Police are trying to establish what happened to the little girl before she was taken to hospital, and have searched two houses - one in a Wanganui suburb and the other in South Taranaki.
Items have been taken from the houses and are to be forensically tested. Both houses were under police guard but have now been released back to the owners.
A post-mortem examination revealed the cause of death to be a head injury which police do not consider to be accidental.
The officer leading the inquiry, Detective Senior Sergeant Neil Forlong, said people should not "leap to conclusions and start pointing the finger prematurely".
He would not discuss the nature of the head injury or what could have caused it.
The detective said Hinekawa had been staying in different homes with members of her extended family before she was killed. "There are a number of people that we now need to speak to in detail, as Hinekawa has recently stayed at both the family home and with extended whanau."
On Ms Topia's Facebook page, a relative posted on January 2 that the babies were cute but they "need to come home now".
The movements of the triplets from the time they came home from hospital will be critical to the inquiry, said Mr Forlong.
"We're trying to establish the movements and background of these triplets from day one ... we want to establish exactly what happened to Hinekawa."
Police were building a picture of what happened by speaking to family members, who were co-operating with the investigation.
He would not say what officers knew about the hours before the dash to the hospital.
"Until we get a fuller picture I'm not prepared to go into that."
Hinekawa's triplet sisters are being assessed in hospital as a precaution and are expected to be released soon.
Police were called to the hospital on Thursday, 20 minutes after the girls and her parents arrived, and began gathering evidence even before the post-mortem examination confirmed the cause of death.
Mr Forlong said 20 staff were working on the inquiry.
"We fully acknowledge the sensitivities of cases such as this and it is important that people do not leap to conclusions and start pointing fingers prematurely," he said.
"We have to be thorough and meticulous and work closely with medical professionals, the pathologist and Hinekawa's family. That process will take time and should not be rushed."
A CYF spokeswoman last night said the family were not known to the agency.
