The facts line up starkly against Ms. Murkowski: The only person ever elected to the United States Senate as a write-in candidate was Strom Thurmond of South Carolina, in 1954. No one in Alaska has ever been elected statewide as a write-in. Joe Miller, the Tea Party star who narrowly defeated Ms. Murkowski in the Republican primary, now has the support of the Washington Republican establishment and is raising money quickly.

So why do plenty of people here, from analysts to many rank-and-file Republicans, think that Ms. Murkowski, 53, who first came to office through the easiest route imaginable (her father, then the governor, appointed her in 2002), could well pull off the impossible?

Because in a matter of weeks, she has morphed from establishment incumbent to renegade underdog. For many, it might seem crushing to go from sitting senator to plaintive write-in, but Ms. Murkowski is using it to her advantage, painting herself as the maverick in this race.

It was Sarah Palin who turned the state on its head in 2006 when she defeated Ms. Murkowski's father, the incumbent, in a landslide in the Republican primary for governor. And of course there is Mr. Miller, a largely unknown lawyer who won last month's primary by portraying Ms. Murkowski as a symbol of broken Washington and himself as the rebel.

Now Ms. Murkowski, already armed with a substantial campaign account and a well-known name (if not one that is easy to spell), has her own dark horse story.

"I failed as a candidate with my campaign in ensuring that Alaskans understood the urgency and why it was important that I retain this seat," the senator told a crowd of supporters who gathered here last week. "And further I failed in defending my record and quite honestly allowed it to be trashed there towards the end."

"I will not let you down again," Ms. Murkowski said to rising applause.