$\newcommand{\hy}{\textrm{-}}$ I'm reading about support $\tau$-tilting modules and their mutations. I'm trying to understand the mutation quiver.
Let $A$ be a finite dimensional algebra over an algebraically closed field, which is $\tau$-tilting infinite. Let $Q(\mathrm{s\tau\hy tilt} A)$ be the mutation quiver of all the basic support $\tau$-tilting modules of $A$. I have found examples where $Q(\mathrm{s\tau\hy tilt} A)$ has two connected components (one component is given by left mutations starting from $A$ and the other component is given by the right mutations starting from $0$).
For a $\tau$-tilting infinite algebra, is it known when $Q(\mathrm{s\tau\hy tilt} A)$ is not connected? How many connected components it has?
I'm wondering if this $Q(\mathrm{s\tau\hy tilt} A)$ can have $3$ or more connected components?
If yes, then what is the "starting" module in these connected components (other than the ones that I've described above)?
What is an example of a $\tau$-tilting infinite algebra $A$ where $Q(\mathrm{s\tau\hy tilt} A)$ has $3$ or more connected components?
For any $k \geq 1$, one can construct an algebra with $2^k$ components in its mutation quiver of support $\tau$-tilting pairs:
Consider the quiver $Q$ with two vertices $1,2$, two arrows $1 \to 2$ and two arrows $2 \to 1$:
Now fix $A := \mathbb{C}Q/r^2$, i.e the quotient of the path algebra over $Q$ where we kill all paths of length 2.
It can be shown that the $\tau$-tilting theory of $A$ separates $(P(1) \oplus P(2),0)$ from the $\tau$-tilting pair $(0,P(1) \oplus P(2))$ in the mutation quiver; there is no finite path between these two tau-tilting pairs. To realize this, one may compare $A$ to the Kronecker algebra $K$, and realize that $A$ essentially inherits the $\tau$-tilting of $K$ and $K^{op}$ at the same time.
So $A$ has exactly two components in its mutation quiver of $\tau$-tilting pairs. Generally, if $B_1$ has $k_1$ components in its mutation quiver and $B_2$ has $k_2$ such components, $B_1 \times B_2$ will have $k_1k_2$ components in its mutation quiver. $A^k$ will therefore have $2^k$ components in its mutation quiver.
One can also find a connected algebra with more than 2 components in its mutation quiver. By adding an arrow to the quiver $Q + Q$, connecting the two components, and killing $r^2$, one obtains an algebra of rank 4 which can be shown to have at least 4 components in its mutation quiver. This is the main result of a preprint I have written, available on the arxiv https://arxiv.org/abs/2109.11464
Lastly, concerning your question on detecting when $Q(s\tau-\text{tilt } A)$ is connected, I want to add that some things are known: For gentle algebras, we know by work of Changjian Fu, Shengfei Geng, Pin Liu and Yu Zhou, that the mutation quiver is always connected: https://www.sciencedirect.com/science/article/pii/S0021869323001187