FCLoopFindSectors[{GLI[...], ...}] analyzes the indices of the GLI integrals in the given list and identifies sectors to which they belong. Notice that only GLIs with integer indices are supported.
If the option GatherBy is set to True (default), the output will be a list of two lists, where the former contains the original integrals sorted w.r.t the identified sectors, while the latter is a list of all available sectors.
For GatherBy->False, the output is a list containing all identified sectors without the original integrals.
Setting the option Last to Truewill return only the top sector.
ints = {
GLI[topo1, {1, 1, 1, 1}],
GLI[topo1, {2, 1, 2, 1}],
GLI[topo2, {1, 0, 1, 1}],
GLI[topo3, {1, 0, 1, -1}]
};FCLoopFindSectors[ints]\left( \begin{array}{ccc} \left\{\{1,0,1,0\},\left\{G^{\text{topo3}}(1,0,1,-1)\right\}\right\} & \left\{\{1,0,1,1\},\left\{G^{\text{topo2}}(1,0,1,1)\right\}\right\} & \left\{\{1,1,1,1\},\left\{G^{\text{topo1}}(1,1,1,1),G^{\text{topo1}}(2,1,2,1)\right\}\right\} \\ \{1,0,1,0\} & \{1,0,1,1\} & \{1,1,1,1\} \\ \end{array} \right)
FCLoopFindSectors[ints, Last -> True]\left\{\{1,1,1,1\},\left\{G^{\text{topo1}}(1,1,1,1),G^{\text{topo1}}(2,1,2,1)\right\}\right\}
FCLoopFindSectors[ints, GatherBy -> False]\left( \begin{array}{cccc} 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ \end{array} \right)
FCLoopFindSectors[ints, GatherBy -> False, Last -> True]\{1,1,1,1\}