Two young children who witnessed their father being stabbed to death had tea towels put over their heads to shield them from his body.
Christopher Nicol, 27, was attacked inside his flat in Maple Road, Greenock, on 26 September.
Det Ch Insp Martin Fergus said Mr Nicol’s partner covered her children’s faces to get them out of the flat after the “frenzied and brutal attack”.
Mr Nicol’s mother said he ” worshipped his children” and was a brilliant dad.
The five-year-old boy and six-year-old girl have been receiving professional support.
Speaking at a police press conference, an emotional Karen Nicol said he “would have done anything for his children”.
“I am devastated by the tragic death of my much-loved son Christopher,” she said. “Our lives have been changed forever. He was a brilliant dad to his two young children who he worshipped and would have done anything for.
“The man responsible has taken one life and ruined the lives of our family. Please, please if you have any information that will help the officers, please call them and Crimestoppers as soon as possible, please.”
Police believe the killer knew who he was targeting, and wrongly thought there was a large amount of cash in the flat. However, Det Ch Insp Fergus said the victim may not have known his killer.
The officer said that just before the attack at 21:05, Mr Nicol had been colouring-in with his five-year-old son in the living room while his six-year-old daughter was helping his partner in the kitchen.
The killer entered the flat after barging past Mr Nicol’s girlfriend when she answered the door.
“This incident is absolutely horrendous,” Det Ch Insp Fergus said. “This has happened during a family scene, within a house in Greenock, a scene that is replicated throughout Britain on any given evening.
“An individual has forced his way in and effectively stabbed Christopher Nicol to death in the presence of his two young children. I don’t think I need to expand any further on how horrific this is.”
He said that after the attacker fled the scene, Mr Nicol’s partner, in an effort to shield the children from what had taken place, put tea towels over their heads to get them out of the flat.
“I don’t think people need to imagine, after such a frenzied and brutal attack, what state the father would have been in and to have tea towels placed over five and six-year-olds heads to shield them from that speaks volumes as to how traumatic that was,” the officer said.
“It’s unimaginable what must have taken place. For a five and six-year-old to have witnessed this speaks for itself.”
Det Ch Insp Fergus said police had received a good response after speaking to people in Greenock, but said he firmly believed there was still crucial information that had not been given to officers.
“We are still needing as much information from the public as we can possibly gather in an effort to piece together the events of that evening,” he said.
“I do believe the answer lies within the community.”
He said even the smallest piece of information could be “critical”.
“On the back of Karen’s emotional plea I don’t think I really need to expand any more on the devastating nature of this horrific crime,” he added.
The killer has been described as white, aged 20-30, about 5ft 9in tall, with a slim build. He had a local accent, and an unkempt, reddish, brown beard and moustache. Officers say he also had a “drawn-in” face and decaying teeth, with some visibly missing.
He was wearing a black beanie hat with a logo, possibly Timberland, a black top and black jeans or bottoms.
Anyone with information has been urged to contact Police Scotland via their non-emergency line.