## Embedding Strings in the Unknot

J. Math. Sci. Univ. Tokyo
Vol. 10 (2003), No. 4, Page 631--660.

In this paper, the possibility of embedding a non trivial string $(\Bbb{R} ^{3},K)$ in the trivial knot $(\Bbb{S}^{3},U)$ is investigated. Uncountably many examples are given. The complementary space in $\Bbb{S}^{3}$ of the image of $\Bbb{R}^{3}$ under the embedding is a continuum. Some well known snake-like continua appear as these residual spaces. The 2-fold coverings of $\Bbb{R}^{3}$ branched over the strings involved are studied. As a consequence, concrete descriptions of the p-adic solenoids are given, and it is shown that the Whitehead continuum is homeomorphic to Bing's snake-like continuum without end points.