Reading this article in the New Yorker I notice this word 'reëxamined'. It's not the only one written like this, any other word with double vowels will be written similarly.
What is the point behind this use ?
This is a matter of the typographic style of the The New Yorker, a publication known for it idiosyncratic, slow-to-change, and close punctuation style. The two dots are called a diaeresis, which appears on the second of a pair of vowels to put the reader on notice that the second vowel starts a separate syllable. Note that a diaeresis is not an umlaut, which serves a different purpose in German. Mary Norris explains all of this in her article "The Curse of the Diaeresis". The rules for its appearance are the publication's own. They don't use it for vacuum, which might lead you to believe the editors consider the word to have only two syllables. But it's not used for zoology, which surely has three. Go figure.
I'm pretty sure (from years of reading The New Yorker) that the magazine adopted the use of diaeresis as an alternative punctuation to using a hyphen in situations where the final letter of a prefix is the same as the first letter of the stem word. That would explain why deadrat's examples of vacuum and zoology are not governed by the style rule.
Thus, The New Yorker prefers reëducate, reëxamined, coöperation, coördinate, and (perhaps) antiïntellectual to the alternative forms re-educate, re-examined, co-operation, co-ordinate, and anti-intellectual. I'm not entirely sure about this last example because The New Yorker's style rules are extremely granular (and often sui generis), and it may be that the house style favoring diaeresis over hyphen applies only to two-letter prefixes where vowel duplication occurs at the interface of prefix and stem.
My guess is that the magazine adopted this approach decades ago, at a time when few publications would simply have used the style reeducate and cooperation (for example). But The New Yorker is an island of usage unto itself; its loyalty to elsewhere-long-abandoned punctuation is unlikely to influence other publications, except perhaps to say "Let's not be as nutty as The New Yorker about this." I, for one, admire the magazine's pertinacity.