Splice of knots

The splice of knots $K_i\subset Y_i$ is obtained by gluing their exteriors so that the meridian of each knot is identified, up to the orientation signs, with the longitude of the other. Thus the gluing exchanges meridians and longitudes.

Let $Y_1^{\prime }=X_1({\lambda }_1)$ and orient its filling core $K_1^{\prime }$ so that its longitude is ${\mu }_1$. Then zero surgery on $K_1\#K_2$ is the splice of $(Y_1^{\prime },K_1^{\prime })$ and $(Y_2,K_2)$. Cutting the connected-sum exterior along its decomposing annulus and applying the annulus-quotient model of Dehn filling identifies ${\lambda }_1$ with ${\lambda }_2$ and ${\mu }_1$ with ${\mu }_2$, which is exactly the splice gluing.