What is the ruling on maintaining ties of kinship with an aunt who lives far away, given the father's refusal to take his daughters to visit her, and the unavailability of a means to communicate with her, and does the obligation of maintaining ties of kinship cease in this case?

Your father should not prevent you from maintaining ties with your paternal aunt through visits and communication; rather, he should help you in this, for it involves cooperation in righteousness, as Allah the Almighty says: "And cooperate in righteousness and piety." If he prevents you, it might be out of concern and fear for you from corruption and mingling with non-mahram men, if there are such men present at your aunt's place. So, try to persuade him kindly to allow you to maintain ties with your aunt in a manner free from sharia prohibitions. Explain to him the importance of maintaining kinship ties, its reward, and the sin and punishment for severing them. If you cannot obtain your aunt's number through him, try to get it from another family member.