362,470 research outputs found
Topological steps toward the Homflypt skein module of the lens spaces via braids
In this paper we work toward the Homflypt skein module of the lens spaces
, , using braids. In particular, we establish the
connection between , the Homflypt skein module of the
solid torus ST, and and arrive at an infinite system,
whose solution corresponds to the computation of . We
start from the Lambropoulou invariant for knots and links in ST, the
universal analogue of the Homflypt polynomial in ST, and a new basis,
, of presented in \cite{DL1}. We show that
is obtained from by considering
relations coming from the performance of braid band moves (bbm) on elements in
the basis , where the braid band moves are performed on any moving
strand of each element in . 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 .
The importance of our approach is that it can shed light to the problem of
computing skein modules of arbitrary c.c.o. -manifolds, since any
-manifold can be obtained by surgery on 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
Spectral problem on graphs and L-functions
The scattering process on multiloop infinite p+1-valent graphs (generalized
trees) is studied. These graphs are discrete spaces being quotients of the
uniform tree over free acting discrete subgroups of the projective group
. As the homogeneous spaces, they are, in fact, identical to
p-adic multiloop surfaces. The Ihara-Selberg L-function is associated with the
finite subgraph-the reduced graph containing all loops of the generalized tree.
We study the spectral problem on these graphs, for which we introduce the
notion of spherical functions-eigenfunctions of a discrete Laplace operator
acting on the graph. We define the S-matrix and prove its unitarity. We present
a proof of the Hashimoto-Bass theorem expressing L-function of any finite
(reduced) graph via determinant of a local operator acting on this
graph and relate the S-matrix determinant to this L-function thus obtaining the
analogue of the Selberg trace formula. The discrete spectrum points are also
determined and classified by the L-function. Numerous examples of L-function
calculations are presented.Comment: 39 pages, LaTeX, to appear in Russ. Math. Sur
Canonical rings of Q-divisors on P^1
The canonical ring of
a divisor D on a curve X is a natural object of study; when D is a Q-divisor,
it has connections to projective embeddings of stacky curves and rings of
modular forms. We study the generators and relations of S_D for the simplest
curve X = P^1. When D contains at most two points, we give a complete
description of S_D; for general D, we give bounds on the generators and
relations. We also show that the generators (for at most five points) and a
Groebner basis of relations between them (for at most four points) depend only
on the coefficients in the divisor D, not its points or the characteristic of
the ground field; we conjecture that the minimal system of relations varies in
a similar way. Although stated in terms of algebraic geometry, our results are
proved by translating to the combinatorics of lattice points in simplices and
cones.Comment: 19 pages, 3 figure
An integrability result for -vectorfields in the plane
We prove that if then the divergence of a -vectorfield on a
2-dimensional domain is the boundary of an integral 1-current, if and
only if can be represented as the rotated gradient for a
-map . Such result extends to exponents the
result on distributional Jacobians of Alberti, Baldo, Orlandi.Comment: 16 pages, some typing errors fixe
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