On Dec. 26, 1996, the Ramseys called police to report their daughter missing; later that day John Ramsey found her body in the basement. Boulder police, feeling the couple were less than forthright, zeroed in on them as suspects. "We were aghast," John told PEOPLE in 2004. Photo by: Patrick Davison / Rocky Mountain News / Polaris
Could This Be JonBenét's Killer?| John Mark Karr, JonBenet Ramsey, John Ramsey, Patsy Ramsey
But do they? Just two months after Patsy died of ovarian cancer, and four months shy of the 10th anniversary of the death of 6-year-old JonBenét Ramsey, came electrifying news: Police in Bangkok were holding a suspect in the murder. John Mark Karr, 41, a teacher-for-hire with a history of bizarre behavior around children, told police he had been in the basement of the Ramseys' Boulder home the night JonBenét was murdered, and admitted she died at his hands. But he also professed his love for her and calmly insisted that her death – she was strangled and had an 8 1/2-in. fracture to her skull – was accidental. The Ramseys "feel their daughter was brutally murdered, and she wasn't," Karr told a Thai official who spoke with PEOPLE. "It looks like that but she wasn't. I want them to hear the truth . . . I need closure and [JonBenét's] family needs closure . . . all of us have gone through enough pain."

Yet almost immediately there were troubling questions about Karr's strange confession. Not long after he was detained U.S. investigators took a saliva swab from Karr to compare his DNA with that found in JonBenét's underwear and under her fingernails; the results of those tests were not available at press time. "Even as kooky as his confession was, if you pair it up with a DNA match, then it's game, set and match for the state," says former Denver prosecutor turned talk show host Craig Silverman, an expert on the case. But without that, investigators will need to tie up a number of puzzling loose ends before one of the most infamous unsolved murders can, at last, be solved. John Karr "isn't normal, and he may even be a child molester," says well-known Denver defense attorney Scott Robinson, who has followed the case from the beginning. "But at this point it's pretty hard to believe that he's a murderer who committed this particular murder."

Perhaps most puzzling of all is that Karr, the twice-married father of three teenage boys, has no apparent connection to the Ramsey family. Neither John nor Patsy had ever heard of him, and Boulder investigators did not have him on their radar until earlier this year, and then only after a University of Colorado journalism teacher who was corresponding with Karr brought him to their attention. What's more, Karr's second wife, Lara Knutson, 33, says Karr spent Christmas Day, 1996 – the day before JonBenét's body was found – with her and their sons either at home in Alabama or with his family in Georgia.

So far there is no proof to back that up, but "we've got a picture of his three children that was taken in his father's house on Christmas Day of 1996," says Georgia attorney Gary Harris, a spokesman for the Karr family. "The fact that his kids were there tells me he was there. His wife would not have come to the Christmas party without him." Karr was also known to be obsessed with the Ramsey case, reveling in its every detail, and was similarly fascinated by the 1993 murder of 12-year-old Polly Klaas, even moving his family to her hometown of Petaluma, Calif.