The incident happened in the Wilton Street area of Bradford shortly after 8pm on October 25.
Police say a "disturbance" left a man aged 30 with a serious hand injury after he tried to defend himself against a knifeman, while another man aged 28 suffered a serious head injury.
Both have since been discharged from hospital, along with a 23-year-old woman, who was also injured.
Police are now keen to trace a car, believed to be dark blue Ford Fiesta with white stripes down the side, which was spotted in the area at the time.
Detective Inspector Mark Taylor, of Bradford District CID, said: "One of the injured males nearly lost one of his hands when he tried to defend himself against a man with a knife. The other suffered a serious head injury.
“This was a nasty attack and our enquiries are continuing into the circumstances behind it.
“As part of those enquiries we are keen to hear from anyone who remembers seeing this vehicle in the area at the time, or anyone who may have seen it in the hours leading up to or shortly after the incident.”
Anyone who may have information about the vehicle is asked to Bradford District CID on 101, quoting crime reference 13180536366, or use the live chat facility on the West Yorkshire Police website.
Information can also be reported anonymously to the independent charity Crimestoppers on 0800 555111.