Special Surfaces Remain Distinct in Four Dimensions | Quanta Magazine | For decades mathematicians have searched for a specific pair of surfaces that can’t be transformed into each other in four-dimensional space. Now they’ve found them.

>**Question**. Let Σ\_0 and Σ\_1 be Seifert surfaces of equal genus for a knot K. After pushing the interiors of both surfaces into B\^4, are Σ\_0 and Σ\_1 isotopic in B\^4?
>
>Let Σ\_0 and Σ\_1 be the surfaces of Figure 1, with interiors pushed slightly into B\^4.
>
>**Theorem 2.1**. Let X\_i be the double cover of B\^4 branched along Σ\_i for i = 0, 1. The manifolds X\_0 and X\_1 are not homeomorphic.
>
>Theorem 2.1 implies $that the answer to the initial question is "not necessarily"$, since a locally flat isotopy from Σ\_0 to Σ\_1 would induce a homeomorphism from X\_0 to X\_1.

The paper's authors then proceed to prove Theorem 2.1 — and thus answer a significant, long-standing topological question — in just one page (!), using a relatively simple proof by contradiction. They even note the exceptional nature of the proof's simplicity directly afterwards:

>**Remark 2.2**. We find the argument of Theorem 2.1 striking in that its methods are extremely elementary and have been known to topologists for fifty years, yet are sufficient to answer a question in the literature that has been open for forty years.

As the article notes, sometimes "these examples really are hiding in plain sight." Makes them all the more invigorating when they fall into place.
Nice to see Chuck Livingston's name here, super good dude. I used to go to his weekly movie nights at IU, we watched all of Kurosawa, Ozu, Hurzog, Truffaut, Godard together, along with many more. Good times.
I hope one day I’ll understand stuff that’s said here, but for now, I love the breathtaking visuals.
nice, ill read this in 5 years
Is it common for math papers to be written in this way? I love how the authors use diagrams.
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As of today, LaTeX-styled maths natively supported in GitHub Markdown (comments, issues, README.md, etc) $n!!$