Title: Unimodal Category and the Monotonicity Conjecture,
Abstract: We completely characterize the unimodal category for functions $f:\mathbb
R\to[0,\infty)$ 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 $f:S^1\to[0,\infty)$ 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.