Mathieu Dutour Sikiric
Perfect but not generating Delaunay polytopes.
Mon, 11/09/2009 - 16:01 — SomNambulAuthors: Mathieu Dutour Sikiric, Konstantin Rybnikov
Subjects: Combinatorics
AbstractIn his seminal 1951 paper "Extreme forms" Coxeter \cite{cox51} observed that
for $n \ge 9$ one can add vectors to the perfect lattice $\sfA_9$ so that the
resulting perfect lattice, called $\sfA_9^2$ by Coxeter, has exactly the same
set of minimal vectors. An inhomogeneous analog of the notion of perfect
lattice is that of a lattice with a perfect Delaunay polytope: the vertices of
a perfect Delaunay polytope are the analogs of minimal vectors in a perfect
lattice. We find a new infinite series $P(n,s)$ for $s\geq 2$ and $n+1\geq 4s$
of $n$-dimensional perfect Delaunay polytopes.4-regular and self-dual analogs of fullerenes.
Thu, 10/29/2009 - 16:01 — SomNambulAuthors: Mathieu Dutour Sikiric, Michel Deza
Subjects: Geometric Topology
AbstractAn i-hedrite is a 4-regular plane graph with faces of size 2, 3 and 4. We do
a short survey of their known properties and explain some new algorithms that
allow their efficient enumeration. Using this we give the symmetry groups of
all i-hedrites and the minimal representative for each. We also review the link
of 4-hedrites with knot theory and the classification of 4-hedrites with simple
central circuits. An i-self-hedrite is a self-dual plane graph with faces and
vertices of size/degree 2, 3 and 4. We give a new efficient algorithm for
enumerating them based on i-hedrites.Wythoff polytopes and low-dimensional homology of Mathieu groups.
Wed, 09/02/2009 - 12:02 — SomNambulAuthors: Mathieu Dutour Sikiric, Graham Ellis
Subjects: Group Theory
AbstractWe describe two methods for computing the low-dimensional integral homology
of the Mathieu simple groups and use them to make computations such as
$H_5(M_{23},\ZZ)=\ZZ_7$ and $H_3(M_{24},\ZZ)=\ZZ_{12}$. One method works via
Sylow subgroups. The other method uses a Wythoff polytope and perturbation
techniques to produce an explicit free $\ZZ M_n$-resolution. Both methods apply
in principle to arbitrary finite groups.Wythoff polytopes and low-dimensional homology of Mathieu groups.
Wed, 09/02/2009 - 12:02 — SomNambulAuthors: Mathieu Dutour Sikiric, Graham Ellis
Subjects: Group Theory
AbstractWe describe two methods for computing the low-dimensional integral homology
of the Mathieu simple groups and use them to make computations such as
$H_5(M_{23},\ZZ)=\ZZ_7$ and $H_3(M_{24},\ZZ)=\ZZ_{12}$. One method works via
Sylow subgroups. The other method uses a Wythoff polytope and perturbation
techniques to produce an explicit free $\ZZ M_n$-resolution. Both methods apply
in principle to arbitrary finite groups.
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