227 research outputs found
Cataloguing PL 4-manifolds by gem-complexity
We describe an algorithm to subdivide automatically a given set of PL
n-manifolds (via coloured triangulations or, equivalently, via
crystallizations) into classes whose elements are PL-homeomorphic. The
algorithm, implemented in the case n=4, succeeds to solve completely the
PL-homeomorphism problem among the catalogue of all closed connected PL
4-manifolds up to gem-complexity 8 (i.e., which admit a coloured triangulation
with at most 18 4-simplices). Possible interactions with the (not completely
known) relationship among different classification in TOP and DIFF=PL
categories are also investigated. As a first consequence of the above PL
classification, the non-existence of exotic PL 4-manifolds up to gem-complexity
8 is proved. Further applications of the tool are described, related to
possible PL-recognition of different triangulations of the K3-surface.Comment: 25 pages, 5 figures. Improvements suggested by the refere
Simple crystallizations of 4-manifolds
Minimal crystallizations of simply connected PL 4-manifolds are very natural
objects. Many of their topological features are reflected in their
combinatorial structure which, in addition, is preserved under the connected
sum operation. We present a minimal crystallization of the standard PL K3
surface. In combination with known results this yields minimal crystallizations
of all simply connected PL 4-manifolds of "standard" type, that is, all
connected sums of , , and the K3 surface. In
particular, we obtain minimal crystallizations of a pair of homeomorphic but
non-PL-homeomorphic 4-manifolds. In addition, we give an elementary proof that
the minimal 8-vertex crystallization of is unique and its
associated pseudotriangulation is related to the 9-vertex combinatorial
triangulation of by the minimum of four edge contractions.Comment: 23 pages, 7 figures. Minor update, replacement of Figure 7. To appear
in Advances in Geometr
A note about complexity of lens spaces
Within crystallization theory, (Matveev's) complexity of a 3-manifold can be
estimated by means of the combinatorial notion of GM-complexity. In this paper,
we prove that the GM-complexity of any lens space L(p,q), with p greater than
2, is bounded by S(p,q)-3, where S(p,q) denotes the sum of all partial
quotients in the expansion of q/p as a regular continued fraction. The above
upper bound had been already established with regard to complexity; its
sharpness was conjectured by Matveev himself and has been recently proved for
some infinite families of lens spaces by Jaco, Rubinstein and Tillmann. As a
consequence, infinite classes of 3-manifolds turn out to exist, where
complexity and GM-complexity coincide.
Moreover, we present and briefly analyze results arising from crystallization
catalogues up to order 32, which prompt us to conjecture, for any lens space
L(p,q) with p greater than 2, the following relation: k(L(p,q)) = 5 + 2
c(L(p,q)), where c(M) denotes the complexity of a 3-manifold M and k(M)+1 is
half the minimum order of a crystallization of M.Comment: 14 pages, 2 figures; v2: we improved the paper (changes in
Proposition 10; Corollary 9 and Proposition 11 added) taking into account
Theorem 2.6 of arxiv:1310.1991v1 which makes use of our Prop. 6(b)
(arxiv:1309.5728v1). Minor changes have been done, too, in particular to make
references more essentia
Random Tensors and Quantum Gravity
We provide an informal introduction to tensor field theories and to their
associated renormalization group. We focus more on the general motivations
coming from quantum gravity than on the technical details. In particular we
discuss how asymptotic freedom of such tensor field theories gives a concrete
example of a natural "quantum relativity" postulate: physics in the deep
ultraviolet regime becomes asymptotically more and more independent of any
particular choice of Hilbert basis in the space of states of the universe.Comment: Section 6 is essentially reproduced from author's arXiv:1507.04190
for self-contained purpose of the revie
A census of genus two 3-manifolds up to 42 coloured tetrahedra
We improve and extend to the non-orientable case a recent result of Karabas,
Malicki and Nedela concerning the classification of all orientable prime
3-manifolds of Heegaard genus two, triangulated with at most 42 coloured
tetrahedra.Comment: 24 pages, 3 figure
Computing Matveev's complexity via crystallization theory: the boundary case
The notion of Gem-Matveev complexity has been introduced within
crystallization theory, as a combinatorial method to estimate Matveev's
complexity of closed 3-manifolds; it yielded upper bounds for interesting
classes of such manifolds. In this paper we extend the definition to the case
of non-empty boundary and prove that for each compact irreducible and
boundary-irreducible 3-manifold it coincides with the modified Heegaard
complexity introduced by Cattabriga, Mulazzani and Vesnin. Moreover, via
Gem-Matveev complexity, we obtain an estimation of Matveev's complexity for all
Seifert 3-manifolds with base and two exceptional fibers and,
therefore, for all torus knot complements.Comment: 27 pages, 14 figure
Triangulations
The earliest work in topology was often based on explicit combinatorial models – usually triangulations – for the spaces being studied. Although algebraic methods in topology gradually replaced combinatorial ones in the mid-1900s, the emergence of computers later revitalized the study of triangulations. By now there are several distinct mathematical communities actively doing work on different aspects of triangulations. The goal of this workshop was to bring the researchers from these various communities together to stimulate interaction and to benefit from the exchange of ideas and methods
Investigations of chiral effects in molecular spectroscopy
A consequence of the chiral weak interaction is that the enantiomers of a chiral molecule will differ in energy by the minute parity-violating energy difference (PVED). The enantiomers of a chiral iron complex were prepared and characterized with various spectroscopies, including x-ray photoelectron spectroscopy and x-ray diffraction. Measurements of the Mossbauer spectra show a small difference in the energy of the two enantiomers (~10-10 eV). This energy difference nears the expected order of magnitude of the parity-violating energy difference for a molecule in which the chiral center is a high Z atom. Sodium chlorate has been known to form chiral crystals from achiral aqueous solutions for over one hundred years. Typically, equal numbers of right- and left-handed crystals are produced in unstirred crystallizations. Data has been taken that show an excess of right-handed crystals are produced when crystallizations occur under the influence of a beta source. The beta particles are spin polarized due to the chiral weak interaction which is responsible for beta decay. Preliminary results indicate that the influence of positrons (which are spin polarized oppositely to beta particles) is in the opposite direction. Finally, measurements of mass resolved resonant and non-resonant multiphoton ionization of the chiral 2-butylamine entrained in a nozzle jet expansion into a linear time-of-flight mass spectrometer constructed in house were obtained using right- and left-circularly polarized laser light. In addition, ratios of ionization rates for linearly and circularly polarized light were measured
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