Can A Three-String Knot be Constructed?

  • Thread starter spill
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In summary, the conversation revolves around the possibility of constructing a knot from three pieces of string where removing any of the pieces will cause the knot to collapse into an unknot. It is mentioned that a knot cannot self-intersect and removing an arc from a knot will result in a non-knot curve. The concept of Brunnian links, which are links that can have any number of components and will collapse into an unlink when any one component is removed, is introduced. The possibility of using Brunnian links to construct such a knot is discussed.
  • #1
spill
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This is a problem that's been troubling me recently. I'm neither a boy scout nor a knot theoretician, so I'm not sure how to progress.

Is it possible for three pieces of string to be tied together without the knot being topologically equivalent to a knot tying two of the pieces together with the third string tied either around one of the other pieces or around the first knot itself?

Oh dear, that wasn't very clear. Let's try again.

Is it possible to construct a knot from three pieces of string where removing any of the pieces will cause the knot to collapse into an unknot?

Help appreciated.
 
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  • #2
Since a knot cannot self-intersect, removing an arc from a knot will make it a curve that is not a knot. If you mean remove an arc and close the curve, then there are many obvious choices of making an unkot from a nontrivial knot. If you meant 3 links and the resulting 2 links collapsing into the unlink, then the Borromean rings is an example of one such link. Such links are called http://www.cs.ubc.ca/nest/imager/contributions/scharein/brunnian/brunnian.html and can have any number of components.
I've probably misunderstood your question, though. :)
 
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  • #3
Brunnian links - exactly what I was looking for. Thanks!
 

1. Can a three-string knot be constructed?

Yes, a three-string knot can be constructed. Knots can be made using any number of strings, including three.

2. How is a three-string knot constructed?

A three-string knot can be constructed by taking three strings and intertwining them in a specific pattern. This pattern can vary depending on the type of knot being made.

3. What types of three-string knots exist?

There are many types of three-string knots, including the overhand knot, the reef knot, the bowline knot, and the square knot, among others.

4. Can a three-string knot be untied?

Yes, a three-string knot can be untied. However, it may be more difficult to untie compared to a knot with fewer strings.

5. What are the uses of a three-string knot?

Three-string knots can be used for various purposes, such as securing objects together, creating decorative patterns, or in various crafts and hobbies. They are also used in sailing, rock climbing, and other outdoor activities where knots are needed for safety and security.

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