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Antoine's Necklace


AntoinesNecklace

Construct a chain C of 2n components in a solid torus V. Now thicken each component of C slightly to form a chain C_1 of 2n solid tori in V, where

 pi_1(V-C_1)=pi_1(V-C)

via inclusion. In each component of C_1, construct a smaller chain of solid tori embedded in that component. Denote the union of these smaller solid tori C_2. Continue this process a countable number of times, then the intersection

 A= intersection _(i=1)^inftyC_i

which is a nonempty compact subset of R^3 is called Antoine's necklace. Antoine's necklace is homeomorphic with the Cantor set.


See also

Alexander's Horned Sphere, Necklace

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References

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 73-74, 1976.

Referenced on Wolfram|Alpha

Antoine's Necklace

Cite this as:

Weisstein, Eric W. "Antoine's Necklace." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AntoinesNecklace.html

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