129,022 research outputs found
Hasse-Weil zeta functions of SL_2-character varieties of arithmetic two bridge link complements
Hasse-Weil zeta functions of SL_2-character varieties of arithmetic two
bridge link groups are determined.
Special values of the zeta functions at s=0,1,2 are also investigated.Comment: 13 pages, v2:Theorem and Table 1 have been fixed, version accepted
for publication in Finite Fields and Their Application
Link homology and equivariant gauge theory
The singular instanton Floer homology was defined by Kronheimer and Mrowka in
connection with their proof that the Khovanov homology is an unknot detector.
We study this theory for knots and two-component links using equivariant gauge
theory on their double branched covers. We show that the special generator in
the singular instanton Floer homology of a knot is graded by the knot signature
mod 4, thereby providing a purely topological way of fixing the absolute
grading in the theory. Our approach also results in explicit computations of
the generators and gradings of the singular instanton Floer chain complex for
several classes of knots with simple double branched covers, such as two-bridge
knots, torus knots, and Montesinos knots, as well as for several families of
two-components links.Comment: 59 pages. Corrected a grading error in Lemma 2.5, which affected
calculations for some of the knot
Twisted Alexander polynomials of hyperbolic links
In this paper we apply the twisted Alexander polynomial to study the fibering
and genus detecting problems for oriented links. In particular we generalize a
conjecture of Dunfield, Friedl and Jackson on the torsion polynomial of
hyperbolic knots to hyperbolic links, and confirm it for an infinite family of
hyperbolic 2-bridge links. Moreover we consider a similar problem for parabolic
representations of 2-bridge link groups.Comment: 19 pages, 3 figures. Title changed. Section 3 (Finiteness theorems
for alternating links) of the old version is removed, since there is a gap in
the proof of Theorem 3.3. A new section on parabolic representations (Section
5) is added in the new versio
Simple loops on 2-bridge spheres in 2-bridge link complements
The purpose of this note is to announce complete answers to the following
questions. (1) For an essential simple loop on a 2-bridge sphere in a 2-bridge
link complement, when is it null-homotopic in the link complement? (2) For two
distinct essential simple loops on a 2-bridge sphere in a 2-bridge link
complement, when are they homotopic in the link complement? We also announce
applications of these results to character varieties and McShane's identity.Comment: 19 pages, 6 figures; Theorem 2.6 revised; to appear in Electron. Res.
Announc. Math. Sc
Trace-free -representations of arborescent links
Given a link , a representation is {\it trace-free} if it sends each meridian to an element
with trace zero. We present a method for completely determining trace-free
-representations for arborescent links. Concrete
computations are done for a class of 3-bridge arborescent links.Comment: 19 pages, 8 figures; to appear on Periodica Mathematica Hungaric
Computing twisted Alexander polynomials for Montesinos links
In recent years, twisted Alexander polynomial has been playing an important
role in low-dimensional topology.
For Montesinos links, we develop an efficient method to compute the twisted
Alexander polynomial associated to any linear representation.
In particular, formulas for multi-variable Alexander polynomials of these
links can be easily derived.Comment: 19 pages, 13 figures, many errors are corrected and several examples
are adde
Equivalence of symmetric union diagrams
Motivated by the study of ribbon knots we explore symmetric unions, a
beautiful construction introduced by Kinoshita and Terasaka 50 years ago. It is
easy to see that every symmetric union represents a ribbon knot, but the
converse is still an open problem. Besides existence it is natural to consider
the question of uniqueness. In order to attack this question we extend the
usual Reidemeister moves to a family of moves respecting the symmetry, and
consider the symmetric equivalence thus generated. This notion being in place,
we discuss several situations in which a knot can have essentially distinct
symmetric union representations. We exhibit an infinite family of ribbon
two-bridge knots each of which allows two different symmetric union
representations.Comment: 19 pages, 20 figures; v2 corrected signs in section
Counting genus one fibered knots in lens spaces
The braid axis of a closed 3-braid lifts to a genus one fibered knot in the
double cover of S^3 branched over the closed braid. Every (null homologous)
genus one fibered knot in a 3-manifold may be obtained in this way. Using this
perspective we answer a question of Morimoto about the number of genus one
fibered knots in lens spaces. We determine the number of genus one fibered
knots up to homeomorphism in any given lens space. This number is 3 in the case
of the lens space L(4,1), 2 for the lens spaces L(m,1) with m>0, and at most 1
otherwise.Comment: 10 pages, 4 figures. V4: Fixed cosmetic errors, slight modification
to proof of Thm 2.
Fast Gradient Attack on Network Embedding
Network embedding maps a network into a low-dimensional Euclidean space, and
thus facilitate many network analysis tasks, such as node classification, link
prediction and community detection etc, by utilizing machine learning methods.
In social networks, we may pay special attention to user privacy, and would
like to prevent some target nodes from being identified by such network
analysis methods in certain cases. Inspired by successful adversarial attack on
deep learning models, we propose a framework to generate adversarial networks
based on the gradient information in Graph Convolutional Network (GCN). In
particular, we extract the gradient of pairwise nodes based on the adversarial
network, and select the pair of nodes with maximum absolute gradient to realize
the Fast Gradient Attack (FGA) and update the adversarial network. This process
is implemented iteratively and terminated until certain condition is satisfied,
i.e., the number of modified links reaches certain predefined value.
Comprehensive attacks, including unlimited attack, direct attack and indirect
attack, are performed on six well-known network embedding methods. The
experiments on real-world networks suggest that our proposed FGA behaves better
than some baseline methods, i.e., the network embedding can be easily disturbed
using FGA by only rewiring few links, achieving state-of-the-art attack
performance
On the volume and the Chern-Simons invariant for the -bridge knot orbifolds
We extend some part of the unpublished paper written by Mednykh and
Rasskazov. Using the approach indicated in this paper we derive the
Riley-Mednykh polynomial for some family of the -bridge knot orbifolds. As a
result we obtain explicit formulae for the volume of cone-manifolds and the
Chern-Simons invariant of orbifolds of the knot with Conway's notation
.Comment: 20 pages, 8 figures, 5 tables. arXiv admin note: text overlap with
arXiv:1601.00723, arXiv:1512.0548
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