In this note, we develop a condition on a closed curve on a surface or in a
3-manifold that implies that the curve has the property that its length
function on the space of all hyperbolic structures on the surface or 3-manifold
completely determines the curve.

The results of Culler and Shalen for 2,3 or 4-free hyperbolic 3-manifolds are
contingent on properties specific to and special about rank two subgroups of a
free group. Here we determine what construction and algebraic information is
required in order to make a geometric statement about $M$, a closed, orientable
hyperbolic manifold with $k$-free fundamental group, for any value of $k$
greater than four.

We consider the "intrinsic" symmetry group of a two-component link $L$,
defined to be the image $\Sigma(L)$ of the natural homomorphism from the
standard symmetry group $\MCG(S^3,L)$ to the product $\MCG(S^3) \cross
\MCG(L)$. This group, first defined by Whitten in 1969, records directly
whether $L$ is isotopic to a link $L'$ obtained from $L$ by permuting
components or reversing orientations; it is a subgroup of $\Gamma_2$, the group
of all such operations.

A TQFT is a functor from a cobordism category to the category of vector
spaces, satisfying certain properties. An important property is that the vector
spaces should be finite dimensional. For the WRT TQFT, the relevant
2+1-cobordism category is built from manifolds which are equipped with an extra
structure such as a p_1-structure, or an extended manifold structure. The
purpose of this paper is to explain that without this extra structure, one
would not get finite dimensionality.

Let Y be a noncompact rank one locally symmetric space of finite volume. Then
Y has a finite number e(Y) > 0 of topological ends. In this paper, we show that
for any natural number n, the Y with e(Y) \leq n that are arithmetic fall into
finitely many commensurability classes. In particular, there is a constant c_n
such that n-cusped arithmetic orbifolds do not exist in dimension greater than
c_n. We make this explicit for one-cusped arithmetic hyperbolic n-orbifolds and
prove that none exist for n \geq 30.

We construct quantum invariants for handlebody-knots in a 3-sphere S^3. A
handlebody-knot is an embedding of a handlebody in a 3-manifold. These
invariants are linear sums of Yokota's invariants for colored spatial graphs
which is defined by using Kauffman bracket. We also give a non-trivial example.

Normal surfaces are a key tool in computational knot theory and 3-manifold
topology, and have featured in significant computational breakthroughs in
recent years. Despite this, there has been little practical progress on
algorithms that use fundamental normal surfaces, which are described in terms
of a Hilbert basis for a pointed rational cone on a high-dimensional integer
lattice. In this paper we develop and implement several algorithms to enumerate
fundamental normal surfaces, by merging domain-specific techniques from normal
surface theory with classical Hilbert basis algorithms.

We give some applications of the Chern Simons gauge theory to the study of
the set ${\rm vol}(N,G)$ of volumes of all representations $\rho\co\pi_1N\to
G$, where $N$ is a closed oriented three-manifold and $G$ is either ${\rm
Iso}_e\t{\rm SL_2(\R)}$, the isometry group of the Seifert geometry, or ${\rm
Iso}_+{\Hi}^3$, the orientation preserving isometry group of the hyperbolic
3-space. We focus on three natural questions:

(1) How to find non-zero values in ${\rm vol}(N, G)$? or weakly how to find
non-zero elements in ${\rm vol}(\t N, G)$ for some finite cover $\t N$ of $N$?

If the tunnel number of knot $K$ is denoted $t(K)$, a pair of knots $K_1,K_2$
is said to be subadditive if $t(K_1)+t(K_2)>t(K_1 # K_2)$. We use a slight
generalization of the concept of $\mu$-primitivity to construct subadditive
pairs of knots of arbitrarily large tunnel number.

We calculate the Kauffman bracket skein module (KBSM) of the complement of
all two-bridge links. For a two-bridge link, we show that the KBSM of its
complement is free over the ring $\BC[t^{\pm 1}]$ and when reducing $t=-1$, it
is isomorphic to the ring of regular functions on the character variety of the
link group.

Using the mapping cone of a rational surgery, we give several obstructions
for Seifert fibered surgeries, including obstructions on the Alexander
polynomial, the knot Floer homology, the surgery coefficient and the Seifert
and four-ball genus of the knot.

We find decomposition series of length at most two for modular
representations in positive characteristic of mapping class groups of surfaces
induced by an integral version of the Witten-Reshetikhin-Turaev SO(3)-TQFT at
the p-th root of unity, where p is an odd prime. The dimensions of the
irreducible factors are given by Verlinde-type formulas.

Trivial links are unique up to number of link components, but they can be
hard to recognize from arbitrary diagrams. We define a measure, the crumple, on
link diagrams and then demonstrate that for trivial links there is a sequence
of moves by which the crumple may be strictly monotonically reduced. By our
definition, the minimum possible crumple over all link diagrams is achieved
only by embedding components disjointly in parallel planes, and so a link will
be able to obtain this crumple if and only if it is trivial.

We give a new proof of the theorem of Birman-Powell that the Torelli subgroup
of the mapping class group of a closed orientable surface of genus at least 3
is generated by simple homeomorphisms known as bounding pair maps. The key
ingredient is a proof that the subcomplex of the curve complex of the surface
spanned by curves within a fixed homology class is connected.

Given an l-component pointed oriented link (L,p) in an oriented
three-manifold Y, one can construct its link Floer chain complex CFL(Y,L,p)
over the polynomial ring F_2[U_1,...,U_l]. Moving the basepoint p_i in the link
component L_i once around induces an automorphism of CFL(Y,L,p). In this paper,
we study an automorphism (a possibly different one) of CFL(Y,L,p) defined
explicitly in terms of holomorphic disks; for links in S^3, we show that these
two automorphisms are the same.

For Legendrian links in the 1-jet space of $S^1$ we show that the 1-graded
ruling polynomial may be recovered from the Kauffman skein module. For such
links a generalization of the notion of normal ruling is introduced. We show
that the existence of such a generalized normal ruling is equivalent to
sharpness of the Kauffman polynomial estimate for the Thurston-Bennequin number
as well as to the existence of an ungraded augmentation of the
Chekanov-Eliashberg DGA. Parallel results involving the HOMFLY-PT polynomial
and 2-graded generalized normal rulings are established.

A homology cylinder over a compact manifold is a homology cobordism between
two copies of the manifold together with a boundary parametrization. We study
abelian quotients of the homology cobordism group of homology cylinders. For
homology cylinders over general surfaces, it was shown by Cha, Friedl and Kim
that their homology cobordism groups have infinitely generated abelian quotient
groups by using Reidemeister torsion invariants. In this paper, we first
investigate their abelian quotients again by using another invariant called the
Magnus representation.

We give an explicit construction of linearly independent families of knots
arbitrarily deep in the (n)-solvable filtration of the knot concordance group
using the \rho^1-invariant. A difference between previous constructions of
infinite rank subgroups in the concordance group and ours is that the deepest
infecting knots in the construction we present are allowed to have vanishing
Tristram-Levine signatures.

This work derives direct and concrete relations between colored Jones
polynomials and the topology of incompressible spanning surfaces in knot and
link complements. Under mild diagrammatic hypotheses that arise naturally in
the study of knot polynomial invariants (A-adequacy), we prove that the growth
of the degree of the colored Jones polynomials is a boundary slope of an
essential surface in the knot complement.

We study cosmetic crossings in knots of genus one and obtain obstructions to
such crossings in terms of knot invariants determined by Seifert matrices. In
particular, we prove that for genus one knots the Alexander polynomial and the
homology of the double cover branching over the knot provide obstructions to
cosmetic crossings. As an application we prove the nugatory crossing conjecture
for twisted Whitehead doubles of non-cable knots. We also verify the conjecture
for several families of pretzel knots and all genus one knots with up to 12
crossings.

We investigate Legendrian graphs in $(\R^3, \xi_{std})$. We extend the
classical invariants, Thurston-Bennequin number and rotation number to
Legendrian graphs. We prove that a graph can be Legendrian realized with all
its cycles Legendrian unknots with $tb=-1$ and $rot=0$ if and only if it does
not contain $K_4$ as a minor. We show that the pair $(tb, rot)$ does not
characterize a Legendrian graph up to Legendrian isotopy if the graph contains
a cut edge or a cut vertex.

We use the knot homology of Khovanov and Lee to construct link concordance
invariants generalizing the Rasmussen $s$-invariant of knots. The relevant
invariant for a link is a filtration on a vector space of dimension $2^{|L|}$.
The basic properties of the $s$-invariant all extend to the case of links; in
particular, any orientable cobordism $\Sigma$ between links induces a map
between their corresponding vector spaces which is filtered of degree
$\chi(\Sigma)$.

We show that for any two Heegaard splittings of genus $p$ and $q$ for the
same closed 3-manifold, there is a common stabilization of genus at most 3/2 p
+ 2q - 1. One may compare this to recent examples of Heegaard splittings whose
smallest common stabilizations have genus at least $p+q$ or $p + 1/2 q$
depending on the notion of equivalence.

This article pursues the study of the knot state asymptotics in the large
level limit initiated in "Knot sate Asymptotics I". As a main result, we prove
the Witten asymptotic expansion conjecture for the Dehn fillings of the figure
eight knot. The state of a knot is defined in the realm of Chern-Simons
topological quantum field theory as a holomorphic section on the
SU(2)-character manifold of the peripheral torus.

We construct an infinite family of hyperbolic, homologically thin knots that
are not quasi-alternating. To establish the latter, we argue that the branched
double-cover of each knot in the family does not bound a negative definite
4-manifold with trivial first homology and bounded second betti number. This
fact depends in turn on information from the correction terms in Heegaard Floer
homology, which we establish by way of a relationship to, and calculation of,
the Turaev torsion.

A well-known conjecture asserts that the mapping class group of a surface
(possibly with punctures/boundary) does not virtually surject onto $\Z$ if the
genus of the surface is large. We prove that if this conjecture holds for some
genus, then it also holds for all larger genera. We also prove that if there is
a counterexample to this conjecture, then there must be a counterexample of a
particularly simple form. We prove these results by relating the conjecture to
a family of linear representations of the mapping class group that we call the
higher Prym representations.

This work uncovers the tropical analogue for measured laminations of the
convex hull construction of decorated Teichmueller theory, namely, it is a
study in coordinates of geometric degeneration to a point of Thurston's
boundary for Teichmueller space. This may offer a paradigm for the extension of
the basic cell decomposition of Riemann's moduli space to other contexts for
general moduli spaces of flat connections on a surface. In any case, this
discussion drastically simplifies aspects of previous related studies as is
explained.

A positive solution of a problem by V.I.Arnol'd about higher analog of the
asymptotic Hopf invariant of divergence-free vector fields is presented. A
higher asymptotic invariant of magnetic fields, which is not expressed from the
asymptotic linking numbers of magnetic lines is constructed and a non-trivial
example of an asymptotic invariant is presented. A definition of an asymptotic
finite-order invariant for classical links is presented.

We iterate Manolescu's unoriented skein exact triangle in knot Floer homology
with coefficients in the fraction field of the group ring (Z/2Z)[Z]. The result
is a spectral sequence which converges to a stabilized version of delta-graded
knot Floer homology. The (E_2,d_2) page of this spectral sequence is an
algorithmically computable chain complex expressed in terms of spanning trees,
and we show that there are no higher differentials. This gives the first
combinatorial spanning tree model for knot Floer homology.

Kanenobu has given infinite families of knots with the same HOMFLY
polynomials. We show that these knots also have the same sl(n) and HOMFLY
homologies, thus giving the first example of an infinite family of knots
undistinguishable by these invariants. This is a consequence of a structure
theorem about the homologies of knots obtained by twisting up the ribbon of a
ribbon knot with one ribbon.

We extend to dimension $n \geq 3$ the concept of $\rho$-pair in a coloured
graph and we prove the existence theorem for minimal rigid crystallizations of
handle-free, closed $n$-manifolds.

Let $M$ be a smooth closed orientable surface, and let $F$ be the space of
Morse functions on $M$ such that at least $\chi(M)+1$ critical points of each
function of $F$ are labeled by different labels (enumerated). Endow the space
$F$ with $C^\infty$-topology. We prove the homotopy equivalence $F\sim
R\times{\widetilde{\cal M}}$ where $R$ is one of the manifolds ${\mathbb
R}P^3$, $S^1\times S^1$ and the point in dependence on the sign of $\chi(M)$,
and ${\widetilde{\cal M}}$ is the universal moduli space of framed Morse
functions, which is a smooth stratified manifold.

Given a free group F and a compact Lie group K we have shown that Hom(F,K)/K
is homotopy equivalent to Hom(F,G)//G, where G is the complexification of K. In
this paper we first generalize this result to the moduli spaces of G-valued and
K-valued quiver representations. We then establish a general criterion for the
moduli of quiver representations with relations to admit such a deformation
retraction by pinching vertices.

The Birman exact sequence describes the effect on the mapping class group of
a surface with boundary of gluing discs to the boundary components. We
construct an analogous exact sequence for the automorphism group of a free
group. For the mapping class group, the kernel of the Birman exact sequence is
a surface braid group. We prove that in the context of the automorphism group
of a free group, the natural kernel is finitely generated.

A functorial semi-norm on singular homology is a collection of semi-norms on
the singular homology groups of spaces such that continuous maps between spaces
induce norm-decreasing maps in homology. Functorial semi-norms can be used to
give constraints on the possible mapping degrees of maps between oriented
manifolds.

Given a finite simplicial complex L and a collection of pairs of spaces
indexed by its vertex set, one can define their polyhedral product. We record a
simple formula for its Euler characteristic. In special cases the formula
simplifies further to one involving the h-polynomial of L.

The "polyhedral product functor" produces a space from a simplicial complex L
and a collection of pairs of spaces, {(A(i),B(i))}, where i ranges over the
vertex set of L. We give necessary and sufficient conditions for the resulting
space to be aspherical. There are two similar constructions, each of which
starts with a space X and a collection of subspaces, {X_i} and then produces a
new space. We give conditions for the results of these constructions to be
aspherical. All three techniques can be used to produce examples of closed
aspherical manifolds.

The triple linking number of an oriented surface link was defined as an
analogical notion of the linking number of a classical link. We consider a
certain $m$-component $T^2$-link ($m \geq 3$) determined from two commutative
pure $m$-braids $a$ and $b$. We present the triple linking number of such a
$T^2$-link, by using the linking numbers of the closures of $a$ and $b$. This
gives a lower bound of the triple point number. In some cases, we can determine
the triple point number, which is a multiple of four.

We present a general theory of fractal transformations and show how it leads
to a new type of method for filtering and transforming digital images. This
work substantially generalizes earlier work on fractal tops. The approach
involves fractal geometry, chaotic dynamics, and an interplay between discrete
and continuous representations. The underlying mathematics is established and
applications to digital imaging are described and exemplified.

We compute the fundamental group of various spaces of Desargues
configurations in complex projective spaces: planar and non-planar
configurations, with a fixed center and also with an arbitrary center.

Homogeneous links were introduced by Peter Cromwell, who proved that the
projection surface of these links, that given by the Seifert algorithm, has
minimal genus. Here we provide a different proof, with a geometric rather than
combinatorial flavor. To do this, we first show a direct relation between the
Seifert matrix and the decomposition into blocks of the Seifert graph.
Precisely, we prove that the Seifert matrix can be arranged in a block
triangular form, with small boxes in the diagonal corresponding to the blocks
of the Seifert graph.

Let $M$ be a closed 4-manifold with a free circle action. If the orbit
manifold $N^3$ satisfies an appropriate fibering condition, then we show how to
represent a cone in $H^2(M;\R)$ by symplectic forms. This generalizes earlier
constructions by Thurston, Bouyakoub and Fern\'andez-Gray-Morgan. In the case
that $M$ is the product 4-manifold $S^1\times N$ our construction complements
the results of \cite{FV08} (arXiv:0805:1234 [math.GT]) and allows us to
completely determine the symplectic cone of such 4-manifolds.

We generalize the Morton-Franks-Williams inequality to the colored
$\mathfrak{sl}(N)$ link homology defined in arXiv:0907.0695, which gives
infinitely many new bounds for the braid index and the self linking number. A
key ingredient of our proof is a composition product for the general MOY graph
polynomial, which generalizes that of Wagner arXiv:math/0702230v1.

A polygon in the hyperbolic plane is cyclic if a single circle contains all
of its vertices; we will say it is "centered" if in addition its interior
contains the center of this circle. We give necessary and sufficient conditions
for a set of real numbers to be the side length collection of a cyclic or
centered polygon. A cyclic polygon is uniquely determined by its collection of
side lengths; its vertex angles vary as C^1 functions of side lengths; and so
does the radius of the circle containing its vertices.

The first part of this paper completes the classification of Whitney towers
in the 4-ball that was started in three related papers. We provide an algebraic
framework allowing the computations of the graded groups associated to
geometric filtrations of classical link concordance by order $n$ (twisted)
Whitney towers in the 4-ball. Higher-order Sato-Levine invariants and
higher-order Arf invariants are defined and shown to be the obstructions to
framing a twisted Whitney tower.

We review the representation theory of the quantum group $U_\epsilon
sl_2\mathbb{C}$ at a root of unity $\epsilon$ of odd order, focusing on
geometric aspects related to the 3-dimensional quantum hyperbolic field
theories (QHFT). Our analysis relies on the quantum coadjoint action of De
Concini-Kac-Procesi, and the theory of Heisenberg doubles of Poisson-Lie groups
and Hopf algebras.

To each three-component link in the 3-sphere, we associate a geometrically
natural characteristic map from the 3-torus to the 2-sphere, and show that the
pairwise linking numbers and Milnor triple linking number that classify the
link up to link homotopy correspond to the Pontryagin invariants that classify
its characteristic map up to homotopy. This can be viewed as a natural
extension of the familiar fact that the linking number of a two-component link
in 3-space is the degree of its associated Gauss map from the 2-torus to the
2-sphere.

We prove the following

Theorem: Let X be a nonempty compact metrizable space, let $l_1 \leq l_2
\leq...$ be a sequence of natural numbers, and let $X_1 \subset X_2 \subset...$
be a sequence of nonempty closed subspaces of X such that for each k in N,
$dim_{Z/p} X_k \leq l_k < \infty$. Then there exists a compact metrizable space
Z, having closed subspaces $Z_1 \subset Z_2 \subset...$, and a surjective
cell-like map $\pi: Z \to X$, such that for each k in N,

(a) $dim Z_k \leq l_k$,

(b) $\pi (Z_k) = X_k$, and

In this paper we define the incidence matrix of a link diagram via its signed
planar graph and its dual graph. With a recent result of Ayaka Shimizu, we show
that a link diagram has one component if and only if the $\mathbb{Z}_2$-rank of
its incidence matrix exactly equals to the crossing number of the diagram. By
studying the effect of region crossing changes on 2-component link diagram we
show that region crossing change on 2-component link diagram is an unknotting
operation if and only if the linking number of the diagram is even.

This paper is an introduction to virtual knot theory and an exposition of new
ideas and constructions, including the parity bracket polynomial, the arrow
polynomial, the parity arrow polynomial and categorifications of the arrow
polynomial. The paper is relatively self-contained and it describes virtual
knot theory both combinatorially and in terms of the knot theory in thickened
surfaces. The arrow polynomial (of Dye and Kauffman) is a natural
generalization of the Jones polynomial, obtained by using the oriented
structure of diagrams in the state sum.

For a knot $K$ the cube number is a knot invariant defined to be the smallest
$n$ for which there is a cube diagram of size $n$ for $K$. There is also a
Legendrian version of this invariant called the \emph{Legendrian cube number}.
We will show that the Legendrian cube number distinguishes the Legendrian left
hand torus knots with maximal Thurston-Bennequin number and maximal rotation
number from the Legendrian left hand torus knots with maximal
Thurston-Bennequin number and minimal rotation number.

Oriented closed curves on an orientable surface with boundary are described
up to continuous deformation by reduced cyclic words in the generators of the
fundamental group and their inverses. By self-intersection number one means the
minimum number of transversal self-intersection points of representatives of
the class. We prove that if a class is chosen at random from among all classes
of $m$ letters, then for large $m$ the distribution of the self-intersection
number approaches the Gaussian distribution.

The WRT invariant of a link L in S2xS1 at sufficiently high values of the
level r can be expresses as an evaluation of a special polynomial invariant of
L at 2r-th root of unity. We categorify this polynomial invariant by
associating to L a bigraded homology whose graded Euler characteristic is equal
to this polynomial. If L is presented as a closure of a tangle in S2xS1, then
the homology of L is defined as the Hochschild homology of the H_n-bimodule
associated to the tangle by M. Khovanov.

In 2000, Croke and Kleiner showed that a CAT(0) group G can admit more than
one boundary. This contrasted with the situation for word hyperbolic groups,
where it was well-known that each such group admitted a unique boundary---in a
very stong sense. Prior to Croke and Kleiner's discovery, it had been observed
by Geoghegan and Bestvina that a weaker sort of uniquness does hold for
boundaries of torsion free CAT(0) groups; in particular, any two such
boundaries always have the same shape. Hence, the boundary really does carry
significant information about the group itself.

This article deals with a natural metric induced by smooth cobordisms between
links. We will show that the cobordism distance of torus links is composed of a
quadratic term predicted by the Thom conjecture and a linear error term. It
turns out that the cobordism distance is determined by the profiles of the
signature functions of torus links, up to a constant factor depending only on
the braid index.

This paper deals with the relation between several classical and well-known
objects: triangle Fuchsian groups, quasi-homogeneous singularities of plane
curves, torus knot complements in the 3-sphere. Torus knots are the only
nontrivial knots whose complements admit transitive Lie group actions. In fact
S^3\K_{p,q} is diffeomorphic to a coset space of the universal covering group
of PSL_2(R) with respect to a discrete subgroup G contained in the preimage of
a (p,q,\infty)-triangle Fuchsian group.

In this paper we introduce a representation of a embedded knotted (sometimes
Lagrangian) tori in $\BR^4$ called a hypercube diagram, i.e., a 4-dimensional
cube diagram. We prove the existence of hypercube homology that is invariant
under 4-dimensional cube diagram moves, a homology that is based on knot Floer
homology. We provide examples of hypercube diagrams and hypercube homology,
including using the new invariant to distinguish (up to cube moves) two "Hopf
linked" tori.

In this paper, we study the existence of high dimensional closed smooth
manifolds whose rational homotopy type resembles that of a projective plane.
Applying rational surgery, the problem can be reduced to finding possible
Pontryagin numbers satisfying the Hirzebruch signature formula and a set of
congruence relations, which turns out to be equivalent to finding solutions to
a system of Diophantine equations.

We give a Khovanov homology proof that hyperbolic twist knots do not admit
non-trivial Dehn surgeries with finite fundamental group.

We define the action of the homology group $H_1(M,\partial M)$ on the sutured
Floer homology $SFH(M,\gamma)$. It turns out that the contact invariant
$EH(M,\gamma,\xi)$ is usually sent to zero by this action. This fact allows us
to refine an earlier result proved by Ghiggini and the author. As a corollary,
we classify knots in $#^n(S^1\times S^2)$ which have simple knot Floer homology
groups: They are essentially the Borromean knots.

We use basic results from shape theory to investigate inverse systems of
covers and the corresponding fundamental pro-groups. Many of the standard
shape-theoretic definitions become simpler in the context of covering systems
and filtered groups, and we develop the theory largely within this context. We
give several applications, including a classification of maps between P-adic
solenoids up to homotopy.

In this note, we introduce a class of cell decompositions of PL manifolds and
polyhedra which are more general than triangulations yet not as general as CW
complexes; we propose calling them PLCW complexes. The main result is an analog
of Alexander's theorem: any two PLCW decompositions of the same polyhedron can
be obtained from each other by a sequence of certain "elementary" moves.

This definition is motivated by the needs of Topological Quantum Field
Theory, especially extended theories as defined by Lurie.

We exhibit an encoding of knots into processes in the {\pi}-calculus such
that knots are ambient isotopic if and only their encodings are weakly
bisimilar.

Homology of the circle with non-trivial local coefficients is trivial. From
this well-known fact we deduce geometric corollaries concerning links of
codimension two. In particular, the Murasugi-Tristram signatures are extended
to invariants of links formed of arbitrary oriented closed codimension two
submanifolds of an odd-dimensional sphere. The novelty is that the submanifolds
are not assumed to be disjoint, but are transversal to each other, and the
signatures are parametrized by points of the whole torus.

We introduce three spectral sequences which give some expressions of colored
Jones polynomials. Each spectral sequence contains a Khovanov-type homology
groups. Two of them are derived from a bicomplex of the colored Jones
polynomial. The other is the spectral sequence that deduces a colored Rasmussen
invariant of links. We also introduce three functors between categories of
nanophrases, generalizations of links, and obtain their applications using
colored Jones polynomials and their categorifications.

With the aid of a computer, we provide a motion picture of the twist-spun
trefoil which exhibits the periodicity well.

We prove that while there are maps $\bT^4\to\#^3(\bS^2\times\bS^2)$ of
arbitrarily large degree, there is no branched cover from $4$-torus to
$\#^3(\bS^2\times \bS^2)$. More generally, we obtain that, as long as $N$
satisfies a suitable cohomological condition, any $\pi_1$-surjective branched
cover $\bT^n \to N$ is a homeomorphism.

This is the second of five papers that construct an isomorphism between the
Seiberg-Witten Floer homology and the Heegaard Floer homology of a given
compact, oriented 3-manifold. The isomorphism is given as a composition of
three isomorphisms; the first of these relates a version of embedded contact
homology on an an auxillary manifold to the Heegaard Floer homology on the
original. This paper describes this auxilliary manifold, its geometry, and the
relationship between the generators of the embedded contact homology chain
complex and those of the Heegaard Floer chain complex.

We determine for which $m$, the complete graph $K_m$ has an embedding in
$S^3$ whose topological symmetry group is isomorphic to one of the polyhedral
groups: $A_4$, $A_5$, or $S_4$.

The Lawrence-Krammer-Bigelow representation of the braid group arises from
the monodromy representation on the twisted homology of the fiber of a certain
fiber bundle in which the base and total space are complements of braid
arrangements, and the fiber is the complement of a discriminantal arrangement.
We present a more general version of this construction and use it to construct
nontrivial bundles on the complement of an arbitrary arrangement \A\ whose
fibers are products of discriminantal arrangements.

Let $M$ be a smooth closed orientable surface and $F=F_{p,q,r}$ be the space
of Morse functions on $M$ having exactly $p$ critical points of local minima,
$q\ge1$ saddle critical points, and $r$ critical points of local maxima,
moreover all the points are fixed. Let $F_f$ be the connected component of a
function $f\in F$ in $F$. By means of the winding number introduced by Reinhart
(1960), a surjection $\pi_0(F)\to{\mathbb Z}^{p+r-1}$ is constructed.

We prove that the group-homological version of the generalized Goncharov
invariant of finite-volume locally rank one symmetric spaces determines their
generalized Neumann-Yang invariant, which is defined using ideal fundamental
cycles.

We establish a necessary and sufficient condition for a heptagonal knot to be
figure-8 knot. The condition is described by a set of Radon partitions formed
by vertices of the heptagon. In addition we relate this result to the number of
nontrivial heptagonal knots in linear embeddings of the complete graph $K_7$
into $\mathbb{R}^3$.

We review several results related to the characterization of polyhedra in
hyperbolic 3-space. In particular we present Rivin's theorem that gives a
characterization of compact convex hyperbolic polyhedra, and Hodgson's proof of
the Adreev's theorem. We also review the analogous characterization of ideal
polyhedra, and give a family of counter-examples that proves that hyperbolic
polyhedra are not determined by edge lengths.

The problem of determining minimal dilatations of pseudo-Anosov mapping
classes was introduced by Penner who proved that the logarithm of the minimal
dilatations on closed surfaces behaved like the inverse of the Euler
characteristic. These results have been expanded to asymptotic behavior of
minimal dilatations of punctured spheres by Hironaka, Kin, and Tsai and genus 1
surfaces by Tsai. For fixed genus Tsai found that the logarithm of the minimal
dilatations behaved differently and asked the question: what is the asymptotic
behavior for other lines in the gn-plane?

A family of coordinates $\psi_h$ for the Teichm\"uller space of a compact
surface with boundary was introduced in \cite{l2}. In the work \cite{m1},
Mondello showed that the coordinate $\psi_0$ can be used to produce a natural
cell decomposition of the Teichm\"uller space invariant under the action of the
mapping class group. In this paper, we show that the similar result also works
for all other coordinate $\psi_h$ for any $h \geq 0$.

Let M be a smooth connected orientable compact surface. Denote by F(M,S^1)
the space of all Morse functions f:M-->S^1 having no critical points on the
boundary of M and such that for every boundary component V of M, the
restriction f|V:V-->S^1 is either a constant map or a covering map. Endow
F(M,S^1) with the C^{\infty}-topology. In this note the connected components of
F(M,S^1) are classified. This result extends the results of S.V.Matveev,
V.V.Sharko, and the author for the case of Morse functions being locally
constant on the boundary of M.

We construct knot invariants categorifying the quantum knot variants for all
representations of quantum groups. We show that these invariants coincide with
previous invariants defined by Khovanov for sl_2 and sl_3 and by
Mazorchuk-Stroppel and Sussan for sl_n. We also suggest an approach to showing
that these knot homologies are functorial. Our technique uses categorifications
of the tensor products of integrable representations of Kac-Moody algebras and
quantum groups, constructed a prequel to this paper.

In this paper we prove another pairing theorem for bordered Floer homology.
Unlike the original pairing theorem, this one is stated in terms of
homomorphisms, not tensor products. The present formulation is closer in spirit
to the usual TQFT framework, and allows a more direct comparison with
Fukaya-categorical constructions. The result also leads to various dualities in
bordered Floer homology.

We consider a knot homotopy as a cylinder in 4-space. An ordinary triple
point $p$ of the cylinder is called {\em coherent} if all three branches
intersect at $p$ pairwise with the same index. A {\em triple unknotting} of a
classical knot $K$ is a homotopy which connects $K$ with the trivial knot and
which has as singularities only coherent triple points. We give a new formula
for the first Vassiliev invariant $v_2(K)$ by using triple unknottings. As a
corollary we obtain a very simple proof of the fact that passing a coherent
triple point always changes the knot type.

Knot Floer homology is an invariant for knots in the three-sphere for which
the Euler characteristic is the Alexander-Conway polynomial of the knot. The
aim of this paper is to study this homology for a class of satellite knots, so
as to see how a certain relation between the Alexander-Conway polynomials of
the satellite, companion and pattern is generalized on the level of the knot
Floer homology. We also use our observations to study a classical geometric
invariant, the Seifert genus, of our satellite knots.

The enumeration of normal surfaces is a key bottleneck in computational
three-dimensional topology. The underlying procedure is the enumeration of
admissible vertices of a high-dimensional polytope, where admissibility is a
powerful but non-linear and non-convex constraint.

We construct a triangulation of a compactification of the Moduli space of a
surface with at least one puncture that is closely related to the
Deligne-Mumford compactification. Specifically, there is a surjective map from
the compactification we construct to the Deligne-Mumford compactification so
that the inverse image of each point is contractible. In particular our
compactification is homotopy equivalent to the Deligne-Mumford
compactification.

A singular point of a smooth map F: M -> N of manifolds is a point in M at
which the rank of the differential dF is less than the minimum of dimensions of
M and N. The classical invariant of the set S of singular points of F of a
given type is defined by taking the fundamental class [\bar{S}]\in H_*(M) of
the closure of S. We introduce and study new invariants of singular sets for
which the classical invariants may not be defined, i.e., for which \bar{S} may
not possess the fundamental class.

We compute the $p$-primary components of the linking pairings of orientable
3-manifolds admitting a fixed-point free $S^1$-action. Using this, we show that
any non-singular linking pairing on a finite abelian group with homogeneous
2-primary summand is realized by such a manifold. However, some pairings on
inhomogeneous 2-groups are not realizable by any Seifert fibred 3-manifold.

A triangulation of a surface is called $q$-equivelar if each of its vertices
is incident with exactly $q$ triangles. In 1972 Altshuler had shown that an
equivelar triangulation of torus has a Hamiltonian Circuit. Here we present a
necessary and sufficient condition for existence of a contractible Hamiltonian
Cycle in equivelar triangulation of a surface.

We describe a diagonal condition on the Khovanov complex of tangles, show
that this condition is satisfied by the Khovanov complex of the single crossing
tangles, and prove that it is preserved by alternating planar algebra
compositions. Hence, this condition is satisfied by the Khovanov complex of all
alternating tangles. Finally, in the case of 0-tangles, that is links, our
condition is equivalent to a well known result which states that the Khovanov
homology of a non-split alternating link is supported in two lines.

The volume conjecture states that for a hyperbolic knot K in the three-sphere
S^3 the asymptotic growth of the colored Jones polynomial of K is governed by
the hyperbolic volume of the knot complement S^3\K. The conjecture relates two
topological invariants, one combinatorial and one geometric, in a very
nonobvious, nontrivial manner.

We associate to a CAT(0)-space a flow space that can be used as the
replacement for the geodesic flow on the sphere tangent bundle of a Riemannian
manifold. We use this flow space to prove that CAT(0)-group are transfer
reducible over the family of virtually cyclic groups. This result is an
important ingredient in our proof of the Farrell-Jones Conjecture for these
groups.

The complete classification of representations of the Trefoil knot group G in
S^{3} and SL(2,R), their affine deformations, and some geometric
interpretations of the results, are given. Among other results, we also obtain
the classification up to conjugacy of the non cyclic groups of affine Euclidean
isometries generated by two isometries $\mu$ and $\nu$ such that
$\mu^{2}=\nu^{3}=1$, in particular those which are crystallographic. We also
prove that there are no affine crystallographic groups in the three dimensional
Minkowski space which are quotients of G.

In this paper we give a complete description of the set of discrete faithful
representations SH(M) uniformizing a compact, orientable, hyperbolizable
3-manifold M with incompressible boundary, equipped with the strong topology,
with the description given in term of the end invariants of the quotient
manifolds. As part of this description, we introduce coordinates on SH(M) that
extend the usual Ahlfors-Bers coordinates. We use these coordinates to show the
local connectivity of SH(M) and study the action of the modular group of M on
SH(M).

These are notes based on a series of talks that the author gave at the
"Interactions between hyperbolic geometry and quantum groups" conference held
at Columbia University in June of 2009.

We introduce Fenchel-Nielsen coordinates on Teicm\"uller spaces of surfaces
of infinite type. The definition is relative to a given pair of pants
decomposition of the surface. We start by establishing conditions under which
any pair of pants decomposition on a hyperbolic surface of infinite type can be
turned into a geometric decomposition, that is, a decomposition into hyperbolic
pairs of pants. This is expressed in terms of a condition we introduce and
which we call Nielsen convexity. This condition is related to Nielsen cores of
Fuchsian groups.

We show that if K is a non-trivial knot inside a homology sphere X, the rank
of the knot Floer homology group associated with K is strictly bigger than the
rank of the Heegaard Floer homology group associated with X.

Bordered Heegaard Floer homology is a three-manifold invariant which
associates to a surface F an algebra A(F) and to a three-manifold Y with
boundary identified with F a module over A(F). In this paper, we establish
naturality properties of this invariant. Changing the diffeomorphism between F
and the boundary of Y tensors the bordered invariant with a suitable bimodule
over A(F). These bimodules give an action of a suitably based mapping class
group on the category of modules over A(F).

This paper initiates a systematic study of the relation of commensurability
of surface automorphisms, or equivalently, fibered commensurability of
3-manifolds fibering over the circle. We show that every hyperbolic fibered
commensurability class contains a unique minimal element, whereas the class of
Seifert manifolds fibering over the circle consists of a single
commensurability class with infinitely many minimal elements. The situation for
non-geometric manifolds is more complicated, and we illustrate a range of
phenomena that can occur in this context.

Following an example discovered by John Berge, we show that there is a
4-component link L \subset (S^1 x S^2)#(S^1 x S^2) so that, generically, the
result of Dehn surgery on L is a 3-manifold with two inequivalent genus 2
Heegaard splittings, and each of these Heegaard splittings is of Hempel
distance 3.

We will construct differential forms on the embedding spaces using
configuration space integral associated with 1-loop graphs. In particular, for
any pair n, j of positive integers with n-j>=2, we show that some linear
combination of such differential forms together with some correction terms
coming from a 1-parameter family of immersions gives a closed form on the space
fEmb(R^j,R^n) of embeddings with 1-parameter families of immersions to the
trivial. We also show that the closed forms obtained detect nontrivial elements
of the real homology of the embedding space fEmb(R^j,R^n).

The minimum number of colors is a challenging knot invariant since, by
definition, its calculation requires taking the minimum over infinitely many
minima. In this article we estimate and in some cases calculate the minimum
number of colors for the Turk's head knots on three strands.

We study properties of the signature function of the torus knot $T_{p,q}$.
First we provide a very elementary proof of the formula for the integral of the
signatures over the circle. We obtain also a closed formula for the
Tristram--Levine signature of a torus knot in terms of Dedekind sums.

Let X be a locally compact Polish space and G a non-discrete Polish ANR
group. By C(X,G), we denote the topological group of all continuous maps f:X
\to G endowed with the Whitney (graph) topology and by C_c(X,G) the subgroup
consisting of all maps with compact support. It is known that if X is compact
and non-discrete then the space C(X,G) is an l_2-manifold.

In a previous paper we introduced a notion of "genericity" for countable sets
of curves in the curve complex of a surface S, based on the Lebesgue measure on
the space of projective measured laminations in S. With this definition we
prove that for each fixed g > 1 the set of irreducible genus g Heegaard
splittings of high distance is generic, in the set of all irreducible Heegaard
splittings. Our definition of "genericity" is different and more intrinsic then
the one given via random walks.

We introduce and study knotoids. Knotoids are represented by diagrams in a
surface which differ from the usual knot diagrams in that the underlying curve
is a segment rather than a circle. Knotoid diagrams are considered up to
Reidemeister moves applied away from the endpoints of the underlying segment.
We show that knotoids in $S^2$ generalize knots in $S^3$ and study the
semigroup of knotoids. We also discuss applications to knots and invariants of
knotoids.

Regular integer lattices are characterized by k unit vectors that build up
their generator matrices. These have rank k for D-lattices, and are
rank-deficient for A-lattices, E_6 and E_7. We count lattice points inside
hypercubes centered at the origin for all three types, as if classified by
maximum infinity norm in the host lattice. The results assume polynomial format
as a function of the hypercube edge length.

We show that for every simple closed curve \alpha, the extremal length and
the hyperbolic length of \alpha are quasi-convex functions along any
Teichmuller geodesic. As a corollary, we conclude that, in Teichmuller space
equipped with the Teichmuller metric, balls are quasi- convex.

We show that sub-surfaces of a Heegaard surface for which the relative Hempel
distance of the splitting is sufficiently high have to appear in any Heegaard
surface of genus bounded by half that distance.

Let $P, Q$ be Heegaard surfaces of a closed orientable 3-manifold. In this
paper, we introduce a method for giving an upper bound of (Hempel) distance of
$P$ by using the Reeb graph derived from a certain horizontal arc in the
ambient space $[0,1]\times[0,1]$ of the Rubinstein-Scharlemann graphic derived
from $P$ and $Q$. This is a refinement of a part of Johnson's arguments used
for determining stable genera required for flipping high distance Heegaard
splittings.

We construct an equivariant colored sl(N)-homology for links, which
generalizes both the colored sl(N)-homology defined by the author and the
equivariant sl(N)-homology defined by Krasner. The construction is a
straightforward generalization of that of the colored sl(N)-homology. The proof
of invariance is based on a simple observation which allows us to translate the
proof of the invariance of the colored sl(N)-homology into the new setting.

This paper gives the first explicit, two-sided estimates on the cusp area of
once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link
complements. The input for these estimates is purely combinatorial data coming
from the Farey tesselation of the hyperbolic plane. The bounds on cusp area
lead to explicit bounds on the volume of Dehn fillings of these manifolds, for
example sharp bounds on volumes of hyperbolic closed 3-braids in terms of the
Schreier normal form of the associated braid word.

In this paper, we investigate the structure of the Gardiner-Masur boundary of
Teichmuller space. Indeed, we will give a geometric description of boundary
comparing to the Duchin-Leininger-Rafi compactification of the space of
singular flat structures. We will obtain the coincidence between the
Gardiner-Masur boundary and the Thurston boundary at the projective classes of
uniquely ergodic measured foliations. We also study the action of the mapping
class group on the Gardiner-Masur boundary and characterize the elements by
fixed points.

We show that the nearest point retraction is a uniform quasi-isometry from
the Thurston metric on a hyperbolic domain in the Riemann sphere to the
boundary of the convex hull of its complement. As a corollary, one obtains
explicit bounds on the quasi-isometry constant of the nearest point retraction
with respect to the Poincare metric when the domain is uniformly perfect. We
also establish Marden and Markovic's conjecture that a hyperbolic domain is
uniformly perfect if and only if the nearest point retraction is Lipschitz with
respect to the Poincare metric.

We give an algorithmic proof of the theorem that an orientable irreducible
and atoroidal 3-manifold has only finitely many Heegaard splittings in each
genus, up to isotopy. The proof gives an algorithm to determine the Heegaard
genus of an atoroidal 3-manifold.

We prove the existence of a polynomial invariant that satisfies the HOMFLY
skein relation for links in a lens space. In the process we also develop a
skein theory of toroidal grid diagrams in a lens space.

In view of the Segal construction each category with operation gives rise to
a cohomology theory. We show that similarly each open stable differential
relation R determines cohomology theories k^* of solutions and h^* of stable
formal solutions of R. We prove that k^* and h^* are equivalent under a mild
condition.

In this article we study a partial ordering on knots in the 3-sphere where
K_1 is greater than or equal to K_2 if there is an epimorphism from the knot
group of K_1 onto the knot group of K_2 which preserves peripheral structure.
If K_1 is a 2-bridge knot and K_1 > K_2, then it is known that K_2 must also be
2-bridge. Furthermore, Ohtsuki, Riley, and Sakuma give a construction which,
for a given 2-bridge knot K_{p/q}, produces infinitely 2-bridge knots K_{p'/q'}
with K_{p'/q'}>K_{p/q}.

This paper initiates the study of topological arbiters, a concept rooted in
Poincare-Lefschetz duality. Given an n-dimensional manifold W, a topological
arbiter associates a value 0 or 1 to codimension zero submanifolds of W,
subject to natural topological and duality axioms. For example, there is a
unique arbiter on $RP^2$, which reports the location of the essential 1-cycle.
In contrast, we show that there exists an uncountable collection of topological
arbiters in dimension 4.

We give bounds on knot signature, the Ozsvath-Szabo tau invariant, and the
Rasmussen s invariant in terms of the Turaev genus of the knot.

We consider two well known constructions of link invariants. One uses skein
theory: you resolve each crossing of the link as a linear combination of things
that don't cross, until you eventually get a linear combination of links with
no crossings, which you turn into a polynomial.

Given a harmonic measure of a hyperbolic lamination on a compact metric
space, a positive harmonic function is defined on the universal cover of a
typical leaves. We discuss some properties of this function. Especially if all
the leaves are hyperbolic, ergodic harmonic measures are divided into two
classes.

We prove that a transversely equicontinuous minimal lamination on a locally
compact metric space $Z$ has a transversely invariant Radon measure. Moreover
if the space $Z$ is compact, then the tranversely invariant Radon measure is
shown to be unique up to a scaling.

Garoufalidis conjectured a relation between the boundary slopes of a knot and
its colored Jones polynomials. According to the conjecture, certain boundary
slopes are detected by the sequence of degrees of the colored Jones
polynomials. We verify this conjecture for adequate knots, a class that vastly
generalizes that of alternating knots.

It is well known that the description of topological and geometric properties
of bisectors in normed spaces is a non-trivial subject. In this paper we
introduce the concept of bounded representation of bisectors in finite
dimensional real Banach spaces. This useful notion combines the concepts of
bisector and shadow boundary of the unit ball, both corresponding with the same
spatial direction.

J. Przytycki has established a connection between the Hochschild homology of
an algebra $A$ and the chromatic graph homology of a polygon graph with
coefficients in $A$. In general the chromatic graph homology is not defined in
the case where the coefficient ring is a non-commutative algebra. In this paper
we define a new homology theory for directed graphs which takes coefficients in
an arbitrary $A-A$ bimodule, for $A$ possibly non-commutative, which on
polygons agrees with Hochschild homology through a range of dimensions.

The W-polynomial is applied in two ways to questions involving the Kauffman
bracket of some families of links. First we find a geometric property of a link
diagram, which is less than or equal to the twist number, that bounds the
Mahler measure of the Kauffman bracket. Second we find a general form for the
Kauffman bracket of a link found by surgering in a single rational tangle along
n unlinked components, all in a particular annulus. We then give a condition
under which the Mahler measure of the Kauffman bracket of such families
diverges. We give examples of the condition in action.

A Chebyshev curve $\cC(a,b,c,\phi)$ has a parametrization of the form $
x(t)=T_a(t); y(t)=T_b(t) ; z(t)= T_c(t + \phi), $ where $a,b,c$ are integers,
$T_n(t)$ is the Chebyshev polynomial of degree $n$ and $\phi \in \RR$. When
$\cC(a,b,c,\phi)$ has no double points, it defines a polynomial knot. We
determine all possible knots when $a$, $b$ and $c$ are given.

We study the relationship between three different approaches to the action of
a pseudo-Anosov mapping class $[F]$ on a surface: the original theorem of
Thurston, its algorithmic proof by Bestvina-Handel, and related investigations
of Penner-Harer.

In this paper, we obtain analogues of Jorgensen's inequality for
non-elementary groups of isometries of quaternionic hyperbolic $n$-space
generated by two elements, one of which is loxodromic. Our result gives some
improvement over earlier results of Kim [10] and Markham [15]}. These results
also apply to complex hyperbolic space and give improvements on results of
Jiang, Kamiya and Parker [7]

The main goal of this paper is to discuss a symplectic interpretation of
Lipshitz, Ozsvath and Thurston's bordered Heegaard-Floer homology in terms of
Fukaya categories of symmetric products and Lagrangian correspondences. More
specifically, we give a description of the algebra A(F) which appears in the
work of Lipshitz, Ozsvath and Thurston in terms of (partially wrapped) Floer
homology for product Lagrangians in the symmetric product, and outline how
bordered Heegaard-Floer homology itself can conjecturally be understood in this
language.

In this paper we introduce the framed pure braid group on $n$ strands of an
oriented surface, a topological generalisation of the pure braid group $P_n$.
We give different equivalents definitions for framed pure braid groups and we
study exact sequences relating these groups with other generalisations of
$P_n$, usually called surface pure braid groups. The notion of surface framed
braid groups is also introduced.