The splicing conjecture for knot complements and L-spaces

For $i\in\{1,2\}$, let $K_i$ be nontrivial knots in $L$-space homology spheres $Y_i$, with meridians $\mu_i$ and Seifert longitudes $\lambda_i$. If $\tau(K_1)>0$, set $t=2\tau(K_1)-1$, while if $\tau(K_1)<0$, set $t=2\tau(K_1)+1$. Let $Y$ be obtained by gluing the exterior of $K_1$ to the exterior of $K_2$ so that

$$\mu_1\text{ is identified with }p\mu_2+q\lambda_2,$$

and

$$\lambda_1+t\mu_1\text{ is identified with }r\mu_2+s\lambda_2.$$

Splicing conjecture. The manifold $Y$ is an $L$-space if and only if all of the following conditions hold:

• $K_1$ and $K_2$ are $L$-space knots;
• if $\tau(K_1)>0$, then $\frac{p}{q}>\frac{r}{s}$, while if $\tau(K_1)<0$, then $\frac{p}{q}<\frac{r}{s}$;
• if $\tau(K_2)>0$, then $\frac{p}{q},\frac{r}{s}\in(2\tau(K_2)-1,\infty)$, while if $\tau(K_2)<0$, then $\frac{p}{q},\frac{r}{s}\in(-\infty,2\tau(K_2)+1)$.

This conjecture proposes an extension of the $L$-space characterization for splicings of integer-framed knot complements to arbitrary gluing maps, including rational framings. Its motivation comes from conjectural criteria for gluing graph manifolds along a torus boundary, but the rationally framed bordered invariant needed to prove it is less well understood.