Noncommutative Counting Invariants and Curve Complexes

In our previous paper, viewing $D^b(K(l))$ as a non-commutative curve, where $K(l)$ is the Kronecker quiver with $l$-arrows, we introduced categorical invariants via counting of non-commutative curves. Roughly, these invariants are sets of subcategories in a given category and their quotients. The non-commutative curve-counting invariants are obtained by restricting the subcategories to be equivalent to $D^b(K(l))$. The general definition defines much larger class of invariants and many of them behave properly with respect to fully faithful functors. Here, after recalling the definition, we focus on examples and extend our studies beyond counting. We enrich our invariants with structures: the inclusion of subcategories makes them partially ordered sets, and considering semi-orthogonal pairs of subcategories as edges amount to directed graphs. In addition to computing the non-commutative curve-counting invariants in $D^b(Q)$ for two affine quivers, for $ A_n$ and $D_4$ we derive formulas for counting of the subcategories of type $D^b(A_k)$ in $D^b(A_n)$, whereas for the two affine quivers and for $D_4$ we determine and count all generated by an exceptional collection subcategories. Estimating the numbers counting non-commutative curves in $D^b({\mathbb P}^2)$ modulo group action we prove finiteness and that an exact determining of these numbers leads to proving (or disproving) of Markov conjecture. Regarding the mentioned structure of a partially ordered set we initiate intersection theory of non-commutative curves. Via the structure of a directed graph we build an analogue to the classical curve complex used in Teichmueller and Thurston theory. The paper contains many pictures of graphs and presents an approach to Markov Conjecture via counting of subgraphs in a graph associated with $D^b(P^2)$. Some of the results proved here were announced in the previous work.