Could you please explain in detail the rules of Nun Qutni.

Question
Could you please explain in detail the rules of Nun Qutni?

Answer
The term Nun Qutni is not an Arabic term, we asked, and it appears to be a Persian word, but is used in the Indian-Pakistan language. 
In the copy of the mushaf printed in Pakistan and some other areas of the world, a small with a vowel is written in when there is a tanween followed by a saakin letter, as the first letter of the next word.   The small is not written over or under the tanween, but is written on the hamzah wasl of the first letter of the next word.  The rule in Arabic is that two saakin letters cannot meet between two words, so one of the letters has to change.  In the case of the saakinah of the tanween, it acquires a vowel.  The vowel to be used is was is written over or under the small added in to the script.   This is not found in the copies of the Qur'an printed in the Arab world, to the best of our knowledge.
An example would be in the two words in red below:

The tanween is followed by a saakin letter as the first pronounced letter of the next word (the ba').  The saakinah of the tanween acquires a kasrah in this case.  In the Indian-Pakistani prints of the Qur'an, a small is written under or over the hamzah wasl with a kasrah. 
Another similar example is the tanween in: .

The problem with this is that this only applies when joining the two words together.  If one is starting one the second red word, "ibn", the hamzah wasl is read with a kasrah, but there is no read there at all.   We were told by some students who formerly used this print of the Qur'an that they were taught to start on the words with a .  This is incorrect and changes the Qur'an. 
The other problem is the misunderstanding that many have that the "noon qutni"  is part of the hamzah wasl or has a special identity, when in fact it is just the saakinah of the tanween that acquired a vowel to eliminate the problem of two saakin letters meeting between two words.

Instead of learning about the noon qutni, it is advised to learn the rules of what to do when two saakin letters meet.  This way, you will know what to do when reading any print of the Qur'an and two saakin letters meet.
This has  been explained in detail in the following older question and answer:
http://www.abouttajweed.com/121003.htm