Conway polynomials of two-bridge links

We give necessary conditions for a polynomial to be the Conway polynomial of a two-bridge link. As a consequence, we obtain simple proofs of the classical theorems of Murasugi and Hartley. We give a modulo 2 congruence for links, which implies the classical modulo 2 Murasugi congruence for knots. We also give sharp bounds for the coefficients of the Conway and Alexander polynomials of a two-bridge link. These bounds improve and generalize those of Nakanishi and Suketa.Comment: 15

Representations of fundamental groups of 3-manifolds into PGL(3,C): Exact computations in low complexity

In this paper we are interested in computing representations of the fundamental group of a 3-manifold into PSL(3;C) (in particular in PSL(2;C); PSL(3;R) and PU(2; 1)). The representations are obtained by gluing decorated tetrahedra of flags. We list complete computations (giving 0-dimensional or 1-dimensional solution sets) for the first complete hyperbolic non-compact manifolds with finite volume which are obtained gluing less than three tetrahedra with a description of the computer methods used to find them

The Complete Characterization of Fourth-Order Symplectic Integrators with Extended-Linear Coefficients

The structure of symplectic integrators up to fourth-order can be completely and analytical understood when the factorization (split) coefficents are related linearly but with a uniform nonlinear proportional factor. The analytic form of these {\it extended-linear} symplectic integrators greatly simplified proofs of their general properties and allowed easy construction of both forward and non-forward fourth-order algorithms with arbitrary number of operators. Most fourth-order forward integrators can now be derived analytically from this extended-linear formulation without the use of symbolic algebra.Comment: 12 pages, 2 figures, submitted to Phys. Rev. E, corrected typo

Quantum Statistical Calculations and Symplectic Corrector Algorithms

The quantum partition function at finite temperature requires computing the trace of the imaginary time propagator. For numerical and Monte Carlo calculations, the propagator is usually split into its kinetic and potential parts. A higher order splitting will result in a higher order convergent algorithm. At imaginary time, the kinetic energy propagator is usually the diffusion Greens function. Since diffusion cannot be simulated backward in time, the splitting must maintain the positivity of all intermediate time steps. However, since the trace is invariant under similarity transformations of the propagator, one can use this freedom to "correct" the split propagator to higher order. This use of similarity transforms classically give rises to symplectic corrector algorithms. The split propagator is the symplectic kernel and the similarity transformation is the corrector. This work proves a generalization of the Sheng-Suzuki theorem: no positive time step propagators with only kinetic and potential operators can be corrected beyond second order. Second order forward propagators can have fourth order traces only with the inclusion of an additional commutator. We give detailed derivations of four forward correctable second order propagators and their minimal correctors.Comment: 9 pages, no figure, corrected typos, mostly missing right bracket

Comparison between Deprit and Dragt-Finn Perturbation Methods

In this paper, the relationship between the Dragt-Finn transform and the classical Lie transform introduced by Deprit is discussed. The relative performance of the algorithms used for the computations of the transformed functions is compared, and the relation between their generators is given. These generators produce the same transform which insures the construction of the same invariants

Relations among Lie-Series Transformations and Isomorphisms between Free Lie Algebras

We study the subgroup generated by the exponentials of formal Lie series. We show three different ways to represent elements of this subgroup. These elements induce Lie-series trans- formations. Relations among these family of transformations furnish algorithms of composition. Starting from the Lazard elimination theorem and the Witt's formula, we show isomorphisms between some submodules of free Lie algebras. Combining different results, we also show that the homogeneous terms of the Hausdorff series H(a,b) freely generate the free Lie algebra L(a,b) without a line

Exhaustive Search of Symplectic Integrators using Computer Algebra

. We find symplectic integrators using universal exponential identities or relations among formal Lie series. We give here general methods to compute such identities in a free Lie algebra. We recover by these methods all the previously known symplectic integrators and some new ones. We list all minimal solutions for integrators of low order. We give some improvement in the case when the Hamiltonian is in form T (p) + V (q). We give also all reversible fourth-order symplectic integrators for the planetary hamiltonian expressed in canonical heliocentric coordinates. 1 Introduction For very long time integration, there has been recently a development of numerical methods preserving the symplectic structure (see for example [7, 18, 19, 20]), which seem to be more efficient with respect to the computational cost. Symplectic integrators may be seen as the time evolution mapping of a slightly perturbed Hamiltonian, that is to say as a Lie transformation that can be represented either by an ..

About approximations of exponentials

Abstract. We look for the approximation of exp(A1 +A2) by a product in form exp(x1A1)exp(y1A2)···exp(xnA1)exp(ynA2). We specially are interested in minimal approximations, with respect to the number of terms. After having shown some isomorphisms between specific free Lie subalgebras, we will prove the equivalence of the search of such approximations and approximations of exp(A1 +···+An). The main result is based on the fact that the Lie subalgebra spanned by the homogeneous components of the Hausdorff series is free.

Relations among Lie Series Transformations and Isomorphisms between free Lie Algebras

. We study the subgroup generated by the exponentials of formal Lie series. We show three different way to represent elements of this subgroup. These elements induce Lie series transformations. Relations among these family of transformations furnish algorithms of composition. Starting from the Lazard elimination theorem and the Witt's formula, we show isomorphisms between some submodules of free Lie algebras. Combining different results, we also show that the homogeneous terms of the Hausdorff series H(a; b) freely generate the free Lie algebra L(a; b) without a line. 1 Introduction Lie series automorphisms or Lie transformations play an important role in classical mechanics. They can be seen, for example, as the time evolution in an hamiltonian system. The product of two such transformations may therefore be seen as the combined effects of two Hamiltonians. The use of this formalism becomes efficient when it becomes easy to manipulate formal Lie series, to compute composition of Lie..