Quasi-ordinary surface germs from the topological viewpoint
In the article we give a self-contained new proof that a normal quasi-ordinary surface germ is analytically isomorphic to a cyclic quotient surface germ.
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arXiv subjects
Claude Weber
Publications and source records attributed to Claude Weber.
On the Visibility of Alternating +Achiral Knots
This article is devoted to the study of prime alternating +achiral knots. In the case of arborescent knots, we prove in +AAA Visibility Theorem 5.1, that the symmetry is visible on a certain projection (not necessarily minimal) and that it is realised by a homeomorphism of order 4. In the general case (arborescent or not), if the prime alternating knot has no minimal projection on which +achirality is visible, we prove that the order of +achirality is necessarily equal to 4.
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Michel Kervaire work on knots in higher dimensions
We present Michel Kervaire work on knots in higher dimensions.
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On the Kawauchi conjecture about the Conway polynomial of achiral knots
We give a counterexample to the Kawauchi conjecture on the Conway polynomial of achiral knots which asserts that the Conway polynomial $C(z)$ of an achiral knot satisfies the splitting property $C(z)=F(z)F(-z)$ for a polynomial $F(z)$ with integer coefficients. We show that the Bonahon-Siebenmann decomposition of an achiral and alternating knot is reflected in the Conway polynomial. More explicitly, the Kawauchi conjecture is true for quasi-arborescent knots and counterexamples in the class of alternating knots must be quasi-polyhedral.
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On malnormal peripheral subgroups in fundamental groups of 3-manifolds
Let $K$ be a non-trivial knot in the 3-sphere, $E_K$ its exterior, $G_K = π_1(E_K)$ its group, and $P_K = π_1(\partial E_K) \subset G_K$ its peripheral subgroup. We show that $P_K$ is malnormal in $G_K$, namely that $gP_Kg^{-1} \cap P_K = \{e\}$ for any $g \in G_K$ with $g \notin P_K$, unless $K$ is in one of the following three classes: torus knots, cable knots, and composite knots; these are exactly the classes for which there exist annuli in $E_K$ attached to $T_K$ which are not boundary parallel (Theorem 1 and Corollary 2). More generally, we characterise malnormal peripheral subgroups in the fundamental group of a compact orientable irreducible 3-manifold with boundary a non-empty union of tori (Theorem 3). Proofs are written with non-expert readers in mind. Half of our paper (Sections 7 to 10) is a reminder of some three-manifold topology as it flourished before the Thurston revolution. In a companion paper [HaWeOs], we collect general facts on malnormal subgroups and Frobenius groups, and we review a number of examples.
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Malnormal subgroups and Frobenius groups: basics and examples
Malnormal subgroups occur in various contexts. We review a large number of examples, and we compare the situation in this generality to that of finite Frobenius groups of permutations. In a companion paper [HaWe], we analyse when peripheral subgroups of knot groups and 3-manifold groups are malnormal.
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A proof of Tait's Conjecture on alternating-achiral knots
In this paper we are interested in symmetries of alternating knots, more precisely in those related to achirality. We call the following statement Tait's Conjecture on alternating -achiral knots: Let K be an alternating -achiral knot. Then there exists a minimal projection Π of K in S^2 \subset S^3 and an involution ϕ:S^3\toS^3 such that: 1) ϕ reverses the orientation of $S^3$; 2) ϕ(S^2) = S^2; 3) ϕ (Π) = Π; 4) ϕ has two fixed points on Π and hence reverses the orientation of K. The purpose of this paper is to prove this statement.
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Link projections and flypes
Let Πbe a link projection in S^2. John Conway and later Francis Bonahon and Larry Siebenmann undertook to split $Π$ into canonical pieces. These pieces received different names: basic or polyhedral diagrams on one hand, rational, algebraic, bretzel, arborescent diagrams on the other hand. This paper proposes a thorough presentation of the theory, known to happy fews. We apply the existence and uniqueness theorem for the canonical decomposition to the classification of Haseman circles and to the localisation of the flypes.
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The boundary of the Milnor fiber for some non-isolated germs of complex surfaces
We study the boundary L_t of the Milnor fiber for the non-isolated singularities in C^3 with equation z^m - g(x,y) = 0 where g(x,y) is a non-reduced plane curve germ. We give a complete proof that L_t is a Waldhausen graph manifold and we provide the tools to construct its plumbing graph. As an example, we give the plumbing graph associated to the germs z^2 - (x^2 - y^3)y^l = 0 with l an interger >1. We prove that the boundary of the Milnor fiber is a Waldhausen manifold new in complex geometry, as it cannot be the boundary of a normal surface singularity.
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