13 research outputs found
Hamiltonian submanifolds of regular polytopes
We investigate polyhedral -manifolds as subcomplexes of the boundary
complex of a regular polytope. We call such a subcomplex {\it -Hamiltonian}
if it contains the full -skeleton of the polytope. Since the case of the
cube is well known and since the case of a simplex was also previously studied
(these are so-called {\it super-neighborly triangulations}) we focus on the
case of the cross polytope and the sporadic regular 4-polytopes. By our results
the existence of 1-Hamiltonian surfaces is now decided for all regular
polytopes.
Furthermore we investigate 2-Hamiltonian 4-manifolds in the -dimensional
cross polytope. These are the "regular cases" satisfying equality in Sparla's
inequality. In particular, we present a new example with 16 vertices which is
highly symmetric with an automorphism group of order 128. Topologically it is
homeomorphic to a connected sum of 7 copies of . By this
example all regular cases of vertices with or, equivalently, all
cases of regular -polytopes with are now decided.Comment: 26 pages, 4 figure
Duality properties of indicatrices of knots
The bridge index and superbridge index of a knot are important invariants in
knot theory. We define the bridge map of a knot conformation, which is closely
related to these two invariants, and interpret it in terms of the tangent
indicatrix of the knot conformation. Using the concepts of dual and derivative
curves of spherical curves as introduced by Arnold, we show that the graph of
the bridge map is the union of the binormal indicatrix, its antipodal curve,
and some number of great circles. Similarly, we define the inflection map of a
knot conformation, interpret it in terms of the binormal indicatrix, and
express its graph in terms of the tangent indicatrix. This duality relationship
is also studied for another dual pair of curves, the normal and Darboux
indicatrices of a knot conformation. The analogous concepts are defined and
results are derived for stick knots.Comment: 22 pages, 9 figure
Combinatorial 3-manifolds with transitive cyclic symmetry
In this article we give combinatorial criteria to decide whether a transitive
cyclic combinatorial d-manifold can be generalized to an infinite family of
such complexes, together with an explicit construction in the case that such a
family exists. In addition, we substantially extend the classification of
combinatorial 3-manifolds with transitive cyclic symmetry up to 22 vertices.
Finally, a combination of these results is used to describe new infinite
families of transitive cyclic combinatorial manifolds and in particular a
family of neighborly combinatorial lens spaces of infinitely many distinct
topological types.Comment: 24 pages, 5 figures. Journal-ref: Discrete and Computational
Geometry, 51(2):394-426, 201
On tight immersions of maximal codimension
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46589/1/222_2005_Article_BF01404629.pd
Convex Partitions with 2-Edge Connected Dual Graphs
It is shown that for every finite set of disjoint convex polygonal obstacles in the plane, with a total of n vertices, the free space around the obstacles can be partitioned into open convex cells whose dual graph (defined below) is 2-edge connected. Intuitively, every edge of the dual graph corresponds to a pair of adjacent cells that are both incident to the same vertex. Aichholzer et al. recently conjectured that given an even number of line-segment obstacles, one can construct a convex partition by successively extending the segments along their supporting lines such that the dual graph is the union of two edge-disjoint spanning trees. Here we present a counterexample to this conjecture, which consists of 16 disjoint line segments, such that the dual graph of any convex partition constructed by this method has a bridge edge, and thus the dual graph cannot be partitioned into two spanning trees. Counterexamples of arbitrarily larger sizes can be constructed similarly. Questions about the dual graph of a convex partition are motivated by the still unresolved conjecture about disjoint compatible geometric matchings by Aichholzer et al.. It has application in the design of fault-tolerant wireless networks in the presence of obstacles (e.g. tall buildings in a city)
Computing Elevation Maxima by Searching the Gauss Sphere ⋆
Abstract. The elevation function on a smoothly embedded 2-manifold in R 3 reflects the multiscale topography of cavities and protrusions as local maxima. The function has been useful in identifying coarse docking configurations for protein pairs. Transporting the concept from the smooth to the piecewise linear category, this paper describes an algorithm for finding all local maxima. While its worst-case running time is the same as of the algorithm used in prior work, its performance in practice is orders of magnitudes superior. We cast light on this improvement by relating the running time to the total absolute Gaussian curvature of the 2-manifold.