The AJ conjecture and connected sums of torus knots

TL;DR

This paper verifies the AJ conjecture for certain connected sums of torus knots, revealing new phenomena in recurrence polynomials and suggesting a necessary modification to the conjecture.

Contribution

It proves the AJ conjecture for connected sums of torus knots with same sign parameters and identifies novel polynomial factorization behaviors.

Findings

Verified the AJ conjecture for connected sums of torus knots with same sign.

Discovered recurrence polynomials with repeated factors after evaluation at t=-1.

Identified the need to modify the AJ conjecture to account for these phenomena.

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 $p\ne a$, the recurrence polynomial $\alpha(t,M,L)$ 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.