 10.1007/s41468-021-00077-z
Popis: We completely characterize the unimodal category for functions $$f:{\mathbb {R}}\rightarrow [0,\infty )$$ using a decomposition theorem obtained by generalizing the sweeping algorithm of Baryshnikov and Ghrist. We also give a characterization of the unimodal category for functions $$f:S^1\rightarrow [0,\infty )$$ and provide an algorithm to compute the unimodal category of such a function in the case of finitely many critical points. We then turn to the monotonicity conjecture of Baryshnikov and Ghrist. We show that this conjecture is true for functions on $${\mathbb {R}}$$ and $$S^1$$ using the above characterizations and that it is false on certain graphs and on the Euclidean plane by providing explicit counterexamples. We also show that it holds for functions on the Euclidean plane whose Morse–Smale graph is a tree using a result of Hickok, Villatoro and Wang.
Databáze: OpenAIRE
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