An alternative basis for the Kauffman bracket skein module of the Solid Torus via braids

In this paper we give an alternative basis, $\mathcal{B}_{\rm ST}$, for the Kauffman bracket skein module of the solid torus, ${\rm KBSM}\left({\rm ST}\right)$. The basis $\mathcal{B}_{\rm ST}$ is obtained with the use of the Tempereley--Lieb algebra of type B and it is appropriate for computing the Kauffman bracket skein module of the lens spaces $L(p, q)$ via braids.Comment: 14 pages, 5 figure

Flat histogram diagrammatic Monte Carlo method

The diagrammatic Monte Carlo (Diag-MC) method is a numerical technique which samples the entire diagrammatic series of the Green's function in quantum many-body systems. In this work, we incorporate the flat histogram principle in the diagrammatic Monte method and we term the improved version "Flat Histogram Diagrammatic Monte Carlo" method. We demonstrate the superiority of the method over the standard Diag-MC in extracting the long-imaginary-time behavior of the Green's function, without incorporating any a priori knowledge about this function, by applying the technique to the polaron problemComment: 7 two-column pages 4 eps figure

Topological steps toward the Homflypt skein module of the lens spaces L(p,1)L(p,1) via braids

In this paper we work toward the Homflypt skein module of the lens spaces $L(p,1)$, $\mathcal{S}(L(p,1))$, using braids. In particular, we establish the connection between $\mathcal{S}({\rm ST})$, the Homflypt skein module of the solid torus ST, and $\mathcal{S}(L(p,1))$ and arrive at an infinite system, whose solution corresponds to the computation of $\mathcal{S}(L(p,1))$. We start from the Lambropoulou invariant $X$ for knots and links in ST, the universal analogue of the Homflypt polynomial in ST, and a new basis, $\Lambda$, of $\mathcal{S}({\rm ST})$ presented in \cite{DL1}. We show that $\mathcal{S}(L(p,1))$ is obtained from $\mathcal{S}({\rm ST})$ by considering relations coming from the performance of braid band moves (bbm) on elements in the basis $\Lambda$, where the braid band moves are performed on any moving strand of each element in $\Lambda$. We do that by proving that the system of equations obtained from diagrams in ST by performing bbm on any moving strand is equivalent to the system obtained if we only consider elements in the basic set $\Lambda$. The importance of our approach is that it can shed light to the problem of computing skein modules of arbitrary c.c.o. $3$-manifolds, since any $3$-manifold can be obtained by surgery on $S^3$ along unknotted closed curves. The main difficulty of the problem lies in selecting from the infinitum of band moves some basic ones and solving the infinite system of equations.Comment: 24 pages, 16 figures. arXiv admin note: text overlap with arXiv:1412.364

Holomorphic automorphic forms and cohomology

We investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least 2 this correspondence is given by the Eichler integral. Knopp generalized this to real weights. We show that for weights that are not an integer at least 2 the generalized Eichler integral gives an injection into the first cohomology group with values in a module of holomorphic functions, and characterize the image. We impose no condition on the growth of the automorphic forms at the cusps. For real weights that are not an integer at least 2 we similarly characterize the space of cusp forms and the space of entire automorphic forms. We give a relation between the cohomology classes attached to holomorphic automorphic forms of real weight and the existence of harmonic lifts. A tool in establishing these results is the relation to cohomology groups with values in modules of "analytic boundary germs", which are represented by harmonic functions on subsets of the upper half-plane. Even for positive integral weights cohomology with these coefficients can distinguish all holomorphic automorphic forms, unlike the classical Eichler theory.Comment: 150 pages. An earlier version appeared as an Oberwolfach Preprint (OWP 2014-07

The Monetary Approach in the Presence of I(2) Components: A Cointegration Analysis of the Official and Black Market for Foreign Currency in Latin America

This paper re-examines the long-run properties of the monetary exchange rate model in the presence of a parallel or black market for U.S. dollars in two Latin American countries under the twin hypotheses that the system contains variables that are I(2) and that a linear trend is required in the cointegrating relations. Using the recent I(2) test by Rahbek et al. (1999) to examine the presence of I(2) and I(1) components in a multivariate context we find that the linear trend hypothesis could not be rejected and we find evidence that the system contains two I(2) variables for each country namely, Chile and Mexico, and this finding is reconfirmed by the estimated roots of the companion matrix (Juselius, 1995). The I(2) component led to the transformation of the estimated model by imposing long-run but not short-run proportionality between domestic and foreign money. Three statistically significant cointegrating vectors were found and, by imposing linear restrictions on each vector as suggested by Johansen and Juselius (1994) and Johansen (1995b), the order and rank conditions for identification are satisfied while the test for overidentifying restrictions was significant for either case. The main findings suggest that we reject the forward-looking version of the monetary model for each country, but the unrestricted monetary model is still a valid framework to explain the long-run movements of the parallel exchange rate in both countries. Furthermore, we test for parameter stability using the tests developed by Hansen and Johansen (1993) and it is shown that the dimension of the cointegration rank is sample dependent while the estimated coeffficients do not exhibit instabilities in recursive estimations.I(2) cointegration, monetary model, parallel foreign exchange market, identification, temporal stabi