54 research outputs found
From Weyl-Heisenberg Frames to Infinite Quadratic Forms
Let , be two fixed positive constants. A function is called a \textit{mother Weyl-Heisenberg frame wavelet} for if
generates a frame for under modulates by and
translates by , i.e., is a frame for
. In this paper, we establish a connection between mother
Weyl-Heisenberg frame wavelets of certain special forms and certain strongly
positive definite quadratic forms of infinite dimension. Some examples of
application in matrix algebra are provided
Weyl-Heisenberg Frame Wavelets with Basic Supports
Let , be two fixed non-zero constants. A measurable set is called a Weyl-Heisenberg frame set for if the function
generates a Weyl-Heisenberg frame for under
modulates by and translates by , i.e.,
is a frame for . It is
an open question on how to characterize all frame sets for a given pair
in general. In the case that and , a result due to Casazza and
Kalton shows that the condition that the set
(where
are integers) is a Weyl-Heisenberg frame set for is equivalent to
the condition that the polynomial does not have
any unit roots in the complex plane. In this paper, we show that this result
can be generalized to a class of more general measurable sets (called basic
support sets) and to set theoretical functions and continuous functions defined
on such sets.Comment: 11 pages, 2 figure
The HOMFLY Polynomial of Links in Closed Braid Form
It is well known that any link can be represented by the closure of a braid.
The minimum number of strings needed in a braid whose closure represents a
given link is called the braid index of the link and the well known
Morton-Frank-Williams inequality reveals a close relationship between the
HOMFLY polynomial of a link and its braid index. In the case that a link is
already presented in a closed braid form, Jaeger derived a special formulation
of the HOMFLY polynomial. In this paper, we prove a variant of Jaeger's result
as well as a dual version of it. Unlike Jaeger's original reasoning, which
relies on representation theory, our proof uses only elementary geometric and
combinatorial observations. Using our variant and its dual version, we provide
a direct and elementary proof of the fact that the braid index of a link that
has an -string closed braid diagram that is also reduced and alternating, is
exactly . Until know this fact was only known as a consequence of a result
due to Murasugi on fibered links that are star products of elementary torus
links and of the fact that alternating braids are fibered.Comment: Substantial revisions: abstract changed relation of main theorems to
earlier results better explaine
Gabor Functional Multiplier in the Higher Dimensions
For two given full-rank lattices and
in , where and are
nonsingular real matrices, a function is called a Parseval Gabor frame generator if
holds for any
. It is known that Parseval Gabor frame
generators exist if and only if . A function is called a functional Gabor frame multiplier if it
has the property that is a Parseval Gabor frame generator for
whenever is. It is conjectured that an if and only if
condition for a function to be a functional
Gabor frame multiplier is that must be unimodular and
$h(\bf{x})\overline{h(\bf{x}-(B^T)^{-1}\bf{k})}=h(\bf{x}-A\bf{l})\overline{h(\bf{x}-A\bf{l}-(B^T)^{-1}\bf{k})},\
\forall\ \bf{x}\in \mathbf{R}^d {\em a.e.} for any \bf{l},\bf{k}\in
\mathbb{Z}^d\bf{k}\not=\bf{0}$. The if part of this conjecture is true and
can be proven easily, however the only if part of the conjecture has only been
proven in the one dimensional case to this date. In this paper we prove that
the only if part of the conjecture holds in the two dimensional case
Writhe-like invariants of alternating links
It is known that the writhe calculated from any reduced alternating link
diagram of the same (alternating) link has the same value. That is, it is a
link invariant if we restrict ourselves to reduced alternating link diagrams.
This is due to the fact that reduced alternating link diagrams of the same link
are obtainable from each other via flypes and flypes do not change writhe. In
this paper, we introduce several quantities that are derived from Seifert
graphs of reduced alternating link diagrams. We prove that they are
"writhe-like" invariants in the sense that they are also link invariants among
reduced alternating link diagrams. The determination of these invariants are
elementary and non-recursive so they are easy to calculate. We demonstrate that
many different alternating links can be easily distinguished by these new
invariants, even for large, complicated knots for which other invariants such
as the Jones polynomial are hard to compute. As an application, we also derive
an if and only if condition for a strongly invertible rational link.Comment: 16 pages, 18 figure
Relative Tutte Polynomials for Colored Graphs and Virtual Knot Theory
We introduce the concept of a relative Tutte polynomial of colored graphs. We
show that this relative Tutte polynomial can be computed in a way similar to
the classical spanning tree expansion used by Tutte in his original paper on
this subject. We then apply the relative Tutte polynomial to virtual knot
theory. More specifically, we show that the Kauffman bracket polynomial (hence
the Jones polynomial) of a virtual knot can be computed from the relative Tutte
polynomial of its face (Tait) graph with some suitable variable substitutions.
Our method offers an alternative to the ribbon graph approach, using the face
graph obtained from the virtual link diagram directly.Comment: 22 pages, 6 figure
The Ropelengths of Knots Are Almost Linear in Terms of Their Crossing Numbers
For a knot or link K, let L(K) be the ropelength of K and Cr(K) be the
crossing number of K. In this paper, we show that there exists a constant a>0
such that L(K) is bounded above by a Cr(K) ln^5 (Cr(K)) for any knot K. This
result shows that the upper bound of the ropelength of any knot is almost
linear in terms of its minimum crossing number.Comment: 53 pages, 28 figure
The number of oriented rational links with a given deficiency number
Let be the set of un-oriented and rational links with crossing number
, a precise formula for was obtained by Ernst and Sumners in 1987.
In this paper, we study the enumeration problem of oriented rational links. Let
be the set of oriented rational links with crossing number and
let be the set of oriented rational links with crossing number
() and deficiency . In this paper, we derive precise formulas
for and for any given and and show that
where is the convolved Fibonacci
sequence.Comment: 16 pages, 8 figures, 1 tabl
Nullification of knots and links
In this paper, we study a geometric/topological measure of knots and links
called the nullification number. The nullification of knots/links is believed
to be biologically relevant. For example, in DNA topology, one can intuitively
regard it as a way to measure how easily a knotted circular DNA can unknot
itself through recombination of its DNA strands. It turns out that there are
several different ways to define such a number. These definitions lead to
nullification numbers that are related, but different. Our aim is to explore
the mathematical properties of these nullification numbers. First, we give
specific examples to show that the nullification numbers we defined are
different. We provide detailed analysis of the nullification numbers for the
well known 2-bridge knots and links. We also explore the relationships among
the three nullification numbers, as well as their relationships with other knot
invariants. Finally, we study a special class of links, namely those links
whose general nullification number equals one. We show that such links exist in
abundance. In fact, the number of such links with crossing number less than or
equal to n grows exponentially with respect to n
Invariants of rational links represented by reduced alternating diagrams
A rational link may be represented by any of the (infinitely) many link
diagrams corresponding to various continued fraction expansions of the same
rational number. The continued fraction expansion of the rational number in
which all signs are the same is called a {\em nonalternating form} and the
diagram corresponding to it is a reduced alternating link diagram, which is
minimum in terms of the number of crossings in the diagram. Famous formulas
exist in the literature for the braid index of a rational link by Murasugi and
for its HOMFLY polynomial by Lickorish and Millet, but these rely on a special
continued fraction expansion of the rational number in which all partial
denominators are even (called {\em all-even form}). In this paper we present an
algorithmic way to transform a continued fraction given in nonalternating form
into the all-even form. Using this method we derive formulas for the braid
index and the HOMFLY polynomial of a rational link in terms of its reduced
alternating form, or equivalently the nonalternating form of the corresponding
rational number
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