We calculate the Chern-Simons invariants of the twist knot orbifolds using the Schläfli formula for the generalized Chern-Simons function on the family of the twist knot cone-manifold structures. Following the general instruction of Hilden, Lozano, and Montesinos-Amilibia, we here present the concrete formulae and calculations. We use the Pythagorean Theorem, which was used by Ham, Mednykh, and Petrov, to relate the complex length of the longitude and the complex distance between the two axes fixed by two generators. As an application, we calculate the Chern-Simons invariants of cyclic coverings of the hyperbolic twist knot orbifolds. We also derive some interesting results. The explicit formula of A-polynomials of twist knots are obtained from the complex distance polynomials. Hence the edge polynomials corresponding to the edges of the Newton polygons of A-polynomials of twist knots can be obtained. In particular, the number of boundary components of every incompressible surface corresponding to slope −4n + 2 appears to be 2.

Condensation is characterized with a single macroscopic condensate whose mass is proportional to a system size N . We demonstrate how important particle interactions are in condensation phenomena. We study a modified version of the zero-range process by including a pair exclusion. Each particle is associated with its own partner and particles of a pair are forbidden to stay at the same site. The pair exclusion is weak in that a particle interacts with only a single one among all others. It turns out that such a weak interaction changes the nature of condensation drastically. There appear a number of mesoscopic condensates: the mass of a condensate scales as m con ∼ N 1/2 and the number of condensates scales as N con ∼ N 1/2 with a logarithmic correction. These results are derived analytically through a mapping to a solvable model under a certain assumption and confirmed numerically.

Let [Formula: see text] be the family of two bridge knots of slope [Formula: see text]. We calculate the volumes of the [Formula: see text] cone-manifolds using the Schläfli formula. We present the concrete and explicit formula of them. We apply the general instructions of Hilden, Lozano and Montesinos-Amilibia and extend the Ham, Mednykh and Petrov’s methods. As an application, we give the volumes of the cyclic coverings over those knots. For the fundamental group of [Formula: see text], we take and tailor Hoste and Shanahan’s. As a byproduct, we give an affirmative answer for their question whether their presentation is actually derived from Schubert’s canonical two-bridge diagram or not.

Abstract. We extend some part of the unpublished paper [30] written by Mednykh and Rasskazov. Using the approach indicated in this paper we derive the Riley-Mednykh polynomial for some family of the 2-bridge knot orbifolds. As a result we obtain explicit formulae for the volume of cone-manifolds and the Chern-Simons invariant of orbifolds of the knot with Conway's notation C(2n, 4).

Abstract. We calculate the Chern-Simons invariants of the hyperbolic orbifolds of the knot with Conway's notation C(2n, 3) using the Schläfli formula for the generalized Chern-Simons function on the family of C(2n, 3) cone-manifold structures. We present the concrete and explicit formula of them. We apply the general instructions of Hilden, Lozano, and Montesinos-Amilibia and extend the Ham and Lee's methods. As an application, we calculate the Chern-Simons invariants of cyclic coverings of the hyperbolic C(2n, 3) orbifolds.