# Definition The Fu-Kane formula is used to compute the topological invariant of certain classes of [[Topological Insulator|topological insulators]], such as [[Time-Reversal Symmetric Topological Insulator|time-reversal symmetric topological insulators (TRS-TIs)]], [[Antiferromagnetic Topological Insulator|antiferromagnetic topological insulators (AFTIs)]] and certain classes of [[Higher-Order Topological Insulator|higher-order topological insulators (HOTIs)]]. The formula for the global invariant $nu_{0}$ is $ (-1)^{\nu_0} = \prod_{n_j = 0}^{1} \delta(\Lambda_{n_1 n_2 n_3}), $ In the case of a TRS-TI, we can also define 3 more invariants: $ (-1)^{\nu_i} = \prod_{n_{j\neq i} = 0; n_i = 1}^{1} \delta(\Lambda_{n_1 n_2 n_3}), \quad (i = 1, \ldots 3) $ where $ \delta(\Lambda_{n_1 n_2 n_3}) \equiv \frac{\text{Pf}[\mathcal{B(\Lambda_{n_1 n_2 n_3})}]}{\sqrt{\text{det}[\mathcal{B(\Lambda_{n_1 n_2 n_3})}]}} $ and $ \Lambda_{n_1 n_2 n_3} = \frac{n_1}{2} \pmb{b}_1 + \frac{n_2}{2} \pmb{b}_2 + \frac{n_3}{2} \pmb{b}_3, \quad n_{1,2,3} = 0, 1 $ are the [[Time-Reversal Invariant Momenta|TRIM]]. $\pmb{b}_i$ are the reciprocal lattice vectors. and $\mathcal{B}$ is the [[Sewing Matrix|sewing matrix]] of the system being studied. $\text{Pf}$ denotes the [[Pfaffian]] of the antisymmetric sewing matrix.