Wow! Thank you Dan. I had to take a few minutes visualizing each stage of your proof outline, but I got it and I see where my intuition had failed me. Math is fun! (And thanks for clarifying my misuse of the terms. I guess I did mean the stages are linked, and it was my uncertainty about whether the necklace itself is "linked" or not.) On 12/6/12, Dan Asimov <dasimov@earthlink.net> wrote:
It's not correct to say "the necklace" remains linked through the stages of its construction; it's the *stages* that are linked.
But it is true that the necklace N is totally disconnected, i.e., its connected components are all single points. And that N is indeed linked with a circle in its complement. (So there is nothing at all that "becomes unlinked".)
Here's a rough sketch of a proof that the necklace is linked with a closed curve in its complement that is linked with the first stage in its construction: a single solid torus.
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