13,713 research outputs found
Quantum automorphism groups of homogeneous graphs
Associated to a finite graph is its quantum automorphism group . The
main problem is to compute the Poincar\'e series of , meaning the series
whose coefficients are multiplicities of 1 into tensor
powers of the fundamental representation. In this paper we find a duality
between certain quantum groups and planar algebras, which leads to a planar
algebra formulation of the problem. Together with some other results, this
gives for all homogeneous graphs having 8 vertices or less.Comment: 30 page
The Clique Problem in Ray Intersection Graphs
Ray intersection graphs are intersection graphs of rays, or halflines, in the
plane. We show that any planar graph has an even subdivision whose complement
is a ray intersection graph. The construction can be done in polynomial time
and implies that finding a maximum clique in a segment intersection graph is
NP-hard. This solves a 21-year old open problem posed by Kratochv\'il and
Ne\v{s}et\v{r}il.Comment: 12 pages, 7 figure
Quantum automorphism groups of small metric spaces
To any finite metric space we associate the universal Hopf \c^*-algebra
coacting on . We prove that spaces having at most 7 points fall into
one of the following classes: (1) the coaction of is not transitive; (2)
is the algebra of functions on the automorphism group of ; (3) is a
simplex and corresponds to a Temperley-Lieb algebra; (4) is a product
of simplexes and corresponds to a Fuss-Catalan algebra.Comment: 22 page
Jordan curves and funnel sections
We study the case when solution of an ODE at a given initial condition fail
to be unique and investigate what are the possible time-1 sections of the
`solution funnel'. Along the way we give construction of a natural complete
metric on the space of Jordan curves and prove that the generic Jordan curve is
nowhere pierceable by arcs of finite length.Comment: 22 pages, 16 figure
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