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1,400 |
Möbius invariance of knot energy
|
math.GT
|
A physically natural potential energy for simple closed curves in $\bold R^3$
is shown to be invariant under M\"obius transformations. This leads to the
rapid resolution of several open problems: round circles are precisely the
absolute minima for energy; there is a minimum energy threshold below which
knotting cannot occur; minimizers within prime knot types exist and are
regular. Finally, the number of knot types with energy less than any constant
$M$ is estimated.
|
math
|
1,401 |
New points of view in knot theory
|
math.GT
|
In this article we shall give an account of certain developments in knot
theory which followed upon the discovery of the Jones polynomial in 1984. The
focus of our account will be recent glimmerings of understanding of the
topological meaning of the new invariants. A second theme will be the central
role that braid theory has played in the subject. A third will be the unifying
principles provided by representations of simple Lie algebras and their
universal enveloping algebras. These choices in emphasis are our own. They
represent, at best, particular aspects of the far-reaching ramifications that
followed the discovery of the Jones polynomial.
|
math
|
1,402 |
Topology of homology manifolds
|
math.GT
|
We construct examples of nonresolvable generalized $n$-manifolds, $n\geq 6$,
with arbitrary resolution obstruction, homotopy equivalent to any simply
connected, closed $n$-manifold. We further investigate the structure of
generalized manifolds and present a program for understanding their topology.
|
math
|
1,403 |
Quasipositivity as an obstruction to sliceness
|
math.GT
|
For an oriented link $L \subset S^3 = \Bd\!D^4$, let $\chi_s(L)$ be the
greatest Euler characteristic $\chi(F)$ of an oriented 2-manifold $F$ (without
closed components) smoothly embedded in $D^4$ with boundary $L$. A knot $K$ is
{\it slice} if $\chi_s(K)=1$. Realize $D^4$ in $\C^2$ as
$\{(z,w):|z|^2+|w|^2\le1\}$. It has been conjectured that, if $V$ is a
nonsingular complex plane curve transverse to $S^3$, then $\chi_s(V\cap
S^3)=\chi(V\cap D^4)$. Kronheimer and Mrowka have proved this conjecture in the
case that $V\cap D^4$ is the Milnor fiber of a singularity. I explain how this
seemingly special case implies both the general case and the ``slice-Bennequin
inequality'' for braids. As applications, I show that various knots are not
slice (e.g., pretzel knots like $\Pscr(-3,5,7)$; all knots obtained from a
positive trefoil $O\{2,3\}$ by iterated untwisted positive doubling). As a
sidelight, I give an optimal counterexample to the ``topologically locally-flat
Thom conjecture''.
|
math
|
1,404 |
Alexander's and Markov's theorems in dimension four
|
math.GT
|
Alexander's and Markov's theorems state that any link type in $R^3$ is
represented by a closed braid and that such representations are related by some
elementary operations called Markov moves. We generalize the notion of a braid
to that in 4-dimensional space and establish an analogue of these theorems.
|
math
|
1,405 |
Extremal length estimates and product regions in Teichmüller space
|
math.GT
|
We study the Teichm\"uller metric on the Teichm\"uller space of a surface of
finite type, in regions where the injectivity radius of the surface is small.
The main result is that in such regions the Teichm\"uller metric is
approximated up to bounded additive distortion by the sup metric on a product
of lower dimensional spaces. The main technical tool in the proof is the use of
estimates of extremal lengths of curves in a surface based on the geometry of
their hyperbolic geodesic representatives.
|
math
|
1,406 |
A User's Guide to the Mapping Class Group: Once Punctured Surfaces
|
math.GT
|
This document is a practical guide to computations using an automatic
structure for the mapping class group of a once-punctured, oriented surface
$S$. We describe a quadratic time algorithm for the word problem in this group,
which can be implemented efficiently with pencil and paper. The input of the
algorithm is a word, consisting of ``chord diagrams'' of ideal triangulations
and elementary moves, which represents an element of the mapping class group.
The output is a word called a ``normal form'' that uniquely represents the same
group element.
|
math
|
1,407 |
Exceptional surgery on knots
|
math.GT
|
Let $M$ be an irreducible, compact, connected, orientable 3-manifold whose
boundary is a torus. We show that if $M$ is hyperbolic, then it admits at most
six finite/cyclic fillings of maximal distance 5. Further, the distance of a
finite/cyclic filling to a cyclic filling is at most 2. If $M$ has a
non-boundary-parallel, incompressible torus and is not a generalized 1-iterated
torus knot complement, then there are at most three finite/cyclic fillings of
maximal distance 1. Further, if $M$ has a non-boundary-parallel, incompressible
torus and is not a generalized 1- or 2-iterated torus knot complement and if
$M$ admits a cyclic filling of odd order, then $M$ does not admit any other
finite/cyclic filling. Relations between finite/cyclic fillings and other
exceptional fillings are also discussed.
|
math
|
1,408 |
On the geometric and topological rigidity of hyperbolic 3-manifolds
|
math.GT
|
A homotopy equivalence between a hyperbolic 3-manifold and a closed
irreducible 3-manifold is homotopic to a homeomorphsim provided the hyperbolic
manifold satisfies a purely geometric condition. There are no known examples of
hyperbolic 3-manifolds which do not satisfy this condition.
|
math
|
1,409 |
The Structure and Enumeration of Link Projections
|
math.GT
|
We define a decomposition of link projections whose pieces we call atoroidal
graphs. We describe a surgery operation on these graphs and show that all
atoroidal graphs can be generated by performing surgery repeatedly on a family
of well known link projections. This gives a method of enumerating atoroidal
graphs and hence link projections by recomposing the pieces of the
decomposition.
|
math
|
1,410 |
Bounds on Volume Increase under Dehn Drilling Operations
|
math.GT
|
In this paper we investigate how the volume of hyperbolic manifolds increases
under the process of removing a curve, that is, Dehn drilling. If the curve we
remove is a geodesic we are able to show that for a certain family of manifolds
the volume increase is bounded above by $\pi \cdot l$ where $l$ is the length
of the geodesic drilled. Also we construct examples to show that there is no
lower bound to the volume increase in terms of a linear function of a positive
power of length and in particular volume increase is not bounded linearly in
length.
|
math
|
1,411 |
Floer homologies for Lagrangian intersections and instantons
|
math.GT
|
In 1985 lectures at MSRI, A. Casson introduced an interesting integer valued
invariant for any oriented integral homology 3-sphere Y via beautiful
constructions on representation spaces (see [1] for an exposition). The Casson
invariant \lambda(Y) is roughly defined by measuring the oriented number of
irreducible representations of the fundamental group \pi_1(Y) in SU(2). Such an
invariant generalized the Rohlin invariant and gives surprising corollaries in
low dimensional topology.
|
math
|
1,412 |
Quasigeodesic Flows in Hyperbolic Three-Manifolds
|
math.GT
|
Any closed, oriented, hyperbolic three-manifold with nontrivial second
homology has many quasigeodesic flows, where quasigeodesic means that flow
lines are uniformly efficient in measuring distance in relative homotopy
classes. The flows are pseudo-Anosov flows which are almost transverse to
finite depth foliations in the manifold. The main tool is the use of a sutured
manifold hierarchy which has good geometric properties.
|
math
|
1,413 |
End sums of irreducible open 3-manifolds
|
math.GT
|
An end sum is a non-compact analogue of a connected sum. Suppose we are given
two connected, oriented $n$-manifolds $M_1$ and $M_2$. Recall that to form
their connected sum one chooses an $n$-ball in each $M_i$, removes its
interior, and then glues together the two $S^{n-1}$ boundary components thus
created by an orientation reversing homeomorphism. Now suppose that $M_1$ and
$M_2$ are also open, i.e. non-compact with empty boundary. To form an end sum
of $M_1$ and $M_2$ one chooses a halfspace $H_i$ (a manifold \homeo\ to ${\bold
R}^{n-1} \times [0, \infty)$) embedded in $M_i$, removes its interior, and then
glues together the two resulting ${\bold R}^{n-1}$ boundary components by an
orientation reversing homeomorphism. In order for this space $M$ to be an
$n$-manifold one requires that each $H_i$ be {\bf end-proper} in $M_i$ in the
sense that its intersection with each compact subset of $M_i$ is compact. Note
that one can regard $H_i$ as a regular neighborhood of an end-proper ray (a
1-manifold \homeo\ to $[0,\infty)$) $\ga_i$ in $M_i$.
|
math
|
1,414 |
Attaching boundary planes to irreducible open 3-manifolds
|
math.GT
|
Given any connected, open 3-manifold $U$ having finitely many ends, a
non-compact 3-manifold $M$ is constructed having the following properties: the
interior of $M$ is homeomorphic to $U$; the boundary of $M$ is the disjoint
union of finitely many planes; $M$ is not almost compact; $M$ is eventually
end-irreducible; there are no proper, incompressible embeddings of $S^1 \times
\bold R$ in $M$; every compact subset of $M$ is contained in a larger compact
subset whose complement is anannular; there is a compact subset of $M$ whose
complement is $\bold P^2$-irreducible.
If $U$ is irreducible it also has the following two properties: every proper,
non-trivial plane in $M$ is boundary-parallel; every proper surface in $M$ each
component of which has non-empty boundary and is non-compact and simply
connected lies in a collar on $\partial M$.
This construction can be chosen so that $M$ admits no homeomorphisms which
take one boundary plane to another or reverse orientation. For the given $U$
there are uncountably many non-homeomorphic such $M$.
Two auxiliary results may be of independent interest. First, general
conditions are given under which infinitely many ``trivial'' compact components
of the intersection of two proper, non-compact surfaces in an irreducible
3-manifold can be removed by an ambient isotopy. Second, $n$ component tangles
in a 3-ball are constructed such that every non-empty union of components of
the tangle has hyperbolic exterior.
|
math
|
1,415 |
Contractible open 3-manifolds which non-trivially cover only non-compact 3-manifolds
|
math.GT
|
Suppose $M$ is a closed, connected, orientable, \irr\ \3m\ such that
$G=\pi_1(M)$ is infinite. One consequence of Thurston's geometrization
conjecture is that the universal covering space $\widetilde{M}$ of $M$ must be
\homeo\ to $\RRR$. This has been verified directly under several different
additional assumptions on $G$. (See, for example, \cite{2}, \cite{3}, \cite{6},
\cite{19}.)
|
math
|
1,416 |
On the homology cobordism group of homology 3-spheres
|
math.GT
|
In this paper we present our results on the homology cobordism group $\Th$ of
the oriented integral homology 3-spheres. We specially emphasize the role
played in the subject by the gauge theory including Floer homology and
invariants by Donaldson and Seiberg -- Witten.
|
math
|
1,417 |
Homotopy Hyperbolic 3-Manifolds are Hyperbolic
|
math.GT
|
This paper introduces a rigorous computer-assisted procedure for analyzing
hyperbolic 3-manifolds. This technique is used to complete the proof of several
long-standing rigidity conjectures in 3-manifold theory as well as to provide a
new lower bound for the volume of a closed orientable hyperbolic 3-manifold.
We prove the following result:
\it\noindent Let $N$ be a closed hyperbolic 3-manifold. Then
\begin{enumerate} \item[(1)] If $f\colon M \to N$ is a homotopy equivalence
where $M$ is a closed irreducible 3-manifold, then $f$ is homotopic to a
homeomorphism. \item[(2)] If $f,g\colon M\to N$ are homotopic homeomorphisms,
then $f$ is isotopic to $g$. \item[(3)] The space of hyperbolic metrics on $N$
is path connected. \end{enumerate}
|
math
|
1,418 |
A Standard Form for Incompressible Surfaces in a Handlebody
|
math.GT
|
Let $\F$ be a compact surface and let $I$ be the unit interval. This paper
gives a standard form for all 2-sided incompressible surfaces in the 3-manifold
$\F \times I$. Since $\F \times I$ is a handlebody when $\F$ has boundary, this
standard form applies to incompressible surfaces in a handlebody.
|
math
|
1,419 |
Cannon-Thurston Maps for Trees of Hyperbolic Metric Spaces
|
math.GT
|
Let (X,d) be a tree (T) of hyperbolic metric spaces satisfying the
quasi-isometrically embedded condition. Let $v$ be a vertex of $T$. Let
$({X_v},d_v)$ denote the hyperbolic metric space corresponding to $v$. Then $i
: X_v \rightarrow X$ extends continuously to a map $\hat{i} : \widehat{X_v}
\rightarrow \widehat{X}$. This generalizes a Theorem of Cannon and Thurston.
The techniques are used to give a new proof of a result of Minsky: Thurston's
ending lamination conjecture for certain Kleinian groups. Applications to
graphs of hyperbolic groups and local connectivity of limit sets of Kleinian
groups are also given.
|
math
|
1,420 |
Homeomorphisms of 3-manifolds and the realization of Nielsen Number
|
math.GT
|
The Nielsen Conjecture for Homeomorphisms asserts that any homeomorphism $f$
of a closed manifold is isotopic to a map realizing the Nielsen number of $f$,
which is a lower bound for the number of fixed points among all maps homotopic
to $f$. The main theorem of this paper proves this conjecture for all
orientation preserving maps on geometric or Haken 3-manifolds. It will also be
shown that on many manifolds all maps are isotopic to fixed point free maps.
The proof is based on the understanding of homeomorphisms on 2-orbifolds and
3-manifolds. Thurston's classification of surface homeomorphisms will be
generalized to 2-dimensional orbifolds, which is used to study fiber preserving
maps of Seifert fiber spaces. Maps on most Seifert fiber spaces are indeed
isotopic to fiber preserving maps, with the exception of four manifolds and
orientation reversing maps on lens spaces or $S^3$. It will also be determined
exactly which manifolds have a unique Seifert fibration up to isotopy. These
informations will be used to deform a map to certain standard map on each piece
of the JSJ decomposition, as well as on the neighborhood of the decomposition
tori, which will make it possible to shrink each fixed point class to a single
point, and remove inessential fixed point classes.
|
math
|
1,421 |
Contractible open 3-manifolds with free covering translation groups
|
math.GT
|
This paper concerns the class of contractible open 3-manifolds which are
``locally finite strong end sums'' of eventually end-irreducible Whitehead
manifolds. It is shown that whenever a 3-manifold in this class is a covering
space of another 3-manifold the group of covering translations must be a free
group. It follows that such a 3-manifold cannot cover a closed 3-manifold. For
each countable free group a specific uncountable family of irreducible open
3-manifolds is constructed whose fundamental groups are isomorphic to the given
group and whose universal covering spaces are in this class and are pairwise
non-homeomorphic.
|
math
|
1,422 |
R^2-irreducible universal covering spaces of P^2-irreducible open 3-manifolds
|
math.GT
|
An irreducible open 3-manifold $W$ is {\bf R}$^2$-irreducible if every proper
plane in $W$ splits off a halfspace. In this paper it is shown that if such a
$W$ is the universal cover of a connected, {\bf P}$^2$-irreducible open
3-manifold $M$ with finitely generated fundamental group, then either $W$ is
homeomorphic to {\bf R}$^3$ or the group is a free product of infinite cyclic
groups and infinite closed surface groups. Given any such finitely generated
group uncountably many $M$ are constructed with that fundamental group such
that their universal covers are {\bf R}$^2$-irreducible, are not homeomorphic
to {\bf R}$^3$, and are pairwise non-homeomorphic. These results are related to
the conjecture that closed, orientable, irreducible, aspherical 3-manifolds are
covered by {\bf R}$^3$.
|
math
|
1,423 |
Maximal Nilpotent Quotients of 3-Manifold Groups
|
math.GT
|
We show that if the lower central series of the fundamental group of a closed
oriented $3$-manifold stabilizes then the maximal nilpotent quotient is a
cyclic group, a quaternion $2$-group cross an odd order cyclic group, or a
Heisenberg group. These groups are well known to be precisely the nilpotent
fundamental groups of closed oriented $3$-manifolds.
|
math
|
1,424 |
Floer homology of Brieskorn homology spheres: solution to Atiyah's problem
|
math.GT
|
In this paper we answer the question posed by M.~Atiyah and give an explicit
formula for Floer homology of Brieskorn homology spheres in terms of their
branching sets over the 3--sphere. We further show how Floer homology is
related to other invariants of knots and 3--manifolds, among which are the
$\bar\mu$--invariant of W.~Neumann and L.~Siebenmann and the Jones polynomial.
Essential progress is made in proving the homology cobordism invariance of our
own $\nu$--invariant.
|
math
|
1,425 |
On two-generator satellite knots
|
math.GT
|
Techniques are introduced which determine the geometric structure of
non-simple two-generator $3$-manifolds from purely algebraic data. As an
application, the satellite knots in the $3$-sphere with a two-generator
presentation in which at least one generator is represented by a meridian for
the knot are classified.
|
math
|
1,426 |
On a computer recognition of 3-manifolds
|
math.GT
|
We describe theoretical backgrounds for a computer program that recognizes
all closed orientable 3-manifolds up to complexity 8. The program can treat
also not necessarily closed 3-manifolds of bigger complexities, but here some
unrecognizable (by the program) 3-manifolds may occur.
|
math
|
1,427 |
Dehn surgery on arboresent knots and links -- a survey
|
math.GT
|
This article is solicited by C.\ Adams for a special issue of {\it Chaos,
Solitons and Fractals\/} devoted to knot theory and its applications. We
present some recent results about Dehn surgeries on arborescent knots and
links.
|
math
|
1,428 |
Non-integral toroidal surgery on hyperbolic knots in $S^3$
|
math.GT
|
We show that on any hyperbolic knot in $S^3$ there is at most one
non-integral Dehn surgery which yields a manifold containing an incompressible
torus.
|
math
|
1,429 |
Combinatorial methods in Dehn surgery
|
math.GT
|
This is an expository paper, in which we give a summary of some of the joint
work of John Luecke and the author on Dehn surgery. We consider the situation
where we have two Dehn fillings $M(\alpha)$ and $M(\beta)$ on a given
3-manifold $M$, each containing a surface that is either essential or a
Heegaard surface. We show how a combinatorial analysis of the graphs of
intersection of the two corresponding punctured surfaces in $M$ enables one to
find faces of these graphs which give useful topological information about
$M(\alpha)$ and $M(\beta)$, and hence, in certain cases, good upper bounds on
the intersection number $\Delta(\alpha, \beta)$ of the two filling slopes.
|
math
|
1,430 |
Sutured manifold hierarchies, essential laminations, and Dehn surgery
|
math.GT
|
We use sutured manifold theory, essential laminations and essential branched
surfaces to establish the upper bounds of distances between certain types of
nonsimple Dehn surgery slopes. This is the revised version of an earlier
preprint {\it Dehn surgery and simple manifolds.}
|
math
|
1,431 |
Finite-volume hyperbolic 4-manifolds that share a fundamental polyhedron
|
math.GT
|
It is known that the volume function for hyperbolic manifolds of dimension
$\geq 3$ is finite-to-one. We show that the number of nonhomeomorphic
hyperbolic 4-manifolds with the same volume can be made arbitrarily large. This
is done by constructing a sequence of finite-sided finite-volume polyhedra with
side-pairings that yield manifolds. In fact, we show that arbitrarily many
nonhomeomorphic hyperbolic 4-manifolds may share a fundamental polyhedron. As a
by-product of our examples, we also show in a constructive way that the set of
volumes of hyperbolic 4-manifolds contains the set of even integral multiples
of $4\pi^2/3$. This is ``half'' the set of possible values for volumes, which
is the integral multiples of $4\pi^2/3$ due to the Gauss-Bonnet formula.
|
math
|
1,432 |
Spin^c structures and homotopy equivalences
|
math.GT
|
We show that a homotopy equivalence between manifolds induces a
correspondence between their spin^c-structures, even in the presence of
2-torsion. This is proved by generalizing spin^c-structures to Poincare
complexes. A procedure is given for explicitly computing the correspondence
under reasonable hypotheses.
|
math
|
1,433 |
Toroidal and annular Dehn fillings
|
math.GT
|
Suppose $M$ is a hyperbolic 3-manifold which admits two Dehn fillings
$M(r_1)$ and $M(r_2)$ such that $M(r_1)$ contains an essential torus and
$M(r_2)$ contains an essential annulus. It is known that $\Delta = \Delta(r_1,
r_2) \leq 5$. We will show that if $\Delta = 5$ then $M$ is the Whitehead
sister link exterior, and if $\Delta = 4$ then $M$ is the exterior of either
the Whitehead link or the 2-bridge link associated to the rational number
$3/10$. There are infinitely many examples with $\Delta = 3$.
|
math
|
1,434 |
Alexander duality, gropes and link homotopy
|
math.GT
|
We prove a geometric refinement of Alexander duality for certain 2-complexes,
the so-called gropes, embedded into 4-space. This refinement can be roughly
formulated as saying that 4-dimensional Alexander duality preserves the
disjoint Dwyer filtration. In addition, we give new proofs and extended
versions of two lemmas of Freedman and Lin which are of central importance in
the A-B-slice problem, the main open problem in the classification theory of
topological 4-manifolds. Our methods are group theoretical, rather than using
Massey products and Milnor \mu-invariants as in the original proofs.
|
math
|
1,435 |
Compactifying sufficiently regular covering spaces of compact 3-manifolds
|
math.GT
|
In this paper it is proven that if the group of covering translations of the
covering space of a compact, connected, $P^2$-irreducible 3-manifold
corresponding to a non-trivial, finitely-generated subgroup of its fundamental
group is infinite, then either the covering space is almost compact or the
subgroup is infinite cyclic and has normalizer a non-finitely-generated
subgroup of the rational numbers. In the first case additional information is
obtained which is then used to relate Thurston's hyperbolization and virtual
bundle conjectures to some algebraic conjectures about certain 3-manifold
groups.
|
math
|
1,436 |
Extension of incompressible surfaces on the boundary of 3-manifolds
|
math.GT
|
An incompressible surface $F$ on the boundary of a compact orientable
3-manifold $M$ is arc-extendible if there is an arc $\gamma$ on $\partial M - $
Int $F$ such that $F \cup N(\gamma)$ is incompressible, where $N(\gamma)$ is a
regular neighborhood of $\gamma$ in $\partial M$. Suppose for simplicity that
$M$ is irreducible, and $F$ has no disk components. If $M$ is a product
$F\times I$, or if $\partial M - F$ is a set of annuli, then clearly $F$ is not
arc-extendible. The main theorem of this paper shows that these are the only
obstructions for $F$ to be arc-extendible.
|
math
|
1,437 |
Algorithmic aspects of homeomorphism problems
|
math.GT
|
We will describe some results regarding the algorithmic nature of
homeomorphism problems for manifolds; in particular, the following theorem.
Theorem 1: Every PL or smooth simply connected manifold M^n of dimension n at
least 5 can be recognized among simply connected manifolds. That is, there is
an algorithm to decide whether or not another simply connected manifold is Top,
PL or Diff isomorphic to M. Moreover, an analogous statement is true for
embeddings in codimension at least three: one can algorithmically recognize any
given embedding of one simply connected manifold in another up to isomorphism
of pairs, or up to isotopy, if the codimension of the embedding is not two.
|
math
|
1,438 |
Nonhyperbolic Dehn fillings on hyperbolic 3-manifolds
|
math.GT
|
We give three infinite families of examples of nonhyperbolic Dehn fillings on
hyperbolic manifolds. A manifold in the first family admits two Dehn fillings
of distance two apart, one of which is toroidal and annular, and the other is
reducible and $\partial$-reducible. A manifold in the second family has
boundary consisting of two tori, and admits two reducible Dehn fillings. A
manifold in the third family admits a toroidal filling and a reducible filling
with distance 3 apart. These examples establish the virtual bounds for
distances between certain types of nonhyperbolic Dehn fillings.
|
math
|
1,439 |
Manifolds not containing Gompf nuclei
|
math.GT
|
In this note we show that there are 4-manifolds not containing Gompf nucleus
$N_2$; in this way we answer Problem 4.98 of Kirby's problem list in the
negative.
|
math
|
1,440 |
Quadrisecants of knots and links
|
math.GT
|
We show that every non-trivial tame knot or link in R^3 has a quadrisecant,
i.e. four collinear points. The quadrisecant must be topologically non-trivial
in a precise sense. As an application, we show that a nonsingular, algebraic
surface in R^3 which is a knotted torus must have degree at least eight.
|
math
|
1,441 |
Real trees in topology, geometry, and group theory
|
math.GT
|
This is a survey of the theory of real trees and their applications.
|
math
|
1,442 |
A new approach to the word and conjugacy problems in the braid groups
|
math.GT
|
A new presentation of the $n$-string braid group $B_n$ is studied. Using it,
a new solution to the word problem in $B_n$ is obtained which retains most of
the desirable features of the Garside-Thurston solution, and at the same time
makes possible certain computational improvements. We also give a related
solution to the conjugacy problem, but the improvements in its complexity are
not clear at this writing.
|
math
|
1,443 |
Equivariant configuration spaces
|
math.GT
|
We use the compression theorem (arxiv:math.GT/9712235) cf section 7, to prove
results for equivariant configuration spaces analogous to the well-known
non-equivariant results of May, Milgram and Segal.
|
math
|
1,444 |
The Tits alternative for Out(F_n) I: Dynamics of exponentially-growing automorphisms
|
math.GT
|
The Tits alternative for Out(F_n) is reduced to the case where all elements
in the subgroup under consideration grow polynomially.
|
math
|
1,445 |
The Tits Alternative for $Out(F_n)$ II: A Kolchin Type Theorem
|
math.GT
|
The proof of the Tits alternative for $Out(F_n)$ is completed. The main tool
is a Kolchin type theorem, proved in this paper. It states that a finitely
generated subgroup of $Out(F_n)$ consisting of unipotent automorphisms can be
conjugated into an upper-triangular subgroup (this is interpreted via
train-tracks).
|
math
|
1,446 |
Solvable subgroups of Out(F_n) are virtually abelian
|
math.GT
|
A companion result of the the Tits alternative for $Out(F_n)$ is proved:
Every solvable subgroup of $Out(F_n)$ is finitely generated and virtually
abelian.
|
math
|
1,447 |
3-manifolds as viewed from the curve complex
|
math.GT
|
A Heegaard diagram for a 3-manifold is regarded as a pair of simplexes in the
complex of curves on a surface and a Heegaard splitting as a pair of
subcomplexes generated by the equivalent diagrams. We relate geometric and
combinatorial properties of these subcomplexes with topological properties of
the manifold and/or the associated splitting. For example we show that for any
splitting of a 3-manifold which is Seifert fibered or which contains an
essential torus the subcomplexes are at a distance at most two apart in the
simplicial distance on the curve complex; whereas there are splittings in which
the subcomplexes are arbitrarily far apart. We also give obstructions,
computable from a given diagram, to being Seifert fibered or to containing an
essential torus.
|
math
|
1,448 |
Pure braids, a new subgroup of the mapping class group and finite type invariants
|
math.GT
|
In the study of the relation between the mapping class group M of a surface
and the theory of finite-type invariants of homology 3-spheres, three subgroups
of the mapping class group play a large role. They are the Torelli group, the
Johnson subgroup K and a new subgroup L, which contains K, defined by a choice
of a Lagrangian subgroup of the homology of the surface. In this work we
determine the quotient L/K, in terms of the precise description of M/K given by
Johnson and Morita. We also study the lower central series of L and K, using
some natural imbeddings of the pure braid group in L and the theory of
finite-type invariants.
|
math
|
1,449 |
Bloch invariants of hyperbolic 3-manifolds
|
math.GT
|
We define an invariant \beta(M) of a finite volume hyperbolic 3-manifold M in
the Bloch group B(C) and show it is determined by the simplex parameters of any
degree one ideal triangulation of M. \beta(M) lies in a subgroup of \B(\C) of
finite \Q-rank determined by the invariant trace field of M. Moreover, the
Chern-Simons invariant of M is determined modulo rationals by \beta(M). This
leads to a simplicial formula and rationality results for the Chern Simons
invariant which appear elsewhere.
Generalizations of \beta(M) are also described, as well as several
interesting examples. An appendix describes a scissors congruence
interpretation of B(C).
|
math
|
1,450 |
Rationality problems for Chern-Simons invariants
|
math.GT
|
This paper makes certain observations regarding some conjectures of Milnor
and Ramakrishnan in hyperbolic geometry and algebraic K-theory. As a
consequence of our observations, we obtain new results and conjectures
regarding the rationality and irrationality of Chern-Simons invariants of
hyperbolic 3-manifolds.
|
math
|
1,451 |
Hilbert's 3rd Problem and invariants of 3-manifolds
|
math.GT
|
This paper is an expansion of my lecture for David Epstein's birthday, which
traced a logical progression from ideas of Euclid on subdividing polygons to
some recent research on invariants of hyperbolic 3-manifolds. This `logical
progression' makes a good story but distorts history a bit: the ultimate aims
of the characters in the story were often far from 3-manifold theory.
We start in section 1 with an exposition of the current state of Hilbert's
3rd problem on scissors congruence for dimension 3. In section 2 we explain the
relevance to 3-manifold theory and use this to motivate the Bloch group via a
refined `orientation sensitive' version of scissors congruence. This is not the
historical motivation for it, which was to study algebraic K-theory of C. Some
analogies involved in this `orientation sensitive' scissors congruence are not
perfect and motivate a further refinement in section 4. Section 5 ties together
various threads and discusses some questions and conjectures.
|
math
|
1,452 |
Canonical decompositions of 3-manifolds
|
math.GT
|
We describe a new approach to the canonical decompositions of 3-manifolds
along tori and annuli due to Jaco-Shalen and Johannson (with ideas from
Waldhausen) - the so-called JSJ-decomposition theorem. This approach gives an
accessible proof of the decomposition theorem; in particular it does not use
the annulus-torus theorems, and the theory of Seifert fibrations does not need
to be developed in advance.
|
math
|
1,453 |
Kleinian Groups Generated by Rotations
|
math.GT
|
We discuss which Kleinian groups are commensurable with Kleinian groups
generated by rotations, with particular emphasis on Kleinian groups that arise
from Dehn surgery on a knot.
|
math
|
1,454 |
Akbulut's corks and h-cobordisms of smooth simply connected 4-manifolds
|
math.GT
|
This is primarily an exposition, combining work of several authors (Curtis,
Hsiang, Freedman, Stong, Matveyev, and Bizaca), of the proof that a smooth
5-dimensional h-cobordism between simply connected 4-manifolds is a product off
of a contractible piece which itself is diffeomorphic to the 5-ball.
|
math
|
1,455 |
The Generalized Smale Conjecture for 3-manifolds with genus 2 one-sided Heegaard splittings
|
math.GT
|
The Generalized Smale Conjecture asserts that if M is a closed 3-manifold
with constant positive curvature, then the inclusion of the group of isometries
into the group of diffeomorphisms is a homotopy equivalence. For the 3-sphere,
this was the classical Smale Conjecture proved by A. Hatcher. N. Ivanov proved
the Generalized Smale Conjecture for the M which contain a 1-sided Klein bottle
and such that no Seifert fibering is nonsingular on the complement of any
vertical Klein bottle. We prove it in all remaining cases containing a
one-sided Klein bottle, except for the lens space L(4,1).
|
math
|
1,456 |
The compression theorem I
|
math.GT
|
This the first of a set of three papers about the Compression Theorem: if M^m
is embedded in Q^q X R with a normal vector field and if q-m > 0, then the
given vector field can be straightened (ie, made parallel to the given R
direction) by an isotopy of M and normal field in Q X R. The theorem can be
deduced from Gromov's theorem on directed embeddings [M Gromov, Partial
differential relations, Springer-Verlag (1986); 2.4.5 C'] and is implicit in
the preceeding discussion. Here we give a direct proof that leads to an
explicit description of the finishing embedding. In the second paper in the
series we give a proof in the spirit of Gromov's proof and in the third part we
give applications.
|
math
|
1,457 |
Ubiquity of geometric finiteness in mapping class groups of Haken 3-manifolds
|
math.GT
|
Mapping class groups of Haken 3-manifolds enjoy many of the homological
finiteness properties of mapping class groups of 2-manifolds of finite type.
For example, H(M) has a torsionfree subgroup of finite index, which is
geometrically finite (i. e. is the fundamental group of a finite aspherical
complex). This was proven by J. Harer for 2-manifolds and by the second author
for Haken 3-manifolds. In this paper we prove that H(M) acts properly
discontinuously on a contractible simplicial complex, with compact quotient.
This implies that every torsionfree subgroup of finite index in H(M) is
geometrically finite. Also, a simplified proof of the fact that torsionfree
subgroups of finite index in H(M) exist is given. All results are proven for
mapping class groups that preserve a boundary pattern in the sense of K.
Johannson. As an application, we show that if F is a nonempty compact
2-manifold in the boundary of M, then the classifying space BDiff(M rel F) of
the diffeomorphism group of M relative to F has the homotopy type of a finite
aspherical complex.
|
math
|
1,458 |
Scharlemann's manifold is standard
|
math.GT
|
In his 1974 thesis, Martin Scharlemann constructed a fake homotopy
equivalence from a closed smooth manifold f:Q -> S^3 x S^1 # S^2 x S^2 and
asked whether the manifold Q itself is diffeomorphic to S^3 x S^1 # S^2 x S^2.
Here we answer this question affirmatively.
|
math
|
1,459 |
Finiteness of classifying spaces of relative diffeomorphism groups of 3-manifolds
|
math.GT
|
The main theorem shows that if M is an irreducible compact connected
orientable 3-manifold with non-empty boundary, then the classifying space
BDiff(M rel dM) of the space of diffeomorphisms of M which restrict to the
identity map on boundary(M) has the homotopy type of a finite aspherical
CW-complex. This answers, for this class of manifolds, a question posed by M
Kontsevich. The main theorem follows from a more precise result, which asserts
that for these manifolds the mapping class group H(M rel dM) is built up as a
sequence of extensions of free abelian groups and subgroups of finite index in
relative mapping class groups of compact connected surfaces.
|
math
|
1,460 |
Genus two Heegaard splittings of orientable three-manifolds
|
math.GT
|
It was shown by Bonahon-Otal and Hodgson-Rubinstein that any two genus-one
Heegaard splittings of the same 3-manifold (typically a lens space) are
isotopic. On the other hand, it was shown by Boileau, Collins and Zieschang
that certain Seifert manifolds have distinct genus-two Heegaard splittings. In
an earlier paper, we presented a technique for comparing Heegaard splittings of
the same manifold and, using this technique, derived the uniqueness theorem for
lens space splittings as a simple corollary. Here we use a similar technique to
examine, in general, ways in which two non-isotopic genus-two Heegard
splittings of the same 3-manifold compare, with a particular focus on how the
corresponding hyperelliptic involutions are related.
|
math
|
1,461 |
Algorithms for recognizing knots and 3-manifolds
|
math.GT
|
This is a survey paper on algorithms for solving problems in 3-dimensional
topology. In particular, it discusses Haken's approach to the recognition of
the unknot, and recent variations.
|
math
|
1,462 |
Higher p invariants
|
math.GT
|
The rho-invariant is an invariant of odd-dimensional manifolds with finite
fundamental group, and lies in the representations modulo the regular
representations (after tensoring with Q). It is a fundamental invariant that
occurs in classifying lens spaces, their homotopy analogues, and is intimately
related to the eta-invariant for the signature operator. The goal of this note
is to use some of the technology developed in studying the Novikov higher
signature conjecture to define an analogous invariant for certain situations
with infinite fundamental group.
|
math
|
1,463 |
An invariant of smooth 4-manifolds
|
math.GT
|
We define a diffeomorphism invariant of smooth 4-manifolds which we can
estimate for many smoothings of R^4 and other smooth 4-manifolds. Using this
invariant we can show that uncountably many smoothings of R^4 support no Stein
structure. (Gompf has constructed uncountably many smoothings of R^4 which do
support Stein structures.) Other applications of this invariant are given.
|
math
|
1,464 |
A Geometric Characteristic Splitting in all Dimensions
|
math.GT
|
We prove the existence of a geometric characteristic submanifold for
non-positively curved manifolds of any dimension greater than or equal to
three. In dimension three, our result is a geometric version of the topological
characteristic submanifold theorem due to Jaco, Shalen and Johannson.
|
math
|
1,465 |
Simple Loops on Surfaces and Their Intersection Numbers
|
math.GT
|
Given a compact orientable surface $\Sigma$, let $\Cal S(\Sigma)$ be the set
of isotopy classes of essential simple loops on $\Sigma$. We determine a
complete set of relations for a function from $\Cal S(\Sigma)$ to $\bold Z$ to
be a geometric intersection number function. As a consequence, we obtain
explicit equations in $\bold R^{\Cal S(\Sigma)}$ and $P(\bold R^{\Cal
S(\Sigma)})$ defining Thurston's space of measured laminations and Thurston's
compactification of the Teichm\"uller space. These equations are not only
piecewise integral linear but also semi-real algebraic.
|
math
|
1,466 |
Hyperbolic Structures on 3-manifolds, I: Deformation of acylindrical manifolds
|
math.GT
|
This is the first in a series of papers showing that Haken manifolds have
hyperbolic structures; this first was published, the second two have existed
only in preprint form, and later preprints were never completed. This eprint is
only an approximation to the published version, which is the definitive form
for part I, and is provided for convenience only. All references and quotations
should be taken from the published version, since the theorem numbering is
different and not all corrections have been incorporated into the present
version.
Parts II and III will be made available as eprints shortly.
|
math
|
1,467 |
Geodesic Length Functions and Teichmüller Spaces
|
math.GT
|
Given a compact orientable surface with finitely many punctures $\Sigma$, let
$\Cal S(\Sigma)$ be the set of isotopy classes of essential unoriented simple
closed curves in $\Sigma$. We determine a complete set of relations for a
function from $\Cal S(\Sigma)$ to $\bold R$ to be the geodesic length function
of a hyperbolic metric with geodesic boundary and cusp ends on $\Sigma$. As a
conse quence, the Teichm\"uller space of hyperbolic metrics with geodesic
boundary and cusp ends on $\Sigma$ is reconstructed from an intrinsic $(\bold
QP^1, PSL(2, \bold Z))$ structure on $\Cal S(\Sigma)$.
|
math
|
1,468 |
A Presentation of the Mapping Class Groups
|
math.GT
|
Using the works of Gervais, Harer, Hatcher and Thurston and others, we show
that the mapping class group of a compact orientable surface has a presentation
so that the generators are the set of all Dehn twists and the relations are
supported in subsurfaces homeomorphic to the one-holed torus or the four-holed
sphere. It turns out that all the relations were discovered by Dehn in 1938.
|
math
|
1,469 |
The Burau matrix and Fiedler's invariant for a closed braid
|
math.GT
|
It is shown how Fiedler's `small state-sum' invariant for a braid can be
calculated from the 2-variable Alexander polynomial of the link which consists
of the closed braid together with the braid axis.
|
math
|
1,470 |
Examples of non-trivial roots of unity at ideal points of hyperbolic 3-manifolds
|
math.GT
|
This paper gives examples of hyperbolic 3-manifolds whose SL(2,C) character
varieties have ideal points whose associated roots of unity are not 1 or -1.
This answers a question of Cooper, Culler, Gillet, Long, and Shalen as to
whether roots of unity other than 1 and -1 occur.
|
math
|
1,471 |
Shapes of polyhedra and triangulations of the sphere
|
math.GT
|
The space of shapes of a polyhedron with given total angles less than 2\pi at
each of its n vertices has a Kaehler metric, locally isometric to complex
hyperbolic space CH^{n-3}. The metric is not complete: collisions between
vertices take place a finite distance from a nonsingular point. The metric
completion is a complex hyperbolic cone-manifold. In some interesting special
cases, the metric completion is an orbifold. The concrete description of these
spaces of shapes gives information about the combinatorial classification of
triangulations of the sphere with no more than 6 triangles at a vertex.
|
math
|
1,472 |
Toroidal and Boundary-Reducing Dehn Fillings
|
math.GT
|
Let M be a simple 3-manifold with a toral boundary component partial_0 M. If
Dehn filling M along partial_0 M one way produces a toroidal manifold and Dehn
filling M along partial_0 M another way produces a boundary-reducible manifold,
then we show that the absolute value of the intersection number on partial_0 M
of the two filling slopes is at most two. In the special case that the
boundary-reducing filling is actually a solid torus and the intersection number
between the filling slopes is two, more is said to describe the toroidal
filling.
|
math
|
1,473 |
A new algorithm for recognizing the unknot
|
math.GT
|
The topological underpinnings are presented for a new algorithm which answers
the question: `Is a given knot the unknot?' The algorithm uses the braid
foliation technology of Bennequin and of Birman and Menasco. The approach is to
consider the knot as a closed braid, and to use the fact that a knot is
unknotted if and only if it is the boundary of a disc with a combinatorial
foliation. The main problems which are solved in this paper are: how to
systematically enumerate combinatorial braid foliations of a disc; how to
verify whether a combinatorial foliation can be realized by an embedded disc;
how to find a word in the the braid group whose conjugacy class represents the
boundary of the embedded disc; how to check whether the given knot is isotopic
to one of the enumerated examples; and finally, how to know when we can stop
checking and be sure that our example is not the unknot.
|
math
|
1,474 |
Fiber-preserving diffeomorphisms and imbeddings
|
math.GT
|
Around 1960, R. Palais and J. Cerf proved a fundamental result relating
spaces of diffeomorphisms and imbeddings of manifolds: If V is a submanifold of
M, then the map from Diff(M) to Imb(V,M) that takes f to its restriction to V
is locally trivial. We extend this and related results into the context of
fibered manifolds, and fiber-preserving diffeomorphisms and imbeddings. That
is, if M fibers over B, with compact fiber, and V is a vertical submanifold of
M, then the restriction from the space FDiff(M) of fiber-preserving
diffeomorphisms of M to the space of imbeddings of V into M that take fibers to
fibers is locally trivial. Also, the map from FDiff(M) to Diff(B) that takes f
to the diffeomorphism it induces on B is locally trivial. The proofs adapt
Palais' original approach; the main new ingredient is a version of the
exponential map, called the aligned exponential, which has better properties
with respect to fiber-preserving maps. Versions allowing certain kinds of
singular fibers are proven, using equivariant methods. These apply to almost
all Seifert-fibered 3-manifolds. As an application, we reprove an unpublished
result of F. Raymond and W. Neumann that each component of the space of Seifert
fiberings of a Haken 3-manifold is weakly contractible.
|
math
|
1,475 |
Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds
|
math.GT
|
This paper proves a theorem about Dehn surgery using a new theorem about
PSL(2, C) character varieties. Confirming a conjecture of Boyer and Zhang, this
paper shows that a small hyperbolic knot in a homotopy sphere having a
non-trivial cyclic slope r has an incompressible surface with non-integer
boundary slope strictly between r-1 and r+1. A corollary is that any small knot
which has only integer boundary slopes has Property P. The proof uses
connections between the topology of the complement of the knot, M, and the
PSL(2, C) character variety of M that were discovered by Culler and Shalen. The
key lemma, which should be of independent interest, is that for certain
components of the character variety of M, the map on character varieties
induced by the inclusion of boundary M into M is a birational isomorphism onto
its image. This in turn depends on a fancy version of Mostow rigidity due to
Gromov, Thurston, and Goldman.
|
math
|
1,476 |
The symplectic Floer homology of composite knots
|
math.GT
|
We develop a method of calculation for the symplectic Floer homology of
composite knots. The symplectic Floer homology of knots defined in \cite{li}
naturally admits an integer graded lifting, and it formulates a filtration and
induced spectral sequence. Such a spectral sequence converges to the symplectic
homology of knots in \cite{li}. We show that there is another spectral sequence
which converges to the $\Z$-graded symplectic Floer homology for composite
knots represented by braids.
|
math
|
1,477 |
Integral Invariants of 3-Manifolds. II
|
math.GT
|
This note is a sequel to our earlier paper of the same title [dg-ga/9710001]
and describes invariants of rational homology 3-spheres associated to acyclic
orthogonal local systems. Our work is in the spirit of the Axelrod-Singer
papers, generalizes some of their results, and furnishes a new setting for the
purely topological implications of their work.
|
math
|
1,478 |
Obstructing 4-torsion in the classical knot concordance group
|
math.GT
|
We prove that if the order of the first homology of the 2-fold branched cover
of a knot K in the 3-sphere is given by pm where p is a prime congruent to 3
mod 4 and gcd(p,m) =1, then K is of infinite order in the knot concordance
group. This provides an obstruction to classical knots being of order 4. In
particular, there are 11 prime knots with 10 or fewer crossings that are of
order 4 in the algebraic concordance group; all are infinite order in
concordance. Another corollary states that any knot with Alexander polynomial
5t^2 - 11t + 5 is of infinite order in concordance; Levine proved that in
higher dimensions all such knots are of order 4.
|
math
|
1,479 |
Comparing Heegaard and JSJ structures of orientable 3-manifolds
|
math.GT
|
The Heegaard genus g of an irreducible closed orientable 3-manifold puts a
limit on the number and complexity of the pieces that arise in the
Jaco-Shalen-Johannson decomposition of the manifold by its canonical tori. For
example, if p of the complementary components are not Seifert fibered, then p <
g. This result generalizes work of Kobayashi. The Heegaard genus g also puts
explicit bounds on the complexity of the Seifert pieces. For example, if the
union of the base spaces of the Seifert pieces has Euler characteristic X and
there are a total of f exceptional fibers in the Seifert pieces, then f - X is
no greater than 3g - 3 - p.
|
math
|
1,480 |
Foliation-preserving Maps Between Solvmanifolds
|
math.GT
|
For i = 1,2, let Gamma_i be a lattice in a simply connected, solvable Lie
group G_i, and let X_i be a connected Lie subgroup of G_i. The double cosets
Gamma_igX_i provide a foliation F_i of the homogeneous space Gamma_i\G_i. Let f
be a continuous map from Gamma_1\G_1 to Gamma_2\G_2 whose restriction to each
leaf of F_1 is a covering map onto a leaf of F_2. If we assume that F_1 has a
dense leaf, and make certain technical technical assumptions on the lattices
Gamma_1 and Gamma_2, then we show that f must be a composition of maps of two
basic types: a homeomorphism of Gamma_1\M_1 that takes each leaf of F_1 to
itself, and a map that results from twisting an affine map by a homomorphism
into a compact group.
We also prove a similar result for many cases where G_1 and G_2 are neither
solvable nor semisimple.
|
math
|
1,481 |
3-Manifolds with irreducible Heegaard splittings of high genus
|
math.GT
|
Non-isotopic Heegaard splittings of non-minimal genus were known previously
only for very special 3-manifolds. We show in this paper that they are in fact
a wide spread phenomenon in 3-manifold theory: We exhibit a large class of
knots and manifolds obtained by Dehn surgery on these knots which admit such
splittings. Many of the manifolds have irreducible Heegaard splittings of
arbitrary large genus. All these splittings are horizontal and are isotopic,
after one stabilization, to a multiple stabilization of certain canonical low
genus vertical Heegaard splittings.
|
math
|
1,482 |
Gauss sums on almost positive knots
|
math.GT
|
Using the Fiedler-Polyak-Viro Gauss diagram formulas we study the Vassiliev
invariants of degree 2 and 3 on almost positive knots. As a consequence we show
that the number of almost positive knots of given genus or unknotting number
grows polynomially in the crossing number, and also recover and extend, inter
alia to their untwisted Whitehead doubles, previous results on the polynomials
and signatures of such knots. In particular, we prove that there are no achiral
almost positive knots and classify all almost positive diagrams of the unknot.
We give an application to contact geometry (Legendrian knots) and property P.
|
math
|
1,483 |
Generic immersions of curves, knots, monodromy and gordian number
|
math.GT
|
Starting from a divide, i.e. a generic immersion of finitely many copies of
the interval [0,1] in the disk, we construct a classical link in the 3-sphere.
We prove that the link's complement fibers over the circle, if the divide is
connected. Moreover, we compute the monodromy diffeomorphism from the
combinatorics of the divide. We added to this version of the paper the theorem
about the gordian number of the link of a divide. The gordian number of the
link of a divide equals the number of double points of the divide.
|
math
|
1,484 |
A survey of 4-manifolds through the eyes of surgery
|
math.GT
|
The title says it all.
|
math
|
1,485 |
Foliations Transverse to Triangulations of 3-Manifolds
|
math.GT
|
We investigate the combinatorial analogues, in the context of normal
surfaces, of taut and transversely measured (codimension 1) foliations of
3-manifolds. We establish that the existence of certain combinatorial
structures, a priori weaker than the existence of the corresponding foliation,
is sufficient to guarantee that the manifold in question satisfies certain
properties, e.g. irreducibility. The finiteness of our combinatorial structures
allows us to make our results quantitative in nature and has (coarse)
geometrical consequences for the manifold. Furthermore, our techniques give a
straightforward combinatorial proof of Novikov's theorem.
|
math
|
1,486 |
Pseudo-Anosov maps and simple closed curves on surfaces
|
math.GT
|
Given a pair of curves C_1 and C_2 on a hyperbolic surface F, when does there
exist a pseudo-Anosov map sending one to another? More generally, one may ask
the same question for C_i to be sets of disjoint simple closed curves. We will
give necessary and sufficient conditions for the existence of such maps.
|
math
|
1,487 |
The Multivariable Alexander Polynomial for a Closed Braid
|
math.GT
|
A simple multivariable version of the reduced Burau matrix is constructed for
any braid. It is shown how the multivariable Alexander polynomial for the
closure of the braid can be found directly from this matrix.
|
math
|
1,488 |
The structure of a solvmanifold's Heegaard splittings
|
math.GT
|
We classify isotopy classes of irreducible Heegaard splittings of
solvmanifolds. If the monodromy of the solvmanifold can be expressed as a 2 x 2
matrix with 0 in the lower right hand corner (as always is true when the
absolute value of the trace is 3), then any irreducible splitting is strongly
irreducible and of genus two. If furthermore the absolute value of the trace is
4 or greater, then any two such splittings are isotopic. If the absolute value
of the trace is 3 then, up to isotopy, there are exactly two irreducible
splittings, their associated hyperelliptic involutions commute, and the product
of the involutions is the central involution of the solvmanifold.
If the monodromy cannot be expressed as a 2 x 2 matrix with 0 in the lower
right hand corner, then the splitting is weakly reducible, of genus three and
unique up to isotopy.
|
math
|
1,489 |
A natural framing of knots
|
math.GT
|
Given a knot K in the 3-sphere, consider a singular disk bounded by K and the
intersections of K with the interior of the disk. The absolute number of
intersections, minimised over all choices of singular disk with a given
algebraic number of intersections, defines the framing function of the knot. We
show that the framing function is symmetric except at a finite number of
points. The symmetry axis is a new knot invariant, called the natural framing
of the knot. We calculate the natural framing of torus knots and some other
knots, and discuss some of its properties and its relations to the signature
and other well-known knot invariants.
|
math
|
1,490 |
Positive links are strongly quasipositive
|
math.GT
|
Let S(D) be the surface produced by applying Seifert's algorithm to the
oriented link diagram D. I prove that if D has no negative crossings then S(D)
is a quasipositive Seifert surface, that is, S(D) embeds incompressibly on a
fiber surface plumbed from positive Hopf annuli. This result, combined with the
truth of the `local Thom Conjecture', has various interesting consequences; for
instance, it yields an easily-computed estimate for the slice euler
characteristic of the link L(D) (where D is arbitrary) that extends and often
improves the `slice-Bennequin inequality' for closed-braid diagrams; and it
leads to yet another proof of the chirality of positive and almost positive
knots.
|
math
|
1,491 |
Braided chord diagrams
|
math.GT
|
The notion of a braided chord diagram is introduced and studied. An
equivalence relation is given which identifies all braidings of a fixed chord
diagram. It is shown that finite-type invariants are stratified by braid index
for knots which can be represented as closed 3-braids. Partial results are
obtained about spanning sets for the algebra of chord diagrams of braid index
3.
|
math
|
1,492 |
Studying surfaces via closed braids
|
math.GT
|
This is a review article on the Bennequin-Birman-Menasco machinery for
studying embedded incompressible surfaces in 3-space via their `braid
foliations'. Two cases are investigated: case (1) The surface has non-empty
boundary; the boundary is a knot or link which is represented as a closed
braid, Case (2) The surface is closed, but it lies in the complement of a knot
or link which is represented as a closed braid. The main results in the area
are established with full proofs, in a systematic fashion, with an eye toward
making them accessible to the beginning reader. There are some new
contributions, described in detail in the introduction.
|
math
|
1,493 |
Milnor and finite type invariants of plat-closures
|
math.GT
|
We show that for an $n$-component, $n$-bridge link and a positive integer
$m$, the following is true: If the longitudes of $L$ lie in the $(m+2)$-th term
of the lower central series of the link group then all the finite type
invariants of orders $\leq m$ for $L$ are the same as these of the
$n$-component unlink.
|
math
|
1,494 |
Regular Seifert surfaces and Vassiliev knot invariants
|
math.GT
|
We show that the Vassiliev invariants of orders $\leq n$ of a knot K, are
obstructions to finding a regular Seifert surface, S, whose complement looks
"simple" (e.g. like the complement of a disc) to the lower central series of
its fundamental group. As a consequence of this, we obtain that the Vassiliev
invariants of the knot $K=\partial S$ are null-concordance obstructions of
certain links that can be obtained from regular spines of S. We also discuss
various generalizations of these results, and we conjecture a geometric
characterization of knots whose invariants of all orders vanish.
|
math
|
1,495 |
Canonical decomposition of manifolds with flat real projective structure into (n-1)-convex manifolds and concave affine manifolds
|
math.GT
|
We try to understand the geometric properties of $n$-manifolds ($n\geq 2$)
with geometric structures modeled on $(\bR P^n, \PGL(n+1, \bR))$, i.e.,
$n$-manifolds with projectively flat torsion free affine connections. We define
the notion of $i$-convexity of such manifolds due to Carri\'ere for integers
$i$, $1 \leq i \leq n-1$, which are generalization of convexity. Given a real
projective $n$-manifold $M$, we show that the failure of an $(n-1)$-convexity
of $M$ implies an existence of a certain geometric object, $n$-crescent, in the
completion $\che M$ of the universal cover $\tilde M$ of $M$. We show that this
further implies the existence of a particular type of affine submanifold in $M$
and give a natural decomposition of $M$ into simpler real projective manifolds,
some of which are $(n-1)$-convex and others are affine, more specifically
concave affine. We feel that it is useful to have such decomposition
particularly in dimension three. Our result will later aid us to study the
geometric and topological properties of radiant affine 3-manifolds leading to
their classification. We get a consequence for affine Lie groups.
|
math
|
1,496 |
Finite type invariants of 3-manifolds
|
math.GT
|
A theory of finite type invariants for arbitrary compact oriented 3-manifolds
is proposed, and illustrated through many examples arising from both classical
and quantum topology. The theory is seen to be highly non-trivial even for
manifolds with large first betti number, encompassing much of the complexity of
Ohtsuki's theory for homology spheres. (For example, it is seen that the
quantum SO(3) invariants, though not of finite type, are determined by finite
type invariants.) The algebraic structure of the set of all finite type
invariants is investigated, along with a combinatorial model for the theory in
terms of trivalent "Feynman diagrams".
|
math
|
1,497 |
The homology of abelian coverings of knotted graphs
|
math.GT
|
Let N be a regular branched cover of a homology 3-sphere M with deck group G
isomorphic to Z_2^d and branch set a trivalent graph Gamma; such a cover is
determined by a coloring of the edges of Gamma with elements of G. For each
index-2 subgroup H of G, M_H = N/H is a double branched cover of M. Sakuma has
proved that the first homology of N is isomorphic, modulo 2-torsion, to the
direct sum of the first homology groups of the M_H, and has shown that H_1(N)
is determined up to isomorphism by the direct sum of the H_1(M_H) in certain
cases; specifically, when d=2 and the coloring is such that the branch set of
each cover M_H -> M is connected, and when d=3 and Gamma is the complete graph
K_4. We prove this for a larger class of coverings: when d=2, for any coloring
of a connected graph; when d=3 or 4, for an infinite class of colored graphs;
and when d=5, for a single coloring of the Petersen graph.
|
math
|
1,498 |
Geometrization of 3-orbifolds of cyclic type
|
math.GT
|
We give a complete proof of Thurston's Orbifold Theorem for very good
3-orbifolds of cyclic type. An orbifold is said to be very good when it has a
finite cover which is a manifold. A 3-orbifold is of cyclic type if the
singular set is a non-empty 1-manifold transverse to the boundary.
|
math
|
1,499 |
Positive knots, closed braids and the Jones polynomial
|
math.GT
|
Using the recent Gauss diagram formulas for Vassiliev invariants of
Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality,
we prove several inequalities for positive knots relating their Vassiliev
invariants, genus and degrees of the Jones polynomial. As a consequence, we
prove that for any of the polynomials of Alexander/Conway, Jones, HOMFLY,
Brandt-Lickorish-Millett-Ho and Kauffman there are only finitely many positive
knots with the same polynomial and no positive knot with trivial polynomial.
We also discuss an extension of the Bennequin inequality, showing that the
unknotting number of a positive knot not less than its genus, which recovers
some recent unknotting number results of A'Campo, Kawamura and Tanaka, and give
applications to the Jones polynomial of a positive knot.
|
math
|
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