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It’s a scene that, as Donald J. Trump describes it, would seem to be seared into the American consciousness.

“I watched when the World Trade Center came tumbling down,” he told a crowd in Birmingham, Ala., on Saturday. “And I watched in Jersey City, New Jersey, where thousands and thousands of people were cheering as that building was coming down.”

No news reports exist of people cheering in the streets, and both police officials and the mayor of Jersey City have said that it did not happen. An Internet rumor about people cheering in the streets, which said it was in Paterson, not Jersey City, has been denied numerous times by city and police officials.

But when pressed on Sunday by George Stephanopoulos in an interview, Mr. Trump emphatically stuck to his story.

“It did happen, I saw it,” Mr. Trump said. “It was on television. I saw it.”

When reminded that police said it didn’t happen, Mr. Trump again insisted that he saw it.

“I know it might be not politically correct for you to talk about it, but there were people cheering as that building came down — as those buildings came down. And that tells you something,” he said. “It was well covered at the time, George.”

In New Hampshire on Sunday, Gov. Chris Christie of New Jersey said he had no recollection of state residents celebrating the terror attacks in 2001.

“I think if it had happened, I would remember it,” Mr. Christie said.

Also on Sunday, Mr. Trump reiterated many of his emphatic, if controversial, positions in the wake of the terrorist attacks in Paris. He said he would not unequivocally rule out a database on all Muslims and that he wants a database of refugees and surveillance of certain mosques.

He also said that he would bring back interrogation techniques like waterboarding.

“We have to be strong,” he said in response to Mr. Stephanopoulos’s question. “You know, they don’t use waterboarding over there; they use chopping off people’s heads. They use drowning people.”

When asked if that meant he would bring back waterboarding, Mr. Trump said, “We have to — I would bring it back, yes. I would bring it back. I think waterboarding is peanuts compared to what they’d do to us.”

The terrorist attacks in Paris dominated the Sunday news shows, as every presidential candidate’s interviews focused nearly exclusively on the attacks. And each candidate tried to point to his record as evidence of why he is best positioned to defeat the Islamic State.

Mr. Christie reiterated, as he often has on the campaign trail, that he is the only candidate “who has actually done this before,” pointing to his tenure as United States attorney for New Jersey, when he “brought two of the most major terrorism cases this country has brought in the aftermath of September 11.”

Mr. Christie called for rebuilding the intelligence programs, in particular the National Security Agency’s metadata collection program, during his interview on CNN’s “State of the Union.”

Senator Marco Rubio of Florida, whose appearance on “Fox News Sunday” was timed with the release of his first television ad, which also focused on the Paris attacks, laid out a detailed plan on how to combat the Islamic State in Syria, including by expanding air strikes and providing arms, training and American troops to help arm “Arab Sunnis” to combat terrorists in Syria.

“There will have to be American operators embedded alongside them,” he said. “Special operators are combat troops. This is not a return to Iraq. We’re not talking about 100,000 people or 50,000 armed soldiers.”

Ben Carson, the retired neurosurgeon, also called for increased intelligence gathering, and pointed to his record as a surgeon to show his readiness in the face of emergencies.

“I would be willing to say that I probably have more 2 a.m.-in-the-morning experience than everybody else running combined, making life and death decisions,” Mr. Carson said in an appearance on “This Week” on ABC.

The focus on the Paris attacks dominated the interviews despite a series of new polls released early Sunday morning, all with Mr. Trump holding a commanding lead in both nationwide and state-specific polling.

But his strong standing among Republicans was not enough for Mr. Trump to again reverse his position on a possible independent run.

“I will see what happens,” he said in response to a question about a potential independent run. “I have to be treated fairly.”