I am looking for a resource to learn about the double branched cover of a knot. How do we construct them? Given a knot in $S^3$, is there a way to get a cell decomposition of the double branched cover? CW? Heegaard splitting?
How do we know double branched covers exist? How do we know they are unique? Can we talk of double branched covers of knots embedded in other three manifolds?
Thanks!