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For a knot $K\subset \mathbb{S}^3$, one can construct the covering space branched over that knot by assigning elements of the symmetric group $S_n$ to each arc of the knot. You can find the knot group in the cover by using Fox's variation of the Reidemeister-Schreier algorithm.

Since you can always turn a knot into a braid, I'm interested in doing this over the braid, and seeing what kinds of braids you end up with. The inspiration is the example 10D10 from Rolfsen, in which he demonstrates pictorially you can get a $(4,2)$ torus knot in a 3-fold branched covering space over the trefoil, a $(3,2)$ torus knot.

NEW IDEA: Trying to do something closer to what Rolfsen does: given the number of cycles in the representation $\pi(\mathbb{S}-K)\to S_n$, you know how many preimages of each component you have in the cover (for a branched cover, this will be less than $n$ for example). Now, for each twist in the braid in the base, there is a twist in the cover which exchanges components in the braid and corresponds to elements of the braid group. Say you do this over a trefoil, you three twists (a $3\pi$ rotation) in the torus knot.

So I think my problem reduces to finding the element of the braid group which corresponds to each exchange of the components in the base, which of course depends on the representation $S_n$.

FURTHER RESTRICTION: Clearly, if the branched cover itself is not simply-connected, you will have some trouble saying what the preimage of the knot is. If we need it, we could consider only those cases in which the cover is simply-connected, which can easily be checked.

PREVIOUS: If I had some kind of map

$\pi(\mathbb{S}^3-K)\to B_n$

whose image was the braid word represented by the knot $K$, I could do this in the cover. Since the knot group doesn't uniquely determine the knot, I guess such a map is pretty unlikely to be well-defined. But maybe you could have an algorithm for it if you just considered the relators at each knot crossing ($a=bcb^{-1}$, etc...) and considered the image of them under such a map.

Anyone know any generic results on the preimages of braids in branched covering spaces over braids?

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