  This is a companion paper to an earlier work of the authors. In this paper,
we provide an axiomatic definition of Floer homology for balanced sutured
manifolds and prove that the graded Euler characteristic of this homology is
fully determined by the axioms we proposed. As a result, we conclude that
$\chi(SHI(M,\gamma))=\chi(SFH(M,\gamma))$ for any balanced sutured manifold
$(M,\gamma)$. In particular, for any link $L$ in $S^3$, the Euler
characteristic $\chi(KHI(S^3,L))$ recovers the multi-variable Alexander
polynomial of $L$, which generalizes the knot case. Combined with the authors'
earlier work, we provide more examples of $(1,1)$-knots in lens spaces whose
$KHI$ and $\widehat{HFK}$ have the same dimension. Moreover, for a rationally
null-homologous knot in a closed oriented 3-manifold $Y$, we construct a
canonical $\mathbb{Z}_2$ on $KHI(Y,K)$, the decomposition of $I^\sharp(Y)$
discussed in the previous paper, and the minus version of instanton knot
homology $\underline{\rm KHI}^-(Y,K)$ introduced by the first author.
