Abstract. The minimum dilatation of pseudo-Anosov 5-braids is shown to be the largest zero λ 5 ≈ 1.72208 of x 4 − x 3 − x 2 − x + 1 which is attained by σ 1 σ 2 σ 3 σ 4 σ 1 σ 2 .
We show that the minimal dilatation of pseudo-Anosov homeomorphisms of a closed oriented genus-two surface is equal to the largest root of x 4 − x 3 − x 2 − x + 1, which is approximately 1.72208.
We calculate the volumes of the hyperbolic twist knot cone-manifolds using the Schläfli formula. Even though general ideas for calculating the volumes of cone-manifolds are around, since there is no concrete calculation written, we present here the concrete calculations. We express the length of the singular locus in terms of the distance between the two axes fixed by two generators. In this way the calculation becomes easier than using the singular locus directly. The volumes of the hyperbolic twist knot cone-manifolds simpler than Stevedore's knot are known. As an application, we give the volumes of the cyclic coverings over the hyperbolic twist knots.
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.
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.
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