Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0311511

From tangle fractions to DNA

Louis H. Kauffman·Sofia Lambropoulou
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper is an introduction to rational tangles, rational knots and links and their applications to DNA. The paper can be read as an introduction to our more technical papers on rational tangles (math.GT/0311499) and on rational knots (math.GT/0212011). The present paper includes a self-contained account of the tangle fraction from the point of view of the bracket polynomial, accounts of the classification theorems for rational tangles and rational knots (and of our new proofs of these results) and a description of the use of rational tangles in studying DNA recombination.

Explore related subjects

math.GTmath.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Louis H. Kauffman, Sofia Lambropoulou. 2003-11-27. From tangle fractions to DNA. https://arxiv.org/abs/math/0311511

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Links and singularities of stable maps from $3$-manifolds to surfaces

For any two links in the $3$-sphere, we provide a visual construction of a stable map $f$ from the $3$-sphere to the Euclidean plane such that $f$ has no cusp points, the set of definite (resp. indefinite) fold points of $f$ is isotopic to the first (resp. second) given link, and $f$ does not have a certain type of singular fibers. Moreover, we obtain a similar result for any $3$-manifold by setting the target space to the $2$-sphere.

math.GT↗
arXiv2026

The Smoothability of Embedded Surfaces in 4-Manifolds Under Stabilizations

We study the smooth stabilization number of a locally flat embedded surface in a smooth $4$-manifold, defined as the minimum number of connected sums with $S^2\times S^2$ needed to make the surface topologically isotopic to a smooth embedding. We prove that a nonsmoothable locally flat surface in a closed, simply connected smooth $4$-manifold becomes smoothable after a single stabilization if its homology class is primitive and noncharacteristic and its complement has abelian fundamental group. This yields infinitely many examples of nonsmoothable surfaces with smooth stabilization number one. We also exhibit examples for which a single stabilization produces infinitely many pairwise smoothly nonisotopic representatives in the same topological isotopy class. In contrast, we construct a locally flat genus-eight surface representing $6h$ in $\mathbb{CP}^2$ whose smooth stabilization number is exactly two. More generally, we show that smooth stabilization numbers of locally flat embedded surfaces in $\mathbb{CP}^2$ are unbounded.

math.GT↗
arXiv2026

Sections, pseudosections, and multisections of Lefschetz fibrations

We show that not every symplectic Lefschetz fibration over the $2$--sphere admits a smooth section, settling a long-standing open problem. To prove this, we develop a method for studying sections via lifts of monodromy factorizations with point-pushing maps and the induced handle decomposition of the total space, and apply it to an infinite family of Lefschetz fibrations. In contrast, we show that every Lefschetz fibration over $S^2$ admits positive multisections. Some of these examples admit no multisections with spherical components. We also construct signature-zero Lefschetz fibrations of every odd genus $g\geq 7$, which in turn give symplectic Lefschetz fibrations of any prescribed signature in each of these genera.

math.GT↗