Can A Three-String Knot be Constructed?

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SUMMARY

The discussion centers on the construction of a three-string knot that collapses into an unknot when any one of the strings is removed. The concept of Brunnian links is introduced as a solution, where the removal of any component results in a trivial link. The Borromean rings serve as a specific example of such a link, demonstrating the properties of nontrivial knots that maintain their structure until a component is removed.

PREREQUISITES
  • Understanding of knot theory and its terminology
  • Familiarity with the concept of links and unlinks
  • Knowledge of Brunnian links and their properties
  • Basic grasp of topological concepts related to knots
NEXT STEPS
  • Research the properties and examples of Brunnian links
  • Study the Borromean rings and their significance in knot theory
  • Explore the mathematical definitions and implications of unlinking
  • Learn about the applications of knot theory in various fields such as biology and chemistry
USEFUL FOR

Mathematicians, knot theorists, and anyone interested in the complexities of knot construction and topology.

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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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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Brunnian links - exactly what I was looking for. Thanks!
 

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