Ja n 20 04 On the classification of rational tangles (original) (raw)
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On the classification of rational tangles
Advances in Applied Mathematics, 2004
In this paper we give two new combinatorial proofs of the classification of rational tangles using the calculus of continued fractions. One proof uses the classification of alternating knots. The other proof uses colorings of tangles. We also obtain an elementary proof that alternating rational tangles have minimal number of crossings. Rational tangles form a basis for the classification of knots and are of fundamental importance in the study of DNA recombination.
2 7 N ov 2 00 3 On the Classification of Rational Knots
2008
In this paper we give combinatorial proofs of the classification of unoriented and oriented rational knots based on the now known classification of alternating knots and the calculus of continued fractions. We also characterize the class of strongly invertible rational links. Rational links are of fundamental importance in the study of DNA recombination. AMS Subject Classification: 57M27
On the Classification of Rational Knots
In this paper we give combinatorial proofs of the classification of unoriented and oriented rational knots based on the now known classification of alternating knots and the calculus of continued fractions. We also characterize the class of strongly invertible rational links. Rational links are of fundamental importance in the study of DNA recombination.
ec 2 00 2 On the Classification of Rational Knots
2009
In this paper we give combinatorial proofs of the classification of unoriented and oriented rational knots based on the now known classification of alternating knots and the calculus of continued fractions. We also characterize the class of strongly invertible rational links. Rational links are of fundamental importance in the study of DNA recombination. AMS Subject Classification: 57M27
Biological and Medical Physics, Biomedical Engineering, 2007
This paper draws a line from the elements of tangle fractions to the tangle model of DNA recombination. In the process, we sketch the classification of rational tangles, unoriented and oriented rational knots and the application of these subjects to DNA recombination.
Invariants of Rational Links Represented by Reduced Alternating Diagrams
2020
A rational link may be represented by any of the (infinitely) many link diagrams corresponding to various continued fraction expansions of the same rational number. The continued fraction expansion of the rational number in which all signs are the same is called a {\em nonalternating form} and the diagram corresponding to it is a reduced alternating link diagram, which is minimum in terms of the number of crossings in the diagram. Famous formulas exist in the literature for the braid index of a rational link by Murasugi and for its HOMFLY polynomial by Lickorish and Millet, but these rely on a special continued fraction expansion of the rational number in which all partial denominators are even (called {\em all-even form}). In this paper we present an algorithmic way to transform a continued fraction given in nonalternating form into the all-even form. Using this method we derive formulas for the braid index and the HOMFLY polynomial of a rational link in terms of its reduced altern...
Determinants of rational knots
Discrete Mathematics Theoretical Computer Science Dmtcs, 2007
We study the Fox coloring invariants of rational knots. We express the propagation of the colors down the twists of these knots and ultimately the determinant of them with the help of finite increasing sequences whose terms of even order are even and whose terms of odd order are odd.