Biopolymer unfolding events are ubiquitous in biology and mechanical unfolding is an established approach to study the structure and function of biomolecules, yet whether and how mechanical unfolding processes depend on native state topology remain unexplored. Here, we investigate how the number of unfolding pathways via mechanical methods depends on the circuit topology of a folded chain, which categorizes the arrangement of intrachain contacts into parallel, crossing, and series. Three unfolding strategies, namely, threading through a pore, pulling from the ends, and pulling by threading, are compared. Considering that some contacts may be unbreakable within the relevant forces, we also study the dependence of the unfolding efficiency on the chain topology. Our analysis reveals that the number of pathways and the efficiency of unfolding are critically determined by topology in a manner that depends on the employed mechanical approach, a significant result for interpretation of the unfolding experiments.