Rational Tangles

Abstract

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

Tangle Analysis of Xer Recombination Reveals only Three Solutions, all Consistent with a Single Three-dimensional Topological Pathway

“…Rational tangles and 4-plats have been classified, allowing them to be represented by integer entry vectors and rational numbers. [25][26][27] In the tangle method, a recombination event is modeled by a system of two tangle equations ( Figure 4):where O, P and R are unknown tangles, and K 1 and K 2 are substrate and product of recombination (respectively). P is the parental tangle and contains only the DNA that is changed during recombination.…”

“…Rational tangles and 4-plats have been classified, allowing them to be represented by integer entry vectors and rational numbers. [25][26][27] In the tangle method, a recombination event is modeled by a system of two tangle equations ( Figure 4):…”

From Tangle Fractions to DNA

“…The tangle fraction is a key ingredient in both the classification of rational knots and in the applications of knot theory to DNA. Proofs of Theorem 1 can be found in [33], [6] p.196, [16] and [25].…”

“…The first definition is due to John Conway in [7] using the Alexander polynomial of the knots N(T ) and D(T ). In [16] an alternate definition is given that uses the bracket polynomial of the knots N(T ) and D(T ), and in [15] the fraction of a tangle is related to the conductance of an associated electrical network. In all these definitions the fraction is by definition an isotopy invariant of tangles.…”

“…Rational knots and links were first considered in [40] and [2]. Treatments of various aspects of rational knots and rational tangles can be found in [3], [7], [46], [6], [42], [16], [27], [31], [34]. A rational tangle is associated in a canonical manner with a unique, reduced rational number or ∞, called the fraction of the tangle.…”