I know that some toolbox such as Dionysus can also return the representative points which are on the boundaries of the cycles (topological features of a point cloud). I can clearly extract these points using the TDA toolbox in R which acts as a wrapper on the Dionysus and several other TDA toolbox. However, I wonder what is the theory behind it. Can someone please explain it in short or guide me to the papers where I can read more about it. I'm familiar with the reduced boundary matrix and how it is possible to compute the persistence homology using this matrix, however, I'm curious if Dionysus takes advantage of this matrix to detect the boundaries.
The simplest example to think about is when add simplices or cells one at a time. One choice for the representative cycle is given a simplex creates a new cycle, take the linear combination which is used to zero out the boundary of the simplex. This is certainly a representative but not the only choice. For example you could also take the boundary of the simplex that corresponds to the death time.
If you look at the paper vineyards: updatilng persistence in linear time by dmitriy morozov - there he explicitly keeps track of cycles.