Not a single Republican represents New York City in Washington. Only three ­Republicans serve in the 51-seat City Council. Only one borough president is a Republican. A special election coming up on Feb. 26 might change the picture.

The office of public advocate is up for grabs that day, and there are indications that one of the three City Council Republicans, Eric Ulrich, might be in a position to clinch it. He is one of 17 candidates, and while it’s bad to be a Republican in a citywide race these days, it may be an ­advantage if you are running against 16 Democrats who are cannibalizing each other.

I’ve seen private polling — not commissioned by Ulrich or any of the campaigns — that suggests Ulrich has a real shot. He leads with 22 percent against the six best-known Democrats. ­Jumaane Williams, another city council member, is at 19 percent, with former Council Speaker Melissa Mark-Viverito at 14 percent.

But here’s the savage irony: ­Ulrich gets that 22 percent when the pollster identifies him as a Republican. The thing is, in this case, he can’t actually run as a Republican. The rules governing special elections mean the candidates aren’t allowed to run on their traditional party lines. ­Instead, Ulrich’s line-cum-motto is “Common Sense.”

When the candidates are listed by their weird line names and not as Democrats or Republicans, ­Ulrich falls to 12 percent, while Mark-Viverito and Williams both get 15. But that still puts Ulrich within the margin of error.

Should Ulrich win, he will ­instantly become the best-known GOP politician in the city.

He will have a tough row to hoe, though, because the person who wins the race will only have the seat through November. In June, there will be a partisan primary, and the November election will feature a lone Democrat against a lone Republican.

If Ulrich succeeds in the special election, he would have the advantage of incumbency, not to mention the benefit of running in a November election in which three people and his mother would be likely to turn out. The problem is those three people are more likely to be Democrats than Republicans. He would want to make a splash to give his candidacy some oomph.

Fortunately, the silly and ill-defined office he is seeking allows him to do just about anything he wants, and if he can come up with a dynamite report on some populist matter that gets attention from the media, that may be enough to put him over the top in November.

Why has the New York City GOP fallen so low? After all, nominal Republicans ran City Hall from 1994 until 2013. But nominal is the operative word here. In 2007, Michael Bloomberg announced he was no longer a Republican — which didn’t prevent him from running on the GOP line in 2009. Now he’s an out-and-out Democrat. Maybe next year he’ll join The Rent Is Too Damn High party. Whatever.

The New York Republican is an ideological outlier and always has been. These days, it’s the fact that the GOP is the nation’s conservative party that makes its politicians as scarce as truffles in winter and its voters starved for truffles.

That was not always the case.

In the middle decades of the 20th century, politicians who ran as Republicans were often more liberal than standard-issue machine Democrats. They joined the GOP because they couldn’t countenance the presence of Southern racist segregationists in the national Democratic electoral alliance or the corruption of Tammany Hall.

This is the iconoclastic tradition from which sprang the very different political careers of Mayors Fiorello La Guardia (in the 1930s and ’40s) and John Lindsay (in the ’60s and ’70s), and it helped provide the votes that sent Jacob Javits to Washington as a congressman and senator from 1946 to 1980.

More recently, Republicans have been elected to statewide office to clean up the mistakes of Democrats — and those Republicans were more ideologically iconoclastic than the ­increasingly conservative national Republican party.

Rudy Giuliani’s mayoralty (1994 to 2002) and George Pataki’s governorship (1995 to 2007) followed public discontent with the performance of their Democratic predecessors. Both were comfortably ­reelected (Pataki twice!) despite the city’s 4-to-1 Democratic registration and the state’s 3-to-2 Democratic advantage. Whatever Giuliani is now, he was on the liberal side of the Republican Party then, and so was Pataki.

Ulrich follows in their iconoclastic footsteps. He’s a supporter of some but not all liberal legislation.

If there is to be any hope of the city getting off the path to perpetual one-party rule, it will have to start somewhere. A ­special election win for a Republican who can’t call himself a Republican might be that somewhere.