Regularized $+1/r$-surgery formula

Let $K$ be a knot, let $r$ be the surgery parameter, and write

$$F_K(x,q)=x^{\frac12}\sum_{j\geq 0}f_j(K;q)x^j.$$

Let

$$f(a,b):=\sum_{n\in\mathbb Z}a^{\frac{n(n+1)}2}b^{\frac{n(n-1)}2}=(-a;ab)_\infty(-b,ab)_\infty(ab;ab)_\infty$$

be the Ramanujan theta function. Regularized $+1/r$-surgery conjecture. When the $+1/r$-surgery formula of Gukov and Manolescu converges, it should be used. When it does not converge, one may use

$$\hat{Z}(S^3_{+1/r}(K))=q^{\frac{r+r^{-1}}4}\sum_{j\geq0}f_j(K;q)\left(q^{-r(j+\frac12-\frac{1}{2r})^2}-q^{-r(j+\frac12+\frac{1}{2r})^2}\right)\left(1-\frac{\sum_{|k|\leq j}(-1)^kq^{\frac{k((2r+1)k+1)}2}}{f(-q^r,-q^{r+1})}\right),$$

provided that this regularization converges. This extends the proposed regularization scheme from positive integer surgeries to reciprocal surgeries; the paper derives it from twist-knot examples, but its general validity remains conjectural.