Embedding Strings in the Unknot
Jose Mar ́ ia · 2003
Dedicated with respect and friendship to Professor Yukio Matsumoto on his 60th birthday Abstract. In this paper, the possibility of embedding a non triv- ial string (R 3 ,K ) in the trivial knot (S 3 ,U ) is investigated. Uncount- ably many examples are given. The complementary space in S 3 of the image of R 3 under the embedding is a continuum. Some well known snake-like continua appear as these residual spaces. The 2-fold coverings of 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.