1. CJM Online first
 Mihara, Tomoki

Cohomological Approach to Class Field Theory in Arithmetic Topology
We establish class field theory for $3$dimensional manifolds
and knots. For this purpose, we formulate analogues of the multiplicative
group, the idÃ¨le class group, and ray class groups in a cocycletheoretic
way. Following the arguments in abstract class field theory,
we construct reciprocity maps and verify the existence theorems.
Keywords:arithmetic topology, class field theory, branched covering, knots and prime numbers Categories:11Z05, 18F15, 55N20, 57P05 

2. CJM Online first
 Scaduto, Christopher W.; Stoffregen, Matthew

The mod two cohomology of the moduli space of rank two stable bundles on a surface and skew Schur polynomials
We compute cup product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function for certain skew Schur polynomials. As an application, we compute the nilpotency degree of a distinguished degree two generator in the mod two cohomology ring. We then give descriptions of the mod two cohomology rings in low genus, and describe the subrings invariant under the mapping class group action.
Keywords:stable bundle, mod two cohomology, skew schur polynomial Categories:14D20, 57R58 

3. CJM 2018 (vol 70 pp. 983)
 Conway, Anthony

An Explicit Computation of the Blanchfield Pairing for Arbitrary Links
Given a link $L$, the Blanchfield pairing $\operatorname{Bl}(L)$ is a pairing
which is defined on the torsion submodule of the Alexander module
of $L$. In some particular cases, namely if $L$ is a boundary
link or if the Alexander module of $L$ is torsion, $\operatorname{Bl}(L)$
can be computed explicitly; however no formula is known in general.
In this article, we compute the Blanchfield pairing of any link,
generalizing the aforementioned results. As a corollary, we obtain
a new proof that the Blanchfield pairing is hermitian. Finally,
we also obtain short proofs of several properties of $\operatorname{Bl}(L)$.
Keywords:link, Blanchfield pairing, Ccomplex, Alexander module Category:57M25 

4. CJM 2018 (vol 70 pp. 1130)
 Rushworth, William

Doubled Khovanov Homology
We define a homology theory of virtual links built out of the
direct sum of the standard Khovanov complex with itself, motivating
the name doubled Khovanov homology. We demonstrate that
it can be used to show that some virtual links are nonclassical,
and that it yields a condition on a virtual knot being the connect
sum of two unknots. Further, we show that doubled Khovanov homology
possesses a perturbation analogous to that defined by Lee in
the classical case and define a doubled Rasmussen invariant.
This invariant is used to obtain various cobordism obstructions;
in particular it is an obstruction to sliceness. Finally, we
show that the doubled Rasmussen invariant contains the odd writhe
of a virtual knot, and use this to show that knots with nonzero
odd writhe are not slice.
Keywords:Khovanov homology, virtual knot concordance, virtual knot theory Categories:57M25, 57M27, 57N70 

5. CJM 2017 (vol 69 pp. 851)
 Pronk, Dorette; Scull, Laura

Erratum: Translation Groupoids and Orbifold Cohomology
We correct an error in the proof of a
lemma in
"Translation Groupoids and Orbifold Cohomology",
Canadian J. Math Vol 62 (3), pp 614645 (2010).
This error was pointed out to the authors
by Li Du of the GeorgAugustUniversitÃ¤t at Gottingen, who
also suggested the outline for the corrected proof.
Keywords:orbifold, equivariant homotopy theory, translation groupoid, bicategory of fractions Category:57S15 

6. CJM 2016 (vol 69 pp. 1201)
 Abe, Tetsuya; Tagami, Keiji

Characterization of Positive Links and the $s$invariant for Links
We characterize positive links in terms of strong quasipositivity,
homogeneity and the value of Rasmussen and BeliakovaWehrli's
$s$invariant.
We also study almost positive links,
in particular, determine the $s$invariants of
almost positive links.
This result suggests that all almost positive links might
be strongly quasipositive.
On the other hand, it implies that
almost positive links are never homogeneous links.
Keywords:knot, $s$invariant, positive link, almost positive link Categories:57M25, 57M27 

7. CJM 2016 (vol 69 pp. 767)
 Choi, Suyoung; Park, Hanchul

Wedge Operations and Torus Symmetries II
A fundamental idea in toric topology is that classes of manifolds
with wellbehaved torus actions (simply, toric spaces) are classified
by pairs of simplicial complexes and (nonsingular) characteristic
maps. The authors in their previous paper provided a new way
to find all characteristic maps on a simplicial complex $K(J)$
obtainable by a sequence of wedgings from $K$. The main idea
was that characteristic maps on $K$ theoretically determine all
possible characteristic maps on a wedge of $K$.
In this work, we further develop our previous work for classification
of toric spaces. For a starshaped simplicial sphere $K$ of dimension
$n1$ with $m$ vertices, the Picard number $\operatorname{Pic}(K)$ of $K$ is
$mn$. We refer to $K$ as a seed if $K$ cannot be obtained
by wedgings. First, we show that, for a fixed positive integer
$\ell$, there are at most finitely many seeds of Picard number
$\ell$ supporting characteristic maps. As a corollary, the conjecture
proposed by V.V. Batyrev in 1991 is solved affirmatively.
Second, we investigate a systematic method to find all characteristic
maps on $K(J)$ using combinatorial objects called (realizable)
puzzles that only depend on a seed $K$.
These two facts lead to a practical way to classify the toric
spaces of fixed Picard number.
Keywords:puzzle, toric variety, simplicial wedge, characteristic map Categories:57S25, 14M25, 52B11, 13F55, 18A10 

8. CJM 2016 (vol 68 pp. 1201)
 Banks, Jessica; Rathbun, Matt

Monodromy Action on Unknotting Tunnels in Fiber Surfaces
In \cite{RatTOFL}, the second author showed that a tunnel of a tunnel
number one, fibered link in $S^3$ can be isotoped to lie as a properly
embedded arc in the fiber surface of the link. In this paper, we
observe that this is true for fibered links in any 3manifold, we
analyze how the arc behaves under the monodromy action, and we show
that the tunnel arc is nearly clean, with the possible exception of
twisting around the boundary of the fiber.
Keywords:fibered, monodromy, tunnel, clean Category:57M25 

9. CJM 2016 (vol 68 pp. 445)
 Martins, Luciana de Fátima; Saji, Kentaro

Geometric Invariants of Cuspidal Edges
We give a normal form of the cuspidal edge
which uses only diffeomorphisms on the source
and isometries on the target.
Using this normal form, we study differential
geometric invariants of
cuspidal edges which determine them up to order three.
We also
clarify relations between these invariants.
Keywords:cuspidal edge, curvature, wave fronts Categories:57R45, 53A05, 53A55 

10. CJM 2016 (vol 68 pp. 655)
 Klartag, Bo'az; Kozma, Gady; Ralli, Peter; Tetali, Prasad

Discrete Curvature and Abelian Groups
We study a natural discrete Bochnertype inequality on graphs,
and explore its merit as a notion of ``curvature'' in discrete
spaces.
An appealing feature of this discrete version of the socalled
$\Gamma_2$calculus (of BakryÃmery) seems to be that it is
fairly
straightforward to compute this notion of curvature parameter
for
several specific graphs of interest  particularly, abelian
groups, slices of the hypercube, and the symmetric group under
various sets of generators.
We further develop this notion by deriving Busertype inequalities
(Ã la Ledoux), relating functional and isoperimetric constants
associated with a graph.
Our derivations provide a tight bound on the Cheeger constant
(i.e., the edgeisoperimetric constant) in terms of
the spectral gap, for graphs with nonnegative curvature, particularly,
the class of abelian Cayley graphs  a result of independent
interest.
Keywords:Ricci curvature, graph theory, abelian groups Categories:53C21, 57M15 

11. CJM 2015 (vol 68 pp. 3)
 Boden, Hans Ulysses; Curtis, Cynthia L

The SL$(2, C)$ Casson Invariant for Knots and the $\hat{A}$polynomial
In this paper, we extend the definition of the ${SL(2, {\mathbb C})}$ Casson
invariant
to arbitrary knots $K$ in integral homology 3spheres and relate
it to the $m$degree of the $\widehat{A}$polynomial of $K$. We
prove a product formula for the $\widehat{A}$polynomial of the connected
sum $K_1 \# K_2$ of two knots in $S^3$ and deduce additivity
of ${SL(2, {\mathbb C})}$ Casson knot invariant under connected sum for a large
class of knots in $S^3$. We also present an example of a nontrivial
knot $K$ in $S^3$ with trivial $\widehat{A}$polynomial and trivial
${SL(2, {\mathbb C})}$ Casson knot invariant, showing that neither of these invariants
detect the unknot.
Keywords:Knots, 3manifolds, character variety, Casson invariant, $A$polynomial Categories:57M27, 57M25, 57M05 

12. CJM 2014 (vol 67 pp. 152)
 Lescop, Christine

On Homotopy Invariants of Combings of Threemanifolds
Combings of compact, oriented $3$dimensional manifolds $M$ are
homotopy classes of nowhere vanishing vector fields.
The Euler class of the normal bundle is an invariant of the combing,
and it only depends on the underlying Spin$^c$structure. A combing
is called torsion
if this Euler class is a torsion element of $H^2(M;\mathbb Z)$. Gompf
introduced a $\mathbb Q$valued invariant $\theta_G$ of torsion combings
on closed $3$manifolds, and he showed that $\theta_G$ distinguishes
all torsion combings with the same Spin$^c$structure.
We give an alternative definition for $\theta_G$ and we express
its variation as a linking number. We define a similar invariant
$p_1$ of combings for manifolds bounded by $S^2$. We relate $p_1$
to the $\Theta$invariant, which is the simplest configuration
space integral invariant of rational homology $3$balls, by the
formula $\Theta=\frac14p_1 + 6 \lambda(\hat{M})$ where $\lambda$
is the CassonWalker invariant.
The article also includes a selfcontained presentation of combings
for $3$manifolds.
Keywords:Spin$^c$structure, nowhere zero vector fields, first Pontrjagin class, Euler class, Heegaard Floer homology grading, Gompf invariant, Theta invariant, CassonWalker invariant, perturbative expansion of ChernSimons theory, configuration space integrals Categories:57M27, 57R20, 57N10 

13. CJM 2013 (vol 66 pp. 141)
 CaillatGibert, Shanti; Matignon, Daniel

Existence of Taut Foliations on Seifert Fibered Homology $3$spheres
This paper concerns the problem of existence of taut foliations among $3$manifolds.
Since the contribution of David Gabai,
we know that closed $3$manifolds with nontrivial second homology group
admit a taut foliation.
The essential part of this paper focuses on Seifert fibered homology $3$spheres.
The result is quite different if they are integral or rational but nonintegral homology $3$spheres.
Concerning integral homology $3$spheres, we can see that all but the $3$sphere and the PoincarÃ© $3$sphere admit a taut foliation.
Concerning nonintegral homology $3$spheres,
we prove there are infinitely many which admit a taut foliation, and infinitely many without taut foliation.
Moreover, we show that the geometries do not determine the existence of taut foliations
on nonintegral Seifert fibered homology $3$spheres.
Keywords:homology 3spheres, taut foliation, Seifertfibered 3manifolds Categories:57M25, 57M50, 57N10, 57M15 

14. CJM 2013 (vol 66 pp. 453)
 Vaz, Pedro; Wagner, Emmanuel

A Remark on BMW algebra, $q$Schur Algebras and Categorification
We prove that the 2variable BMW algebra
embeds into an algebra constructed from the HOMFLYPT polynomial.
We also prove that the $\mathfrak{so}_{2N}$BMW algebra embeds in the $q$Schur algebra
of type $A$.
We use these results
to suggest a schema providing categorifications of the $\mathfrak{so}_{2N}$BMW algebra.
Keywords:tangle algebras, BMW algebra, HOMFLYPT Skein algebra, qSchur algebra, categorification Categories:57M27, 81R50, 17B37, 16W99 

15. CJM 2012 (vol 65 pp. 575)
 Kallel, Sadok; Taamallah, Walid

The Geometry and Fundamental Group of Permutation Products and Fat Diagonals
Permutation products and their various ``fat diagonal'' subspaces are
studied from the topological and geometric point of view. We describe
in detail the stabilizer and orbit stratifications related to the
permutation action, producing a sharp upper bound for its depth and
then paying particular attention to the geometry of the diagonal
stratum. We write down an expression for the fundamental group of any
permutation product of a connected space $X$ having the homotopy type
of a CW complex in terms of $\pi_1(X)$ and $H_1(X;\mathbb{Z})$. We then
prove that the fundamental group of the configuration space of
$n$points on $X$, of which multiplicities do not exceed $n/2$,
coincides with $H_1(X;\mathbb{Z})$. Further results consist in giving
conditions for when fat diagonal subspaces of manifolds can be
manifolds again. Various examples and homological calculations are
included.
Keywords:symmetric products, fundamental group, orbit stratification Categories:14F35, 57F80 

16. CJM 2011 (vol 64 pp. 102)
 Ishii, Atsushi; Iwakiri, Masahide

Quandle Cocycle Invariants for Spatial Graphs and Knotted Handlebodies
We introduce a flow of a spatial graph and see how invariants for
spatial graphs and handlebodylinks are derived from those for flowed
spatial graphs.
We define a new quandle (co)homology by introducing a subcomplex of the
rack chain complex.
Then we define quandle colorings and quandle cocycle invariants for
spatial graphs and handlebodylinks.
Keywords:quandle cocycle invariant, knotted handlebody, spatial graph Categories:57M27, 57M15, 57M25 

17. CJM 2010 (vol 63 pp. 436)
 Mine, Kotaro; Sakai, Katsuro

Simplicial Complexes and Open Subsets of NonSeparable LFSpaces
Let $F$ be a nonseparable LFspace homeomorphic to
the direct sum $\sum_{n\in\mathbb{N}} \ell_2(\tau_n)$,
where $\aleph_0 < \tau_1 < \tau_2 < \cdots$.
It is proved that
every open subset $U$ of $F$ is homeomorphic to the product $K \times F$
for some locally finitedimensional simplicial complex $K$ such that
every vertex $v \in K^{(0)}$ has the star $\operatorname{St}(v,K)$
with $\operatorname{card} \operatorname{St}(v,K)^{(0)} < \tau = \sup\tau_n$
(and $\operatorname{card} K^{(0)} \le \tau$),
and, conversely, if $K$ is such a simplicial complex,
then the product $K \times F$ can be embedded in $F$ as an open set,
where $K$ is the polyhedron of $K$ with the metric topology.
Keywords:LFspace, open set, simplicial complex, metric topology, locally finitedimensional, star, small box product, ANR, $\ell_2(\tau)$, $\ell_2(\tau)$manifold, open embedding, $\sum_{i\in\mathbb{N}}\ell_2(\tau_i)$ Categories:57N20, 46A13, 46T05, 57N17, 57Q05, 57Q40 

18. CJM 2010 (vol 62 pp. 1387)
19. CJM 2010 (vol 62 pp. 994)
 Breslin, William

Curvature Bounds for Surfaces in Hyperbolic 3Manifolds
A triangulation of a hyperbolic $3$manifold is \emph{Lthick} if each
tetrahedron having all vertices in the thick part of $M$ is
$L$bilipschitz diffeomorphic to the standard Euclidean tetrahedron.
We show that there exists a fixed constant $L$ such that every
complete hyperbolic $3$manifold has an $L$thick geodesic
triangulation. We use this to prove the existence of universal bounds on
the principal curvatures of $\pi_1$injective surfaces and strongly
irreducible Heegaard surfaces in hyperbolic $3$manifolds.
Category:57M50 

20. CJM 2009 (vol 62 pp. 614)
 Pronk, Dorette; Scull, Laura

Translation Groupoids and Orbifold Cohomology
We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps, giving a mechanism for transferring results from equivariant homotopy theory to the orbifold category. As an application, we use this result to define orbifold versions of a couple of equivariant cohomology theories: Ktheory and Bredon cohomology for certain coefficient diagrams.
Keywords:orbifolds, equivariant homotopy theory, translation groupoids, bicategories of fractions Categories:57S15, 55N91, 19L47, 18D05, 18D35 

21. CJM 2009 (vol 62 pp. 284)
 Grbić, Jelena; Theriault, Stephen

SelfMaps of Low Rank Lie Groups at Odd Primes
Let G be a simple, compact, simplyconnected Lie group localized at an odd prime~p. We study the group of homotopy classes of selfmaps $[G,G]$ when the rank of G is low and in certain cases describe the set of homotopy classes of multiplicative selfmaps $H[G,G]$. The low rank condition gives G certain structural properties which make calculations accessible. Several examples and applications are given.
Keywords:Lie group, selfmap, Hmap Categories:55P45, 55Q05, 57T20 

22. CJM 2008 (vol 60 pp. 1240)
 Beliakova, Anna; Wehrli, Stephan

Categorification of the Colored Jones Polynomial and Rasmussen Invariant of Links
We define a family of formal Khovanov brackets
of a colored link depending on two parameters.
The isomorphism classes of these brackets are
invariants of framed colored links.
The BarNatan functors applied to these brackets
produce Khovanov and Lee homology theories categorifying the colored
Jones polynomial. Further,
we study conditions under which
framed colored link cobordisms induce chain transformations between
our formal brackets. We conjecture that
for special choice of parameters, Khovanov and Lee homology theories
of colored links are functorial (up to sign).
Finally, we extend the Rasmussen invariant to links and give examples
where this invariant is a stronger obstruction to sliceness
than the multivariable LevineTristram signature.
Keywords:Khovanov homology, colored Jones polynomial, slice genus, movie moves, framed cobordism Categories:57M25, 57M27, 18G60 

23. CJM 2008 (vol 60 pp. 164)
 Lee, Sangyop; Teragaito, Masakazu

Boundary Structure of Hyperbolic $3$Manifolds Admitting Annular and Toroidal Fillings at Large Distance
For a hyperbolic $3$manifold $M$ with a torus boundary component,
all but finitely many Dehn fillings yield hyperbolic $3$manifolds.
In this paper, we will focus on the situation where
$M$ has two exceptional Dehn fillings: an annular filling and a toroidal filling.
For such a situation, Gordon gave an upper bound of $5$ for the distance between such slopes.
Furthermore, the distance $4$ is realized only by two specific manifolds, and $5$
is realized by a single manifold.
These manifolds all have a union of two tori as their boundaries.
Also, there is a manifold with three tori as its boundary which realizes the distance $3$.
We show that if the distance is $3$ then the boundary of the manifold consists of at most three tori.
Keywords:Dehn filling, annular filling, toroidal filling, knot Categories:57M50, 57N10 

24. CJM 2007 (vol 59 pp. 418)
 Stoimenow, A.

On Cabled Knots and Vassiliev Invariants (Not) Contained in Knot Polynomials
It is known that the BrandtLickorishMillettHo polynomial $Q$
contains Casson's knot invariant. Whether there are (essentially)
other Vassiliev knot invariants obtainable from $Q$ is an open
problem. We show that this is not so up to degree $9$. We also
give the (apparently) first examples of knots not distinguished
by 2cable HOMFLY polynomials which are not mutants. Our calculations
provide evidence of a negative answer to the question whether Vassiliev
knot invariants of degree $d \le 10$ are determined by the HOMFLY and
Kauffman polynomials and their 2cables, and for the existence of
algebras of such Vassiliev invariants not isomorphic to the algebras
of their weight systems.
Categories:57M25, 57M27, 20F36, 57M50 

25. CJM 2006 (vol 58 pp. 673)
 Bart, Anneke; Scannell, Kevin P.

The Generalized Cuspidal Cohomology Problem
Let $\Gamma \subset \SO(3,1)$ be a lattice.
The well known \emph{bending deformations}, introduced by
\linebreak Thurston
and Apanasov, can be used
to construct nontrivial curves of representations of $\Gamma$
into $\SO(4,1)$ when $\Gamma \backslash \hype{3}$ contains
an embedded totally geodesic surface. A tangent vector to such a
curve is given by a nonzero group cohomology class
in $\H^1(\Gamma, \mink{4})$. Our main result generalizes this
construction of cohomology to the context of ``branched''
totally geodesic surfaces.
We also consider a natural generalization of the famous
cuspidal cohomology problem for the Bianchi groups
(to coefficients in nontrivial representations), and
perform calculations in a finite range.
These calculations lead directly to an interesting example of a
link complement in $S^3$
which is not infinitesimally rigid in $\SO(4,1)$.
The first order deformations of this link complement are supported
on a piecewise totally geodesic $2$complex.
Categories:57M50, 22E40 
