The AJ conjecture and connected sums of torus knots

The set of isotopy classes of nontrivial torus knots $T(p,q)$ in $S^3$ is in bijection with the set of coprime integer pairs $(p,q)$ satisfying $\mathrm{\mid }p\mathrm{\mid }>q\ge 2$. We verify the AJ conjecture for the connected sums $T(p,q)\mathrm{\#}T(a,b)$ when $p$ and $a$ have the same sign. Notably, in cases where $pq=ab$ but $p\mathrm{\ne }a$, the recurrence polynomial $\alpha (t,M,L)$ of $T(p,q)\mathrm{\#}T(a,b)$ has repeated factors involving the variable $L$ after evaluation at $t=-1$. These appear to be the first examples of knots exhibiting this phenomenon. Therefore, the AJ conjecture requires a slight modification to accommodate this possibility.