I have been working on my research as a student at Wilbur Wright College on Topological Complexity. We solved the problem of two robots moving on a circle and letter $T$ using Farber's theorem but having difficulty finding $TC$ for two robots moving on a number $8$. Our group founded $2 \leq TC \leq 3$ using Farber's theorem. 


What will be the exact $TC$ for one/two robots moving on a number $8$ using Farber's theorem?


(Note: We founded configuration space for two robots moving on $8$ is a bouquet of seven circles(seven petals)



Topological Complexity $TC$ of two robots moving on number $8$
The topological complexity of $X$ is the minimum number of open sets needed to cover $X\times X$, on each of which the path fibration $X^I\to X\times X$ admits a local section. If I understood your question, then you are asking about the cases $X=S^1\vee S^1$, and $Y=F(S^1\vee S^1,2)$, the $2$-point ordered configuration space.


The wedge of circles $X$ has LS-category $2$ (by Jeff Strom's argument), and hence $TC(X)\le 2\,cat(X)-1 = 3$. The space $Y$ is the configuration space of a graph with one essential vertex, therefore $TC(Y)\le 3$ also: this follows from the theorem of Michael Farber you mention.


To see that both of these space have topological complexity $3$, you can apply a result of Greg Lupton, John Oprea and myself:


*Grant, Mark; Lupton, Gregory; Oprea, John*, [**Spaces of topological complexity one**](http://dx.doi.org/10.4310/HHA.2013.v15.n2.a4), Homology Homotopy Appl. 15, No. 2, 73-81 (2013). [ZBL1277.55001](https://zbmath.org/?q=an:1277.55001).


The main result of the cited paper is that if $TC(Z)=2$ then $Z$ has the homotopy type of an odd sphere (the "one" in the title of the paper is because we are using the reduced version of topological complexity which is one less than your definition). Thus if your space has $2\le TC(Z)\le 3$ and is not an odd sphere, then in fact $TC(Z)=3$. The arguments use little more than the standard zero-divisors cup-length lower bound.


The space $X$ is clearly not an odd sphere (it has nonabelian $\pi\_1$, for example). You seem to have convinced yourselves that $Y$ is not an odd sphere, either.


