Two-fold branched cover of a knot
= Two-fold branched cover of a knot
{title2=$\Sigma_2(K)$}
The two-fold branched cover of a knot is the double cover of $S^3$ branched along the knot. Its first homology has a nonsingular linking form and, when finite, has order $\det K$.
Away from the knot this is a two-sheeted <covering space>; at the knot it has the standard local model of a <branched covering>.