Silatul Arham is an act that results in family ties. Its importance in Islam is to the extent that cutting ties even with a kin who is not a believer isn’t permissible. Of course maintaining ties with a blood relative is not allowed if a he/she intentionally is harming Islam. It was due to such an exception that surah Lahab which is about Abu Lahab and his wife; was revealed, through which the prophet (pbuh) was ordered to discontinue his ties with them.
The word “Silah” in Arabic means to connect or to stay in touch and “Rahemm” (which literally means ‘womb’) refers to kinsfolk and close relatives.
Maintaining one’s relationship with his/her blood relatives has great value and bears high importance. In response to a person asking him: “Some of my relatives have different faiths and customs than mine. Do they have a right over me?” Imam Sadiq (as) said: “Yes, nothing overcomes the right of kinship. And if they share the same faith and custom with you; then they have two rights over you; one is the right of kinship and the other is Islam and being a Muslim.”
Other traditions also say: “Be good with your kinsfolk and close relatives even if they have cut ties with you; don’t cut ties with them.” Imam Ali (A) says: “Create a bond with your kinsfolk even if they cut ties with you.” The question here is, does Islam also emphasize on keeping ties with those relatives whose intentions are harming Islam?
For instance, did the holy Prophet (pbuh) maintain his ties with his uncle, Abu Lahab, who unfortunately was a pagan and a very stubborn polytheist? His enmity, crimes and tyranny against the holy Prophet (pbuh) and the Muslims are undeniable.
In response we have to say “No because of two reasons: 1- Allah (SWT) says: “It is not fitting, for the prophet and those who believe, that they should pray for forgiveness for pagans, even thought they be of kin, after it is clear to them that they are companions of the fire”.
Obviously, asking for forgiveness and praying for a relative are all instances of silatul arham (so when the Quran is telling the prophet (pbuh) not to pray for them or ask Allah (swt) for their forgiveness, it is saying to cut all ties and not practice silatul arham with them).
There are minimums mentioned in some traditions for observing this Islamic tradition and ruling. For example, greetings (saying Salam) or serving a drink of water means that you have met your duty of maintaining ties with family. Also, there are traditions mentioning that in the case of relatives being far away, one could pray for them and that will serve as silatul arham and maintaining ties with them.
2- In the holy Quran it is stated: “And Abraham prayed for his father’s forgiveness only because of a promise he had made to him. But when it became clear to him that he was an enemy to Allah, he dissociated himself from him”
It is mentioned in this verse that Prophet Ibrahim (A) prayed for his uncle which is one way for practicing silatul arham but after he realized that he was an enemy to Allah, he cut ties with him and showed his repulsion for him.
When the holy prophet of Islam, during the third year of his prophetic mission invited his relatives, Abu Lahab stood against him as an enemy but the Prophet did not rush towards disassociating himself from him. But later on when Abu Lahab increased his enmity and made his stand against the prophet and his message clear and public, and when the specific verse was revealed about him and his wife, the holy Prophet cut ties with him.
3- Imam Sajjad (A) in his blessings upon Muhammad and his household says: “and showed open enmity towards his kin in the way of inviting towards you, fought against his tribe for your pleasure, distanced from his family in the way of reviving your religion, separated from his family members who had denied you, brought near those whom he had no family ties with because of their response to you, showed friendship to the most distant for your sake, and displayed enmity toward the nearest for your sake”
In his commentary on the Sahifah Sajjadiyyah, the late Sayed Ali Khan explains: Cut ties with them means he really dissociated and cursed them and tore away from them and discontinued doing them good. The reaction of the holy Prophet towards those who called him a liar and stood against him to turn off the light of guidance especially those from his tribe and those related was to dissociate, stand against and fight them; he killed most of them during the battles of Badr and Uhud. He took them as prisoners and did not show them mercy because of them being the enemy of Allah (not that he bothered or tortured them, but he didn’t show any mercy as he would if they weren’t enemies of Islam). There is no doubt that Abu Lahab was one of these enemies. Therefore, we reach a conclusion that in Islam associating with polytheists and unbelievers that are enemies of Allah isn't allowed; even if they are of one’s kin and his/her close relatives.
 Mohammad Rey Shahri, Mizanul-Hikamah, vol. 4, pg. 83.
 صلوا ارحامکم و إن قطعوکم Wasa’ilul-Shiah, vol. 11, pg. 175.
 Tawbah:113 ما کانَ لِلنَّبِیِّ وَ الَّذینَ آمَنُوا أَنْ یَسْتَغْفِرُوا لِلْمُشْرِکینَ وَ لَوْ کانُوا أُولی قُرْبى مِنْ بَعْدِ ما تَبَیَّنَ لَهُمْ أَنَّهُمْ أَصْحابُ الْجَحیمِ See: Mohammad Husein Tabatabai, Al-Mizan, vols. 9 and 10, pg. 282.
 The Holy prophet of Islam says: “صلوا ارحامکم فی الدنیا و لو بالسلام” Biharul-Anwar, vol. 71, pg. 104.
 Imam Sadiq (as): صل رحمک و لو بشربة من ماء Biharul-Anwar, vol. 74, pg. 117.
 Tawbah:114 وَ ما کانَ اسْتِغْفارُ إِبْراهیمَ ِلأَبیهِ إِلاّ عَنْ مَوْعِدَةٍ وَعَدَها إِیّاهُ فَلَمّا تَبَیَّنَ لَهُ أَنَّهُ عَدُوُّ لِلّهِ تَبَرَّأَ مِنْهُ إِنَّ إِبْراهیمَ َلأَوّاهٌ حَلیمٌ/ Mumtahinah:4 قَدْ کانَتْ لَکُمْ أُسْوَةٌ حَسَنَةٌ فی إِبْراهیمَ وَ الَّذینَ مَعَهُ إِذْ قالُوا لِقَوْمِهِمْ إِنّا بُرَآؤُا مِنْکُمْ وَ مِمّا تَعْبُدُونَ مِنْ دُونِ اللّهِ کَفَرْنا بِکُمْ وَ بَدا بَیْنَنا وَ بَیْنَکُمُ الْعَداوَةُ وَ الْبَغْضاءُ أَبَدًا حَتّى تُؤْمِنُوا بِاللّهِ وَحْدَهُ إِلاّ قَوْلَ إِبْراهیمَ ِلأَبیهِ َلأَسْتَغْفِرَنَّ لَکَ وَ ما أَمْلِکُ لَکَ مِنَ اللّهِ مِنْ شَیْءٍ رَبَّنا عَلَیْکَ تَوَکَّلْنا وَ إِلَیْکَ أَنَبْنا وَ إِلَیْکَ الْمَصیرُ
 Sahifah Sajjadiyyah, pg. 48 و کاشف فی الدعاء الیک حامّتَهُ * وحارب فی رضاک أسْرَتَهُ * وقطع فی احیاء دینک رِحَمَهُ * وأقصی الادْنینَ علی جُحودهم * وقَرَّبَ الاقصین علی استجابتهم لک * و والی فیک الابعدین * وعادی فیک الاقربین *.
 Seyyid Ali Khan Huseini Mudarresi, Riyadul-Salekin, vol. 1, pp. 465-466.