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This paper gives an elementary and self-contained proof of Conway's Basic Theorem on rational tangles. This theorem states that two rational tangles are topologically equivalent if and only if they have the same associated rational fraction. Our proof divides into a geometric half that relates the arithmetic of continued fractions to the topology of tangles and an algebraic part that defines the fraction of any tangle via the bracket model for the Jones polynomial. We present an application to molecular biology. ᮊ 1997 Academic Press

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