36,078 research outputs found
Area Inequalities for Embedded Disks Spanning Unknotted Curves
We show that a smooth unknotted curve in R^3 satisfies an isoperimetric
inequality that bounds the area of an embedded disk spanning the curve in terms
of two parameters: the length L of the curve and the thickness r (maximal
radius of an embedded tubular neighborhood) of the curve. For fixed length, the
expression giving the upper bound on the area grows exponentially in 1/r^2. In
the direction of lower bounds, we give a sequence of length one curves with r
approaching 0 for which the area of any spanning disk is bounded from below by
a function that grows exponentially with 1/r. In particular, given any constant
A, there is a smooth, unknotted length one curve for which the area of a
smallest embedded spanning disk is greater than A.Comment: 31 pages, 5 figure
Taut ideal triangulations of 3-manifolds
A taut ideal triangulation of a 3-manifold is a topological ideal
triangulation with extra combinatorial structure: a choice of transverse
orientation on each ideal 2-simplex, satisfying two simple conditions. The aim
of this paper is to demonstrate that taut ideal triangulations are very common,
and that their behaviour is very similar to that of a taut foliation. For
example, by studying normal surfaces in taut ideal triangulations, we give a
new proof of Gabai's result that the singular genus of a knot in the 3-sphere
is equal to its genus.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol4/paper12.abs.htm
Conforming restricted Delaunay mesh generation for piecewise smooth complexes
A Frontal-Delaunay refinement algorithm for mesh generation in piecewise
smooth domains is described. Built using a restricted Delaunay framework, this
new algorithm combines a number of novel features, including: (i) an
unweighted, conforming restricted Delaunay representation for domains specified
as a (non-manifold) collection of piecewise smooth surface patches and curve
segments, (ii) a protection strategy for domains containing curve segments that
subtend sharply acute angles, and (iii) a new class of off-centre refinement
rules designed to achieve high-quality point-placement along embedded curve
features. Experimental comparisons show that the new Frontal-Delaunay algorithm
outperforms a classical (statically weighted) restricted Delaunay-refinement
technique for a number of three-dimensional benchmark problems.Comment: To appear at the 25th International Meshing Roundtabl
The robust assembly of small symmetric nano-shells
Highly symmetric nano-shells are found in many biological systems, such as
clathrin cages and viral shells. Several studies have shown that symmetric
shells appear in nature as a result of the free energy minimization of a
generic interaction between their constituent subunits. We examine the physical
basis for the formation of symmetric shells, and using a minimal model we
demonstrate that these structures can readily grow from identical subunits
under non equilibrium conditions. Our model of nano-shell assembly shows that
the spontaneous curvature regulates the size of the shell while the mechanical
properties of the subunit determines the symmetry of the assembled structure.
Understanding the minimum requirements for the formation of closed nano-shells
is a necessary step towards engineering of nano-containers, which will have far
reaching impact in both material science and medicine.Comment: 12 pages, 12 figure
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