The case has sent a shiver through the city, both because of its echoes of the Central Park jogger case, one of the city’s most infamous crimes, but also because five days after Ms. Vetrano’s death, another New York woman, Vanessa Marcotte, was killed during a jog near her mother’s home in a small town in Massachusetts.

Ms. Vetrano’s death has also resonated because of poignant details about her relationship with her father, Philip Vetrano, whose grief-stricken public appearances since her death have fueled more news media attention. Mr. Vetrano, a retired firefighter, usually jogged through the preserves with his daughter. But he had recently injured his back, so his daughter had been jogging there without him, despite his concerns about the safety of doing so.

At a news conference last week, Mr. Vetrano said he believed that the investigation was progressing toward a resolution. “We have a number of very, very strong leads, and there are a few suspects that are very interesting to the police,” he said, according to news reports. Mr. Vetrano called on the family of the killer to come forward, noting that there was considerable reward money, more than $250,000 so far.

But the same day as Mr. Vetrano’s news conference, however, the police commissioner, William J. Bratton, denied that the police had any suspects in their sight, according to news reports.

A call on Monday to Mr. Vetrano’s phone number was answered by a man who identified himself as a family member. Before hanging up, he insisted that there remained “a number of leads and a number of suspects.”