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On the classification of rational tangles

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arxiv math/0311499 v2 pith:VFJ23LDL submitted 2003-11-27 math.GT math.NT

On the classification of rational tangles

classification math.GT math.NT
keywords rationaltanglesclassificationcoloringknotslinksprooftangle
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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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  1. Analogue of Goeritz matrices for computation of bipartite HOMFLY-PT polynomials

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    A modified Goeritz matrix is defined for bipartite link diagrams that reduces HOMFLY-PT computation for any N to matrix algebra.