566 research outputs found
Geometric Permutations of Non-Overlapping Unit Balls Revisited
Given four congruent balls in that have disjoint
interior and admit a line that intersects them in the order , we show
that the distance between the centers of consecutive balls is smaller than the
distance between the centers of and . This allows us to give a new short
proof that interior-disjoint congruent balls admit at most three geometric
permutations, two if . We also make a conjecture that would imply that
such balls admit at most two geometric permutations, and show that if
the conjecture is false, then there is a counter-example of a highly degenerate
nature
Line transversals to disjoint balls
We prove that the set of directions of lines intersecting three disjoint
balls in in a given order is a strictly convex subset of . We then
generalize this result to disjoint balls in . As a consequence, we can
improve upon several old and new results on line transversals to disjoint balls
in arbitrary dimension, such as bounds on the number of connected components
and Helly-type theorems.Comment: 21 pages, includes figure
New Beauville surfaces and finite simple groups
In this paper we construct new Beauville surfaces with group either
\PSL(2,p^e), or belonging to some other families of finite simple groups of
Lie type of low Lie rank, or an alternating group, or a symmetric group,
proving a conjecture of Bauer, Catanese and Grunewald. The proofs rely on
probabilistic group theoretical results of Liebeck and Shalev, on classical
results of Macbeath and on recent results of Marion.Comment: v4: 18 pages. Final version, to appear in Manuscripta Mat
Numerical calculation of three-point branched covers of the projective line
We exhibit a numerical method to compute three-point branched covers of the
complex projective line. We develop algorithms for working explicitly with
Fuchsian triangle groups and their finite index subgroups, and we use these
algorithms to compute power series expansions of modular forms on these groups.Comment: 58 pages, 24 figures; referee's comments incorporate
New perspectives on self-linking
We initiate the study of classical knots through the homotopy class of the
n-th evaluation map of the knot, which is the induced map on the compactified
n-point configuration space. Sending a knot to its n-th evaluation map realizes
the space of knots as a subspace of what we call the n-th mapping space model
for knots. We compute the homotopy types of the first three mapping space
models, showing that the third model gives rise to an integer-valued invariant.
We realize this invariant in two ways, in terms of collinearities of three or
four points on the knot, and give some explicit computations. We show this
invariant coincides with the second coefficient of the Conway polynomial, thus
giving a new geometric definition of the simplest finite-type invariant.
Finally, using this geometric definition, we give some new applications of this
invariant relating to quadrisecants in the knot and to complexity of polygonal
and polynomial realizations of a knot.Comment: 26 pages, 17 figure
Geometric Permutations of Disjoint Unit Spheres
http://www.elsevier.com/locate/comgeoWe show that a set of disjoint unit spheres in admits at most two distinct geometric permutations if , and at most three if . This result improves a Helly-type theorem on line transversals for disjoint unit spheres in : if any subset of size of a family of such spheres admits a line transversal, then there is a line transversal for the entire family
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