More specifically, we consider holomorphic integrands, which implies that the degree of $N(z)$ should satisfy $3+2\mathrm{deg}N\le \mathrm{deg}P$.

Note that Feynman integrals will also in general depend on the integral geometries associated with the maximal cut of their subtopologies.

We define ${\mathrm{tr}}_{\pm }(abcd)\equiv \mathrm{tr}(\frac{1\pm {\gamma }_5}{2}\cancel{a}\cancel{b}\cancel{c}\cancel{d})$.

This genus can also be confirmed by feeding the seven on-shell equations directly into singular [103].

Consider the field of rational functions on the curve, $\mathbb{C}(y,z)$: rational functions of $y$ and $z$ such that $y$ satisfies equation (7). Viewing $\mathbb{C}(y,z)$ as a field extension of $\mathbb{C}$ we can consider the group of automorphisms of this extension: $\mathrm{Aut}(\mathbb{C}(y,z)\cancel{\mathbb{C}})$, which is $\mathrm{Aut}(\mathcal{H})$. This is the set of automorphisms of $\mathbb{C}(y,z)$ that act as the identity on $\mathbb{C}$.

If a Möbius transformation exists to bring a curve into the form of equation (11), it will not be unique; alternate transformations exist that lead to the same projective curve in different affine charts.

For curves of genus 3 defined over a field of characteristic 0, similar invariants were found by Shioda [87, 104]. We include code for calculating these genus 2 and genus 3 absolute invariants in the Supplementary Material [89].

In the example of the nonplanar crossed box, the maximal cut from equation (4) gets mapped by (16) to a linear combination of only genus 2 periods, since all non-integer powers of $w$ drop out of the numerator.

This reflects the fact that the Jacobian variety $\mathrm{Jac}(\mathcal{H})$, which corresponds to the $2g$ dimensional lattice spanned by the column vectors of $\mathcal{P}$, is in this case isogenous to $\mathrm{Jac}({\mathcal{H}}_1)\times \mathrm{Jac}({\mathcal{H}}_2)$ [86, 105].

Notably, the double box integral was also recently shown to be hyperelliptic for external kinematics in more than four dimensions [22]; however, the double box exhibits much more kinematic complexity, making it a less-than-ideal example.