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Show that any simple closed polygon in $\mathbb{R^2}$ belongs to the trivial knot type.

Could anyone give me a hint for the solution?

1 Answers1

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The Jordan-Schoenflies theorem extends the Jordan Curve theorem: Every simple-closed curve in the plane is homeomorphic to the circle. The interior and exterior can be mapped homeomorphically to the respective complements.

https://en.wikipedia.org/wiki/Schoenflies_problem

This clearly does not extend to $\mathbb R^3$ which is why there are non-trivial knots in $\mathbb R^3 $

MSIS
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  • https://math.stackexchange.com/questions/2664730/is-a-knot-in-mathbbr3-that-lies-in-a-plane-necessarily-trivial?rq=1 is this question answering my question also? –  Jun 26 '19 at 21:08
  • Also is this answering some of my question ?https://math.stackexchange.com/questions/2664793/is-a-knot-in-mathbbr3-that-can-be-untied-necessarily-trivial?rq=1 –  Jun 26 '19 at 21:10
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    @Smart: For the first link see the second answer , making reference to Schoenflies theorem. – MSIS Jun 26 '19 at 21:32
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    @Smart, I don't see off-hand the connection to this answer for your bottom link, will reread. – MSIS Jun 26 '19 at 21:40
  • Is there a place where I can find a proof for Jordan-Schoenflies theorem? – Intuition Jul 01 '19 at 13:47
  • Try this: http://eretrandre.org/rb/files/Cairns1951_193.pdf – MSIS Jul 03 '19 at 22:22
  • @Secretly: Sorry for the delay. Please see the Wiki page for proofs: https://en.wikipedia.org/wiki/Schoenflies_problem – MSIS Sep 16 '22 at 19:05