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The other day, my wife found an old dress that had somehow got twisted. She gave it to me to straighten it. After numerous tries, I concluded that it wasn't possible and that it must be by design or a manufacturing defect. The label being off-center supports my 'by design' theory.

But, she swears she has worn it and it used to be perfectly fine. Here is a photo of the top part of the dress. The rest of the dress is absolutely normal and plain.

enter image description here

Topographically, there are 5 holes in the dress. Neck, 2 arms, one at the back and the bottom hole. As you can see, there is a full twist at the top, above the hole for the back.

Is is possible to "untwist" the dress? Or can it be proven that it is by design and it cannot be straightened?

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  • $\begingroup$ Can you move the twist around to other parts of the dress? $\endgroup$ – Michael McGovern Jul 4 '17 at 19:17
  • $\begingroup$ Yes, partially, but it looks a lot more messed up. If you try straightening, it always returns to the top. $\endgroup$ – navigator Jul 4 '17 at 19:28
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    $\begingroup$ At first glance this seems like a fun opportunity to apply some knot theory ;-) $\endgroup$ – Stefan Mesken Jul 4 '17 at 19:39
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    $\begingroup$ I think I found the dress elsewhere. Looks like the twist is there by design! $\endgroup$ – Peter Woolfitt Jul 4 '17 at 19:51
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    $\begingroup$ Fun with a soft topology question! :-D $\endgroup$ – md2perpe Jul 4 '17 at 19:55
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If you cut a strip around the neck hole, you get a strip, closed with a full (360 degree) twist; similar to a Moebius strip which would be 180 degree twisted.

If you consider the boundary of this twisted strip, it's two linked circles in $\mathbb{R}^3$. After untwisting, the strip would become an untwisted strip, and its boundary consists of two unlinked circles in $\mathbb{R}^3$.

Those two pairs of circles are not homotopic (what surely someone here can show...)

So your wife is the proud owner of a twisted-by-design dress.

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