On the Classification of Rational Knots (original) (raw)

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

ec 2 00 2 On the Classification of Rational Knots

2009

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.

Ja n 20 04 On the classification of rational tangles

2004

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.

Some Properties of Polynomials G and N, and Tables of Knots and Links

2013

In [1], we have constructed a polynomial invariant of regular isotopy, , for oriented knot and link diagrams L. From by multiplying it by normalizing factor, we obtained an ambient isotopy invariant, , for oriented knotsand links. In this paper, we give some properties of these polynomials. Wealso calculate the polynomials and of the knots through nine crossings and thetwo-component links through eight crossing

Unknotting numbers of alternating knot and link families

Publications de l'Institut Mathematique, 2014

After proving a theorem about the general formulae for the signature of alternating knot and link families, we distinguished all families of knots obtained from generating alternating knots with at most 10 crossings and alternating links with at most 9 crossings, for which the unknotting (unlinking) number can be confirmed by using the general formulae for signatures.

On the Classification of Rational Knots (original) (raw)

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