For a pseudoholomorphic polygon $u:\Sigma\to X$ with inputs at $p_{1},\ldots p_{k}$ and an output at $q$, arranged in clockwise order, the index is $\deg q-\sum_{1}^{k}\deg p_{k}$, where the degree convention is (63). The index amounts to a Maslov number computation, and an alternative topological description is as follows: take a section $s$ of the complex line bundle $\Lambda^{n}TM\to\Sigma$, which restricts on $\partial\Sigma$ to a section of the real line bundle $\Lambda^{n}TL$. (When several Lagrangians are involved, it is understood that $TL$ refers to the appropriate Lagrangian on the portion of $\partial\Sigma$.) We assume $s$ has isolated zeros up to the boundary and the corner (aka. strip like ends). We may also regard $s$ as a function $\Omega(s)$ on the polygon, by contraction with $\Omega$. Then

$$\text{Index}=2\sum(\text{interior zeros})+\sum(\text{boundary zeros})+\sum(\text{excess corner zeros})+n.$$

(23)

Here the order of zeros is computed from winding numbers, and for general sections $s$ may take positive and negative values. At a corner where $\partial\Sigma$ passes from $L_{+}$ to $L_{-}$ in the clockwise direction, we can put the tangent spaces $TL_{\pm}\subset TX$ into the standard form respecting the complex structure

$$L_{+}=\mathbb{R}^{n},\quad L_{-}=(e^{i\phi_{1}},\ldots e^{i\phi_{n}})\mathbb{R}^{n},\quad 0<\phi_{i}<\pi,\quad TX=\mathbb{R}^{n}\otimes\mathbb{C},$$

(24)

so if $\Omega(s)\sim z^{\alpha}$ in the complex coordinate of the upper half plane model, the excess vanishing order at the corner is $\frac{1}{\pi}(\alpha-\sum_{1}^{n}\phi_{i})$. Formula (23) is equivalent to the standard index formula by a topological version of the Cauchy residue formula.