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Can a trefoil knot be stretched to look like a triangle with three knots at the vertices, like in the right side of the image below, or is that transformation impossible to happen? If possible, what is the nature of the crossings that has to occur at the three corners of the triangle like figure on the right hand side? enter image description here

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    $\begingroup$ The trefoil knot is a prime knot, i.e. it cannot be split to 3 (or any number $\geq2$) successive knots $\endgroup$
    – user8268
    Apr 29, 2015 at 18:34
  • $\begingroup$ @user8268. Can you extend on your statement and infer on if the right hand image is possible or not? Thanks again. Am not a topologist! $\endgroup$
    – hearse
    Apr 29, 2015 at 18:38
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    $\begingroup$ see e.g. wikipedia for the definition of prime knots. Trefoil being prime implies that the right hand image is impossible (provided the knots at at least 2 of the 3 vertices of the triangle are nontrivial :) [I'm not a topologist either] $\endgroup$
    – user8268
    Apr 29, 2015 at 19:36
  • $\begingroup$ Thanks! I'd need a microscope to zoom if these vertices are non-trivial :) $\endgroup$
    – hearse
    Apr 30, 2015 at 3:01

1 Answer 1

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enter image description here

$$\boldsymbol{\operatorname{Figure:}} \text{ The Stretchfoil}$$

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    $\begingroup$ Okay, I'm being a bit flippant here. But is this what you mean? $\endgroup$
    – Kyle
    Apr 30, 2015 at 14:59
  • $\begingroup$ Yes! Indeed .... $\endgroup$
    – hearse
    Apr 30, 2015 at 16:17
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    $\begingroup$ Good! Then the answer to the question you've posted is yes. A knot type is defined up to "ambient isotopy", which basically means continuously pushing or pulling the knot any which way, as long as you don't break the string by tearing it open or passing part of one strand through another. $\endgroup$
    – Kyle
    Apr 30, 2015 at 16:28
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    $\begingroup$ This made me burst out laughing. $\endgroup$
    – user98602
    Dec 18, 2015 at 0:50

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