Classifying and Applying Rational Knots and Rational Tangles
نویسندگان
چکیده
In this survey paper we sketch new combinatorial proofs of the classification of rational tangles and of unoriented and oriented rational knots, using the classification of alternating knots and the calculus of continued fractions. We continue with the classification of achiral and strongly invertible rational links, and we conclude with a description of the relationships among tangles, rational knots and DNA recombination.
منابع مشابه
On the classification of rational tangles
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 kno...
متن کاملVassiliev Invariants and Rational Knots of Unknotting Number One
Introducing a way to modify knots using n-trivial rational tangles, we show that the number of n-trivial rational knots of at most k crossings is for any n asymptotically at least C(lnk)2 for any C < 2 ln2 pe. The method also extends a recent result of Ohyama, Taniyama and Yamada, showing that Vassiliev invariants are unrelated to the unknotting number. The same holds for 4genera, signatures an...
متن کاملAbout Some Infinite Family of 2-bridge Knots and 3-manifolds
We construct an infinite family of 3-manifolds and show that these manifolds have cyclically presented fundamental groups and are cyclic branched coverings of the 3-sphere branched over the 2-bridge knots ( +1)2 or ( +1)1, that are the closure of the rational (2 −1)/( −1)–tangles or (2 −1)/ –tangles, respectively.
متن کاملRational Structure on Algebraic Tangles and Closed Incompressible Surfaces in the Complements of Algebraically Alternating Knots and Links
Let F be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle (B, T ). Then F separates the strings of T in B and the boundary slope of F is uniquely determined by (B, T ) and hence we can define the slope of the algebraic tangle. In addition to the Conway’s tangle sum, we define a natural product of two tangles....
متن کاملRational Structure on Algebraic Tangles and Closed Incompressible Surfaces in Algebraically Alternating Knots and Links
Let F be an incompressible, meridionally incompressible and not boundary-parallel surface in the complement of an algebraic tangle (B, T ). Then F separates the strings of T in B and the boundary slope of F is uniquely determined by (B, T ) and hence we can define the slope of the algebraic tangle. In addition to the Conway’s tangle sum, we define a natural product of two tangles. The slopes an...
متن کامل