The factorization conjecture for Alexander polynomials of equivariant slice knots

Let $K$ be a $q$-equivariant slice knot with Murasugi polynomial $\Delta_{{\bf Z}/q}(g,t)$. The polynomial ring ${\bf Z}[{\bf Z}/q\times {\bf Z}]$ is used, and the factorization condition is that there are nonzero polynomials $a(g,t)$ and $b(g,t)$ satisfying

$$\Delta_{{\bf Z}/q}(g,t)b(g,t)b(g^{-1},t^{-1})=a(g,t)a(g^{-1},t^{-1}),$$

and $a(g,1)=b(g,1)=1$.

Factorization conjecture. Alexander polynomials of equivariant slice knots satisfy this factorization condition with $b(g,t)=1$.

The preceding theorem proves the displayed factorization with an auxiliary factor $b(g,t)$ for equivariant slice knots, while the corresponding condition with $b(g,t)=1$ is known for equivariant ribbon knots. The conjecture asserts that the extra factor is unnecessary for all equivariant slice knots; the paper gives an example showing that algebraic considerations alone do not force this conclusion.