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_{c}to the central energy density ρ

_{c}, higher than σ = 0.677. In the trapping zones of such polytropes, the effective potential of the axial gravitational wave perturbations resembles those related to the ultracompact uniform density objects, giving thus similar long-lived axial gravitational modes. These long-lived linear perturbations are related to the stable circular null geodesic and due to additional non-linear phenomena could lead to conversion of the trapping zone to a black hole. We give in the eikonal limit examples of the long-lived gravitational modes, their oscillatory frequencies and slow damping rates, for the trapping zones of the polytropes with N in (2.138,4). However, in the trapping polytropes the long-lived damped modes exist only for very large values of the multipole number l > 50, while for smaller values of l the numerical calculations indicate existence of fast growing unstable axial gravitational modes. We demonstrate that for polytropes with N >= 3.78, the trapping region is by many orders smaller than extension of the polytrope, and the mass contained in the trapping zone is about 10

^{-3}of the total mass of the polytrope. Therefore, the gravitational instability of such trapping zones could serve as a model explaining creation of central supermassive black holes in galactic halos or galaxy clusters.

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_{q}ρ with the quintessential parameter ω_qin (-1;-1/3). We derive the geometry of quintessential rotating black holes, generalizing thus the Kerr spacetimes. Then we study the quintessential rotating black hole spacetimes with the special value of ω

_{q}= -2/3 when the resulting formulae are simple and easily tractable. We show that such special spacetimes can exist for the dimensionless quintessential parameter c < 1/6 and determine the critical rotational parameter a

_{0}separating the black hole and naked singularity spacetime in dependence on the quintessential parameter c . For the spacetimes with ω

_{q}= -2/3 we give all the black hole characteristics and demonstrate local thermodynamical stability. We present the integrated geodesic equations in separated form and study in details the circular geodetical orbits. We give radii and parameters of the photon circular orbits, marginally bound and marginally stable orbits. We stress that the outer boundary on the existence of circular geodesics, given by the so-called static radius where the gravitational attraction of the black hole is balanced by the cosmic repulsion, does not depend on the dimensionless spin of the rotating black hole, similarly to the case of the Kerr-de Sitter spacetimes with vacuum dark energy. We also give restrictions on the dimensionless parameters c and a of the spacetimes allowing for existence of stable circular geodesics. Finally, using numerical methods we generalize the discussion of the circular geodesics to the black holes with arbitrary quintessential parameter ω

_{q}.

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_{c}to the central energy density ρ

_{c}, can contain a region of trapped null geodesics. Such trapping polytropes can exist for n >2.138 , and they are generally much more extended and massive than the observed neutron stars. We show that in the n - σ parameter space, the region of allowed trapping increases with the polytropic index for intervals of physical interest, 2.138 <n <4 . Space extension of the region of trapped null geodesics increases with both increasing n and σ >0.677 from the allowed region. In order to relate the trapping phenomenon to astrophysically relevant situations, we restrict the validity of the polytropic configurations to their extension r

_{extr}corresponding to the gravitational mass M ̃2 M

_{☉}of the most massive observed neutron stars. Then, for the central density ρ

_{c}̃1 0

^{15}g cm

^{-3}, the trapped regions are outside r

_{extr}for all values of 2.138 <n <4 ; for the central density ρ

_{c}̃5 ×1 0

^{15}g cm

^{-3}, the whole trapped regions are located inside r

_{extr}for 2.138 <n <3.1 ; while for ρ

_{c}̃1 0

^{16}g cm

^{-3}, the whole trapped regions are inside r

_{extr}for all values of 2.138 <n <4 , guaranteeing astrophysically plausible trapping for all considered polytropes. The region of trapped null geodesics is located close to the polytrope center and could have a relevant influence on the cooling of such polytropes or binding of gravitational waves in their interior.

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