On the classification of rational tangles

Details On the classification of rational tangles On the classification of rational tangles

2003-11-27

arxiv.org/abs/math/0311499v2

Mathematics

Geometric Topology

49 pages, 23 figures, LaTeX document

Scientific paper

This paper gives two new combinatorial topological proofs of the classification of rational tangles. Each proof rests on an elegant lemma showing that rational tangles are isotopic to canonical alternating rational tangles. The first proof defines the tangle fraction from the canonical form and uses flyping to prove invariance. The second proof defines the fraction of a rational tangle via integral coloring of the tangle. The coloring method is then used to prove the Kauffman-Harary coloring conjecture for alternating knots and links in the case of rational knots and links (closures of rational tangles). This paper forms the basis for a sequel on the classification of rational knots and links.

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