0 like 0 dislike
0 like 0 dislike
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.
by

5 Answers

0 like 0 dislike
0 like 0 dislike
What I love about this result is how straightforward the proof is. From the paper:

>**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.
0 like 0 dislike
0 like 0 dislike
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.
0 like 0 dislike
0 like 0 dislike
I hope one day I’ll understand stuff that’s said here, but for now, I love the breathtaking visuals.
0 like 0 dislike
0 like 0 dislike
nice, ill read this in 5 years
0 like 0 dislike
0 like 0 dislike
Is it common for math papers to be written in this way? I love how the authors use diagrams.
by

Related questions

0 like 0 dislike
0 like 0 dislike
5 answers
BrianDenver7 asked Jun 21, 2022
Is there a nice way to recast riemannian geometry in terms of principal bundles?
BrianDenver7 asked Jun 21, 2022
0 like 0 dislike
0 like 0 dislike
12 answers
joixlee asked Jun 21, 2022
Importance of the role of First Order Logic (FOL) in understanding higher level mathematics
joixlee asked Jun 21, 2022
by joixlee
0 like 0 dislike
0 like 0 dislike
3 answers
militarypoisons asked Jun 21, 2022
what are some surprising applications of math in co.puter science like weird/obscure concepts?
militarypoisons asked Jun 21, 2022
0 like 0 dislike
0 like 0 dislike
2 answers
hiroki_noda asked Jun 21, 2022
Is anyone familiar with the book, “Topics in the Theory of Numbers,” by Erdős and Surányi?
hiroki_noda asked Jun 21, 2022
0 like 0 dislike
0 like 0 dislike
11 answers
djmatchstick asked Jun 21, 2022
As of today, LaTeX-styled maths natively supported in GitHub Markdown (comments, issues, README.md, etc) $n!!$
djmatchstick asked Jun 21, 2022

33.4k questions

135k answers

0 comments

33.7k users

OhhAskMe is a math solving hub where high school and university students ask and answer loads of math questions, discuss the latest in math, and share their knowledge. It’s 100% free!