89,088 research outputs found
Knot Floer homology in cyclic branched covers
In this paper, we introduce a sequence of invariants of a knot K in S^3: the
knot Floer homology groups of the preimage of K in the m-fold cyclic branched
cover over K. We exhibit the knot Floer homology in the m-fold branched cover
as the categorification of a multiple of the Turaev torsion in the case where
the m-fold branched cover is a rational homology sphere. In addition, when K is
a 2-bridge knot, we prove that the knot Floer homology of the lifted knot in a
particular Spin^c structure in the branched double cover matches the knot Floer
homology of the original knot K in S^3. We conclude with a calculation
involving two knots with identical knot Floer homology in S^3 for which the
knot Floer homology groups in the double branched cover differ as Z_2-graded
groups.Comment: This is the version published by Algebraic & Geometric Topology on 25
September 200
The distribution of points on superelliptic curves over finite fields
We give the distribution of points on smooth superelliptic curves over a
fixed finite field, as their degree goes to infinity. We also give the
distribution of points on smooth m-fold cyclic covers of the line, for any m,
as the degree of their superelliptic model goes to infinity. This builds on
previous work of Kurlberg, Rudnick, Bucur, David, Feigon, and Lalin for p-fold
cyclic covers, but the limits taken differ slightly and the resulting
distributions are interestingly different
Hodge theory of cyclic covers branched over a union of hyperplanes
Suppose that Y is a cyclic cover of projective space branched over a
hyperplane arrangement D, and that U is the complement of the ramification
locus in Y. The first theorem implies that the Beilinson-Hodge conjecture holds
for U if certain multiplicities of D are coprime to the degree of the cover.
For instance this applies when D is reduced with normal crossings. The second
theorem shows that when D has normal crossings and the degree of the cover is a
prime number, the generalized Hodge conjecture holds for any toroidal
resolution of Y. The last section contains some partial extensions to more
general nonabelian covers.Comment: 18 pages; final revision; to appear in Can. J. Mat
Deriving Good LDPC Convolutional Codes from LDPC Block Codes
Low-density parity-check (LDPC) convolutional codes are capable of achieving
excellent performance with low encoding and decoding complexity. In this paper
we discuss several graph-cover-based methods for deriving families of
time-invariant and time-varying LDPC convolutional codes from LDPC block codes
and show how earlier proposed LDPC convolutional code constructions can be
presented within this framework. Some of the constructed convolutional codes
significantly outperform the underlying LDPC block codes. We investigate some
possible reasons for this "convolutional gain," and we also discuss the ---
mostly moderate --- decoder cost increase that is incurred by going from LDPC
block to LDPC convolutional codes.Comment: Submitted to IEEE Transactions on Information Theory, April 2010;
revised August 2010, revised November 2010 (essentially final version).
(Besides many small changes, the first and second revised versions contain
corrected entries in Tables I and II.
The Lattice of Cyclic Flats of a Matroid
A flat of a matroid is cyclic if it is a union of circuits. The cyclic flats
of a matroid form a lattice under inclusion. We study these lattices and
explore matroids from the perspective of cyclic flats. In particular, we show
that every lattice is isomorphic to the lattice of cyclic flats of a matroid.
We give a necessary and sufficient condition for a lattice Z of sets and a
function r on Z to be the lattice of cyclic flats of a matroid and the
restriction of the corresponding rank function to Z. We define cyclic width and
show that this concept gives rise to minor-closed, dual-closed classes of
matroids, two of which contain only transversal matroids.Comment: 15 pages, 1 figure. The new version addresses earlier work by Julie
Sims that the authors learned of after submitting the first versio
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