Brothers Kyson and Bryson, aged four, and Leyton and Logan, aged three, were alone when they died in a fire in Collingwood Road, Sutton, south London, last week.
A 27-year-old woman who was arrested on suspicion of child neglect has been bailed.
She will return to a south London police station in mid-January.
The boys’ father, Dalton Hoath, has paid tribute to them and said he can still hear them telling him, “Daddy, I love you.”
He said he does not think the news will sink in for him and has chosen to call Christmas Day just “December 25” following his boys’ deaths.
Mr Hoath told ITV News: “My sons were loving, kind, caring unique boys and didn’t deserve to be involved in such a tragic accident that nobody could have expected.
“They will forever be the biggest part of my heart and carry us through in these hard times.
“Bryson, he was the first one born. He had a cute little birthmark on his face and he was just so clever. Kyson, he was just such a gentle soul. He was so shy but so outgoing at the same time, he was so beautiful. Leyton, he’s one of my youngest twins.
“The younger two, they were more boisterous and would play fight with each other, and do sword fights, Nerf guns, they liked all stuff like that, same as Logan, but they all liked superheroes, that’s the one thing they shared together, they used to love dressing up and playing with each other and just pretending to be all of these superheroes. It was just lovely.”
Mr Hoath said he has kept the presents – including trainers, clothes and superhero figures – he bought for his sons underneath his Christmas tree, but will donate them after Christmas Day.
He said: “I feel that I should donate them maybe to a hospital or charities and at least some children somewhere can get to use them and make my sons proud because they’re going to sit there and they’re not going to get used.
“I just feel it’d be better if I gave them to someone who can enjoy them.”
Mr Hoath said he was “overwhelmed” by the public support since the boys’ death.
On Thursday evening, hundreds of balloons were released into the air in their memory, as neighbours, friends and well-wishers gathered to pay their respects.
A sea of blue could be seen from the ground as blue balloons were let into the air for Kyson, Bryson, Leyton and Logan.