It is of increasing contemporary interest to preserve ambient isotopy during geometric modeling. Bézier curves are pervasive in computer aided geometric design, as one of the fundamental computational representations for geometric modeling. For Bézier curves, subdivision algorithms create control polygons as piecewise linear (PL) approximations that converge under Hausdorff distance. A natural ...

SOLVMANIFOLD DARYL COOPER AND MARTIN SCHARLEMANN Abstract. We classify isotopy classes of Heegaard splittings of solvmanifolds. If the monodromy of the solvmanifold can be expressed as m 1 1 0 ; for some m 3 (as always is true when the trace of the monodromy is 3), then any irreducible splitting is strongly irreducible and of genus two. If m 4 any two such splittings are isotopic. If m = 3 then...

We investigate the following problem: Given two embeddings G1 and G2 of the same abstract graph G on an orientable surface S, decide whether G1 and G2 are isotopic; in other words, whether there exists a continuous family of embeddings between G1 and G2. We provide efficient algorithms to solve this problem in two models. In the first model, the input consists of the arrangement of G1 (resp., G...

Four mutually dependent facts are proven. • A smooth saddle sphere in S has at least four inflection arches. • Each hyperbolic hérisson H generates an arrangement of disjoint oriented great semicircles on the unit sphere S. On the one hand, the semicircles correspond to the horns of the hérisson. On the other hand, they correspond to the inflection arches of the graph of the support function hH...

We define the contact homology for Legendrian submanifolds in standard contact (2n + 1)-space using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex n-space. This homology provides new invariants of Legendrian isotopy which indicate that the theory of Legendrian isotopy is very rich. Indeed, in [4] the homology is used to detect infinite families of pairwise non...

If a continuous map f : X → Q is approximable arbitrary closely by embeddings X →֒ Q, can some embedding be taken onto f by a pseudo-isotopy? This question, called Isotopic Realization Problem, was raised by Ščepin and Akhmet’ev. We consider the case where X is a compact n-polyhedron, Q a PL m-manifold and show that the answer is ‘generally no’ for (n,m) = (3, 6); (1, 3), and ‘yes’ when: 1) m > ...

We define enhanced presentations of quandles via generators and relations with additional information comprising signed operations and an order structure on the set of generators. Such a presentation determines a virtual link diagram up to virtual moves. We list formal Reidemeister moves in which Tietze moves on the presented quandle are accompanied by corresponding changes to the order structu...

A contactomorphism is a diffeomorphism ψ : M → M which preserves the contact structure ξ. This means that ψα = eα for some smooth function h : M → . A contact isotopy is a smooth family of contactomorphisms ψs : M → M such that ψ sα = e sα for all s. Any such isotopy with ψ0 = id and h0 = 0 is generated by Hamiltonian functions Hs : M → as follows. Each Hamiltonian function Hs determines a Hami...

We resolve a question of Fuchs and Tabachnikov by showing that there is a Legendrian knot in standard contact R with zero Maslov number which is not Legendrian isotopic to its mirror. The proof uses the differential graded algebras of Chekanov. A Legendrian knot in standard contact R is a knot which is everywhere tangent to the two-plane distribution induced by the contact one-form dz− y dx. Tw...