Top coefficients of ADO invariants are Murasugi-sum multiplicative

Abstract

The Akutsu-Deguchi-Ohtsuki (ADO) invariants of links are a non-semisimple version of the colored Jones polynomials. We prove that the top coefficients of the ADO invariants of links in $S^3$ are multiplicative under Murasugi-sum along connected minimal genus Seifert surfaces. More generally, for any simple complex Lie algebra $\mathfrak{g}$ of type ADE and any $r\geq 3$, we prove that the top coefficient of the $r$-th $\mathfrak{g}$-ADO invariant is multiplicative under Murasugi-sum. As a corollary, we get that all the $\mathfrak{g}$-ADO invariants of fibred links are $q$-monic, generalizing a previous result of the authors for $\mathfrak{sl}_2$. Moreover, whenever $\mathfrak{u}_q(\mathfrak{g})$ is ribbon, we find an expression for the top coefficient of the $r$-th $\mathfrak{g}$-ADO invariant in terms of Hopf algebra integrals of universal $\mathfrak{u}_q(\mathfrak{g})$-invariants of bottom tangles.