Two-fold branched cover of a knot (${\Sigma }_2(K)$)

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.

For a rational homology three-sphere $Y$, the linking form is the nonsingular pairingReversing the orientation of $Y$ negates this form.

Here the paired elements lie in the torsion subgroup of first homology.