The geometry of continued fractions and the topology of surface singularities

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 λ > 1 and λ λ−1 .

Non-integer surgery and branched double covers of alternating knots

“…The following formula allows us to switch between the two types of continued fraction (for example, see [20]): Since we may write p q in the form p q = n − 1 + q−r q , we have a 0 = n − 1 and q q − r = [a 1 , a 2 , . .…”

The Shortest Path Problem for the Distant Graph of the Projective Line Over the Ring of Integers

“…By results of Sect. 5 the matrix given by (7) represents at least one shortest path fromx toȳ. There may be other matrices representing such a path.…”

“…There are at most two matrices representing the shortest paths fromx toȳ, among them there is always a matrix A given by (7).…”

On the smoothings of non-normal isolated surface singularities

“…In this section we recall the basic properties and classes of normal surface singularities which are needed in the sequel. More detailed introductions to the study of normal surface singularities are contained in [41], [28], [29], [51], [38].…”