168,171 research outputs found
Prime component-preservingly amphicheiral link with odd minimal crossing number
For every odd integer , we raise an example of a prime
component-preservingly amphicheiral link with the minimal crossing number .
The link has two components, and consists of an unknot and a knot which is
-amphicheiral with odd minimal crossing number. We call the latter knot a
{\it Stoimenow knot}. We also show that the Stoimenow knot is not invertible by
the Alexander polynomials.Comment: 25 pages, 14 figure
Condorcet Domains, Median Graphs and the Single Crossing Property
Condorcet domains are sets of linear orders with the property that, whenever
the preferences of all voters belong to this set, the majority relation has no
cycles. We observe that, without loss of generality, such domain can be assumed
to be closed in the sense that it contains the majority relation of every
profile with an odd number of individuals whose preferences belong to this
domain.
We show that every closed Condorcet domain is naturally endowed with the
structure of a median graph and that, conversely, every median graph is
associated with a closed Condorcet domain (which may not be a unique one). The
subclass of those Condorcet domains that correspond to linear graphs (chains)
are exactly the preference domains with the classical single crossing property.
As a corollary, we obtain that the domains with the so-called `representative
voter property' (with the exception of a 4-cycle) are the single crossing
domains.
Maximality of a Condorcet domain imposes additional restrictions on the
underlying median graph. We prove that among all trees only the chains can
induce maximal Condorcet domains, and we characterize the single crossing
domains that in fact do correspond to maximal Condorcet domains.
Finally, using Nehring's and Puppe's (2007) characterization of monotone
Arrowian aggregation, our analysis yields a rich class of strategy-proof social
choice functions on any closed Condorcet domain
Removing Even Crossings
An edge in a drawing of a graph is called if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly stronger statement. We show two applications of this strengthened result: an easy proof of a theorem of Hanani and Tutte (not using Kuratowski's theorem), and the result that the odd crossing number of a graph equals the crossing number of the graph for values of at most . We begin with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte
Wigner-Poisson statistics of topological transitions in a Josephson junction
The phase-dependent bound states (Andreev levels) of a Josephson junction can
cross at the Fermi level, if the superconducting ground state switches between
even and odd fermion parity. The level crossing is topologically protected, in
the absence of time-reversal and spin-rotation symmetry, irrespective of
whether the superconductor itself is topologically trivial or not. We develop a
statistical theory of these topological transitions in an N-mode quantum-dot
Josephson junction, by associating the Andreev level crossings with the real
eigenvalues of a random non-Hermitian matrix. The number of topological
transitions in a 2pi phase interval scales as sqrt(N) and their spacing
distribution is a hybrid of the Wigner and Poisson distributions of
random-matrix theory.Comment: 12 pages, 15 figures; v2 to appear in PRL, with appendix in the
supplementary materia
Thrackles: An improved upper bound
A thrackle is a graph drawn in the plane so that every pair of its edges meet exactly once: either at a common end vertex or in a proper crossing. We prove that any thrackle of n vertices has at most 1.3984n edges. Quasi-thrackles are defined similarly, except that every pair of edges that do not share a vertex are allowed to cross an odd number of times. It is also shown that the maximum number of edges of a quasi-thrackle on n vertices is 3/2(n-1), and that this bound is best possible for infinitely many values of n. (C) 2019 Published by Elsevier B.V
Thrackles: An improved upper bound
A thrackle is a graph drawn in the plane so that every pair of its edges meet exactly once: either at a common end vertex or in a proper crossing. We prove that any thrackle of n vertices has at most 1.3984n edges. Quasi-thrackles are defined similarly, except that every pair of edges that do not share a vertex are allowed to cross an odd number of times. It is also shown that the maximum number of edges of a quasi-thrackle on n vertices is 3/2(n-1), and that this bound is best possible for infinitely many values of n. © Springer International Publishing AG 2018
Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle
We show that the Wannier obstruction and the fragile topology of the nearly
flat bands in twisted bilayer graphene at magic angle are manifestations of the
nontrivial topology of two-dimensional real wave functions characterized by the
Euler class. To prove this, we examine the generic band topology of two
dimensional real fermions in systems with space-time inversion
symmetry. The Euler class is an integer topological invariant classifying real
two band systems. We show that a two-band system with a nonzero Euler class
cannot have an -symmetric Wannier representation. Moreover, a two-band
system with the Euler class has band crossing points whose total
winding number is equal to . Thus the conventional Nielsen-Ninomiya
theorem fails in systems with a nonzero Euler class. We propose that the
topological phase transition between two insulators carrying distinct Euler
classes can be described in terms of the pair creation and annihilation of
vortices accompanied by winding number changes across Dirac strings. When the
number of bands is bigger than two, there is a topological invariant
classifying the band topology, that is, the second Stiefel Whitney class
(). Two bands with an even (odd) Euler class turn into a system with
() when additional trivial bands are added. Although the
nontrivial second Stiefel-Whitney class remains robust against adding trivial
bands, it does not impose a Wannier obstruction when the number of bands is
bigger than two. However, when the resulting multi-band system with the
nontrivial second Stiefel-Whitney class is supplemented by additional chiral
symmetry, a nontrivial second-order topology and the associated corner charges
are guaranteed.Comment: 23 pages, 13 figure
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