In this paper we discuss the issues of recurrence and transience of random walks in multiply connected spaces and, in particular, in the presence of knots. Such problems are of interest in polymer physics since almost all sufficiently long linear polymer chains are (quasi) knotted at least at theta solvent quality conditions and, hence, the transience/recurrence provides mathematically rigorous definition of the entanglement concept. These problems are also of biological interest since they clarify the role of topology in problems involving molecular recognition. Obtained results extend and generalize the results of earlier published Rapid Communication (Kholodenko, Phys. Rev. E 58 (1998) R5213). They also provide some answers to problems which were just mentioned in our earlier published report (Kholodenko, Phys. Reports 298 (1998) 251).
