My question is:
What's a closed form of a $k$-sheeted covering map $p:X\to S$, where $S=\Bbb S^1\vee\cdots\vee\Bbb S^1$, is the bouquet of $n$ circles.
I have always seen the interesting example of a triple-sheeted covering of $\Bbb S^1\vee\Bbb S^1$. But everywhere it was only sketched without any motivation or hints about how to write down an actual covering map that does exactly this. I have tried searching online as well as here on MSE but found nothing similar. So, I'm guessing it is just either impossible or too tedious to define a map that does this? Is this true? If yes, then why and do we even know that such a covering map even exists? Maybe I'm missing a reference that does the tedious steps? TIA.