Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0506432

The geometry of continued fractions and the topology of surface singularities

Patrick Popescu-Pampu
View original source ↗Read source document ↗Export metadata ↓

Abstract

We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a canonical plumbing structure on the abstract boundaries (also called links) of normal surface singularities. The duality between supplementary cones gives in particular a geometric interpretation of a duality discovered by Hirzebruch between the continued fraction expansions of two numbers l >1 and l/(l - 1).

Explore related subjects

math.GTmath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Patrick Popescu-Pampu. 2006-05-16. The geometry of continued fractions and the topology of surface singularities. https://arxiv.org/abs/math/0506432

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

KEEP EXPLORING

Related papers

arXiv2026

Cut pairs and Morse splitting of finitely generated groups

Bowditch's theorem for hyperbolic groups establishes a fundamental correspondence between splittings over two-ended subgroups and the existence of local cut points in the Gromov boundary. While analogous results have been obtained for CAT(0) and relatively hyperbolic groups, no general theorem of this type exists for arbitrary finitely generated groups. The Morse boundary, introduced by Charney-Sultan and extended by Cordes, provides a quasi-isometry invariant boundary for any finitely generated group that naturally generalizes the Gromov boundary. In this paper, we prove that a splitting of a finitely generated group with connected Morse boundary over a two-ended Morse subgroup gives rise to a separating pair of points in the Morse boundary.

math.GT↗
arXiv2026

Khovanov Homology in Connected Sums

Khovanov homology is an invariant for links in the three sphere that categorizes the Jones polynomial. We extend Khovanov's construction to links in 3-manifolds that are connected sums of orientable interval bundles over surfaces. Cutting the 3-manifold along a separating sphere, we construct type D and type A structures that are invariants of tangles in the two halves following the work of Roberts. Gluing the type D and type A structures along the common boundary recovers the Khovanov homology of the link.

math.GT↗
arXiv2026

Fox-Milnor condition for concordant knots in homology 3-spheres

This paper will show that the Alexander polynomial of a knot, which is of slice type in an oriented homology 3-sphere, obeys the Fox-Milnor polynomial condition. A relation between Alexander polynomial of concordant knots in an oriented homology 3-sphere is established.

math.GT↗