BOULDER, Colorado (CNN) -- Despite John Mark Karr's startling confession in the JonBenet Ramsey murder case, many questions remain to be answered before authorities can say the case has been solved.
One of those questions was raised by Karr's former wife, Lara Karr, who says Karr was with her in Alabama or Georgia on December 26, 1996, the day JonBenet's body was found in the basement of the Ramsey home in Boulder, Colorado.
Lara Karr's attorney, Michael Rains, said his client "sincerely believes there was no Christmas ... anytime between 1989 and 2000 when they were married ... when her husband was not with her and her family at Christmastime." (Watch forensic pathologist describe some of the holes in Karr's story -- 4:13)
In his confession in Bangkok, Thailand, Karr said: "I loved JonBenet and she died accidentally."
But evidence from the Colorado crime scene suggests otherwise. JonBenet's death was violent, investigators found. She was struck in the head and strangled.
And at least publicly, Karr has not given details about how he got into the Ramsey house and, if he did so, how he found his away around the sprawling house, which has been described as a maze of rooms, hallways and closets.
Even police overlooked the storage room where JonBenet's body was found during their first search of the residence, after the girl was reported missing.
Karr's connection to Colorado is also vague. Although his e-mail correspondence with a University of Colorado journalism professor led to his arrest in Thailand, authorities said, they have produced no evidence to show Karr was ever in Colorado.
The Rocky Mountain News published on its Web site Friday excerpts of what investigators say they believe are e-mails between Karr and professor Michael Tracey, including one that read: "JonBenet, my love, my life. I love you and shall forever love you. I pray that you can hear my voice calling out to you from my darkness..." (Read paper's account of e-mails)
Another question is Karr's familiarity with the Ramsey family.
JonBenet's father, John Ramsey, said in an interview Wednesday that he did not know Karr. But a ransom note, found in the Boulder home after JonBenet went missing, addresses John Ramsey in a familiar tone, repeatedly using his first name and calling him a "fat cat."
The ransom note demanded $118,000, the amount he had received in a company bonus before his daughter was killed.
At a news conference Thursday, Boulder County District Attorney Mary Lacy said the case is still under investigation. "There is much more work that needs to be done now that the suspect is in custody."
"John Karr is presumed innocent," Lacy said. "Do not jump to judgment. Do not speculate. Let the justice system take its course."
The professor who put authorities on the trail of Karr took a similar tack.
"I don't know that he's guilty," Tracey told The Associated Press. "Obviously, I went to the district attorney for a reason, but let him have his day in court and let JonBenet have her day in court and let's see how it plays out."
Karr started work as teacher Tuesday
JonBenet's body was found in the basement of her family's Colorado home on December 26, 1996. Autopsy results showed she received a massive blow to the head and was strangled with a rope that the killer tightened by twisting an attached paintbrush handle. An autopsy did not detect any drugs in her system.
U.S. and Thai officials are working on clearing Karr's return to the United States, although no date for the transfer has been set.
U.S. Immigration and Customs Enforcement agents and Thai authorities worked closely on the case for two months before Karr's arrest on Wednesday, Hurst said.
Officials in Bangkok said he had traveled to Thailand -- a country notorious for its child sex trade -- five times in the past two years.
Karr on Tuesday started a new job as a second-grade teacher at a school in Bangkok.