Fig. 5
Numerical example of homotopic relations between loops. a Two similar loops \({\bigcirc{{\hskip -6.7pt}{\scriptstyle{1}}}}{\hskip 2pt}\) and \({\bigcirc{{\hskip -6.7pt}{\scriptstyle{2}}}}{\hskip 2pt}\) encircle the exceptional points EP1 and EP'1. The two loops are non-homotopic for any starting point including κ0 (gray point) (which is considered for the example), since they cannot be deformed into one another without crossing EP3. Their corresponding matrix product is M1M2M1M2 and I, respectively. This is confirmed by their eigenvalue trajectories as shown in b and c. d The two similar loops \({\bigcirc{{\hskip -6.7pt}{\scriptstyle{3}}}}{\hskip 2pt}\) and \({\bigcirc{{\hskip -6.7pt}{\scriptstyle{4}}}}{\hskip 2pt}\) are non-homotopic for the starting point κ0 but homotopic for κ'0. This is also reflected in the exchange relations of the eigenvalues as shown in e and f. Black dots represent EPs, red lines are the BCs, and the blue loops are the encircling trajectories
