The AJ conjecture and connected sums of torus knots

Xingru Zhang

paper

The AJ conjecture and connected sums of torus knots

arXiv:2603.10235

Abstract

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 $∣ p ∣ > q \geq 2$ . We verify the AJ conjecture for the connected sums $T (p, q) # T (a, b)$ when $p$ and $a$ have the same sign. Notably, in cases where $pq = ab$ but $p \neq = a$ , the recurrence polynomial $α (t, M, L)$ of $T (p, q) # T (a, b)$ 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.