A combinatorial algebraic approach for the modified second-generation time-delay interferometry

We generalize the combinatorial algebraic approach first proposed by Dhurandhar et al. to construct various classes of modified second-generation time-delay interferometry (TDI) solutions. The main idea behind the algorithm is to enumerate, in a given order, a specific type of commutator between two monomials defined by the products of particular time-displacement operators. On the one hand, the above commutators can be systematically rewritten as the elements of a left ideal, defined by the l.h.s. of the relevant equation for the TDI solution. On the other hand, these commutators are shown to vanish if we only keep up the first-order contributions regarding the rate of change of armlengths. In other words, each commutator furnishes a valid TDI solution pertaining to the given type of modified second-generation combinations. In this work, the original algorithm, which only involves time-delay operators, is extended by introducing the time-advance ones and then utilized to seek solutions of the Beacon, Relay, Monitor, Sagnac, and fully symmetric Sagnac types. We discuss the relation between the present scheme's solutions and those obtained by the geometric TDI approach, a well-known method of exhaustion of virtual optical paths. In particular, we report the results on novel Sagnac-inspired solutions that cannot be straightforwardly obtained using the geometric TDI algorithm. The average response functions, floor noise power spectral densities, and sensitivity functions are evaluated for the obtained solutions.