clarified so that related mathematical theories, including Poincaré - Hopf theorem [@Poincare:1890], Conley index theory [@Conley:1978], Floer homology, etc., and algorithms developed in computer science can be better applied to molecular sciences.

In additional to differential topology, algebraic topology, specifically, persistent homology, has drawn much attention in recent years. Persistent homology has been developed as a new multiscale representation of topological features. The 0th dimensional version was originally introduced for computer vision applications under the name “size function" [@Fro90; @Frosini:1999] and the idea was also studied by Robins [@Robins:1999]. Persistent homology theory was formulated, together with an algorithm given, by Edelsbrunner et al. [@Edelsbrunner:2002], and a more general theory was developed by Zomorodian and Carlsson [@Zomorodian:2005]. There has since been significant theoretical development [@BH11; @CEH07; @CEH09; @CEHM09; @CCG09; @CGOS11; @Carlsson:2009theory; @CSM09; @SMV11; @zigzag], as well as various computational algorithms [@OS13; @DFW14; @Mischaikow:2013; @javaPlex; @Perseus; @Dipha]. Often, persistent homology can be visualized through barcodes [@CZOG05; @Ghrist:2008], in which various horizontal line segments or bars are the homology generators that survive over filtration scales. Persistence diagrams are another equivalent representation [@edelsbrunner:2010]. [ Computational homology and persistent homology have been applied to a variety of domains, including image analysis [@Carlsson:2008; @Pachauri:2011; @Singh:2008; @Bendich:2010; 