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Antoine's Necklace

\begin{figure}\begin{center}\BoxedEPSF{AntoinesNecklace.epsf scaled 800}\end{center}\end{figure}

Construct a chain $C$ of $2n$ components in a solid Torus $V$. Now form a chain $C_1$ of $2n$ solid tori in $V$, where

\pi_1(V-C_1) \cong \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=\bigcap_{i=1}^\infty C_i

which is a nonempty compact Subset of $\Bbb{R}^3$ is called Antoine's necklace. Antoine's necklace is Homeomorphic with the Cantor Set.

See also Alexander's Horned Sphere, Necklace


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

© 1996-9 Eric W. Weisstein