1. CMB Online first
 Hong, Kyungpyo; Oh, Seungsang

Bounds on multiple selfavoiding polygons
A selfavoiding polygon is a lattice polygon consisting of a
closed selfavoiding walk on a square lattice.
Surprisingly little is known rigorously about the enumeration
of selfavoiding polygons,
although there are numerous conjectures that are believed to
be true
and strongly supported by numerical simulations.
As an analogous problem of this study,
we consider multiple selfavoiding polygons in a confined region, as a model for multiple ring polymers in physics.
We find rigorous lower and upper bounds of the number $p_{m \times
n}$
of distinct multiple selfavoiding polygons in the $m \times
n$ rectangular grid on the square lattice.
For $m=2$, $p_{2 \times n} = 2^{n1}1$.
And, for integers $m,n \geq 3$,
$$2^{m+n3}
\left(\tfrac{17}{10}
\right)^{(m2)(n2)} \ \leq \ p_{m \times n} \ \leq \
2^{m+n3}
\left(\tfrac{31}{16}
\right)^{(m2)(n2)}.$$
Keywords:ring polymer, selfavoiding polygon Categories:57M25, 82B20, 82B41, 82D60 

2. CMB 2017 (vol 60 pp. 830)
 Motegi, Kimihiko; Teragaito, Masakazu

Generalized Torsion Elements and Biorderability of 3manifold Groups
It is known that a biorderable group has no generalized torsion
element,
but the converse does not hold in general.
We conjecture that the converse holds for the fundamental groups
of $3$manifolds,
and verify the conjecture for nonhyperbolic, geometric $3$manifolds.
We also confirm the conjecture for some infinite families of
closed hyperbolic $3$manifolds.
In the course of the proof,
we prove that each standard generator of the Fibonacci group
$F(2, m)$ ($m \gt 2$) is a generalized torsion element.
Keywords:generalized torsion element, biordering, 3manifold group Categories:57M25, 57M05, 06F15, 20F05 

3. CMB 2016 (vol 59 pp. 472)
 Clay, Adam; Desmarais, Colin; Naylor, Patrick

Testing Biorderability of Knot Groups
We investigate the biorderability of twobridge knot groups
and the groups of knots with 12 or fewer crossings by applying
recent theorems of Chiswell, Glass and Wilson.
Amongst all knots with 12 or fewer crossings (of which there
are 2977), previous theorems were only able to determine biorderability
of 499 of the corresponding knot groups. With our methods we
are able to deal with 191 more.
Keywords:knots, fundamental groups, orderable groups Categories:57M25, 57M27, 06F15 

4. CMB 2015 (vol 59 pp. 159)
 MacColl, Joseph

Rotors in Khovanov Homology
Anstee, Przytycki, and Rolfsen introduced the idea of rotants,
pairs of links related by a generalised form of link mutation.
We exhibit infinitely many pairs of rotants which can be distinguished
by Khovanov homology, but not by the Jones polynomial.
Keywords:geometric topology, knot theory, rotants, khovanov homology, jones polynomial Categories:57M27, 57M25 

5. CMB 2014 (vol 57 pp. 431)
 Tagami, Keiji

The Rasmussen Invariant, Fourgenus and Threegenus of an Almost Positive Knot Are Equal
An oriented link is positive if it has a link diagram whose crossings are all positive.
An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing.
It is known that the Rasmussen invariant, $4$genus and $3$genus of a positive knot are equal.
In this paper, we prove that the Rasmussen invariant, $4$genus and $3$genus of an almost positive knot are equal.
Moreover, we determine the Rasmussen invariant of an almost positive knot in terms of its almost positive knot diagram.
As corollaries, we prove that all almost positive knots are not homogeneous, and there is no almost positive knot of $4$genus one.
Keywords:almost positive knot, fourgenus, Rasmussen invariant Categories:57M27, 57M25 

6. CMB Online first


Leftorderable fundamental group and Dehn surgery on the knot $5_2$
We show that the resulting manifold by $r$surgery on the knot $5_2$, which is
the twobridge knot corresponding to the rational number $3/7$, has leftorderable
fundamental group if the slope $r$ satisfies $0\le r \le 4$.
Keywords:leftordering, Dehn surgery Categories:57M25, 06F15 

7. CMB Online first


Leftorderable fundamental group and Dehn surgery on the knot $5_2$
We show that the resulting manifold by $r$surgery on the knot $5_2$, which is
the twobridge knot corresponding to the rational number $3/7$, has leftorderable
fundamental group if the slope $r$ satisfies $0\le r \le 4$.
Keywords:leftordering, Dehn surgery Categories:57M25, 06F15 

8. CMB 2013 (vol 57 pp. 310)
9. CMB 2012 (vol 56 pp. 850)
 Teragaito, Masakazu

Leftorderability and Exceptional Dehn Surgery on Twist Knots
We show that any exceptional nontrivial Dehn surgery on a twist knot, except the trefoil,
yields a $3$manifold whose fundamental group is leftorderable.
This is a generalization of a result of Clay, Lidman and Watson, and
also gives a new supporting evidence for a conjecture of Boyer, Gordon and Watson.
Keywords:leftordering, twist knot, Dehn surgery Categories:57M25, 06F15 

10. CMB 2010 (vol 54 pp. 556)
 Teragaito, Masakazu

Cyclic Surgery Between Toroidal Surgeries
We show that there is an infinite family of hyperbolic knots such that
each knot admits a cyclic surgery $m$ whose adjacent surgeries $m1$
and $m+1$ are toroidal. This gives an affirmative answer to a
question asked by Boyer and Zhang.
Keywords:cyclic surgery, toroidal surgery Category:57M25 

11. CMB 2009 (vol 52 pp. 257)
 Ikeda, Toru

Essential Surfaces in Graph Link Exteriors
An irreducible graph manifold $M$ contains an essential torus if
it is not a special Seifert manifold.
Whether $M$ contains a closed essential surface of
negative Euler characteristic or not
depends on the difference of Seifert fibrations from the two sides
of a torus system which splits $M$ into Seifert manifolds.
However,
it is not easy to characterize geometrically the class of
irreducible graph manifolds which contain such surfaces.
This article studies this problem in the case of graph link exteriors.
Keywords:Graph link, Graph manifold, Seifert manifold, Essential surface Category:57M25 

12. CMB 2007 (vol 50 pp. 390)
13. CMB 2006 (vol 49 pp. 624)
 Teragaito, Masakazu

On NonIntegral Dehn Surgeries Creating NonOrientable Surfaces
For a nontrivial knot in the $3$sphere,
only integral Dehn surgery can create a closed $3$manifold containing a projective plane.
If we restrict ourselves to hyperbolic knots, the corresponding claim for a Klein bottle is still true.
In contrast to these, we show that nonintegral surgery on a hyperbolic knot
can create a closed nonorientable surface of any genus greater than two.
Keywords:knot, Dehn surgery, nonorientable surface Category:57M25 

14. CMB 2006 (vol 49 pp. 55)
 Dubois, Jérôme

Non Abelian Twisted Reidemeister Torsion for Fibered Knots
In this article, we give an explicit formula to compute the
non abelian twisted signdeter\mined Reidemeister torsion of the
exterior of a fibered knot in terms of its monodromy. As an
application, we give explicit formulae for the non abelian
Reidemeister torsion of torus knots and of the figure eight knot.
Keywords:Reidemeister torsion, Fibered knots, Knot groups, Representation space, $\SU$, $\SL$, Adjoint representation, Monodromy Categories:57Q10, 57M27, 57M25 

15. CMB 2005 (vol 48 pp. 32)
 Dąbkowski, Mieczysław K.; Przytycki, Józef H.; Togha, Amir A.

NonLeftOrderable 3Manifold Groups
We show that several torsion free 3manifold groups
are not leftorderable.
Our examples are groups of cyclic branched coverings of $S^3$
branched along links.
The figure eight knot provides simple
nontrivial examples. The groups arising in these examples are known
as Fibonacci groups which we show not to be leftorderable.
Many other examples of nonorderable groups are obtained by taking
3fold branched covers of $S^3$ branched along various hyperbolic
2bridge knots.
%with various hyperbolic 2bridge knots as branched sets.
The manifold obtained in such a way from the $5_2$ knot
is of special interest as it is conjectured to be the hyperbolic
3manifold with the smallest volume.
Categories:57M25, 57M12, 20F60 

16. CMB 2003 (vol 46 pp. 356)
17. CMB 2002 (vol 45 pp. 231)
 Hironaka, Eriko

Erratum:~~The Lehmer Polynomial and Pretzel Links
Erratum to {\it The Lehmer Polynomial and Pretzel Links},
Canad. J. Math. {\bf 44}(2001), 440451.
Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups Categories:57M05, 57M25, 11R04, 11R27 

18. CMB 2001 (vol 44 pp. 440)
 Hironaka, Eriko

The Lehmer Polynomial and Pretzel Links
In this paper we find a formula for the Alexander polynomial
$\Delta_{p_1,\dots,p_k} (x)$ of pretzel knots and links with
$(p_1,\dots,p_k, \nega 1)$ twists, where $k$ is odd and
$p_1,\dots,p_k$ are positive integers. The polynomial $\Delta_{2,3,7}
(x)$ is the wellknown Lehmer polynomial, which is conjectured to have
the smallest Mahler measure among all monic integer polynomials. We
confirm that $\Delta_{2,3,7} (x)$ has the smallest Mahler measure among
the polynomials arising as $\Delta_{p_1,\dots,p_k} (x)$.
Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups Categories:57M05, 57M25, 11R04, 11R27 

19. CMB 2000 (vol 43 pp. 145)
 Chang, JaeHo; Lee, Sang Youl; Park, ChanYoung

On the 2Parallel Versions of Links
In this paper, we show that the absolute value of the signature of
the $2$parallel version of a link is less than or equal to the
nullity of it and show that the signature, nullity, and Minkowski
units of the $2$parallel version of a certain class of links are
always equal to $0$, $2$, and $1$ respectively.
Keywords:braid, Goeritz matrix, Minkowski unit, nullity, signature, 2parallel version Category:57M25 

20. CMB 2000 (vol 43 pp. 79)
21. CMB 1999 (vol 42 pp. 257)
22. CMB 1999 (vol 42 pp. 149)
 Boyer, S.; Zhang, X.

A Note on Finite Dehn Fillings
Let $M$ be a compact, connected, orientable 3manifold whose
boundary is a torus and whose interior admits a complete hyperbolic
metric of finite volume. In this paper we show that if the minimal
CullerShalen norm of a nonzero class in $H_1(\partial M)$ is
larger than $8$, then the finite surgery conjecture holds for $M$.
This means that there are at most $5$ Dehn fillings of $M$ which
can yield manifolds having cyclic or finite fundamental groups and
the distance between any slopes yielding such manifolds is at most
$3$.
Categories:57M25, 57R65 

23. CMB 1998 (vol 41 pp. 140)
24. CMB 1997 (vol 40 pp. 309)
 Hillman, J. A.; Sakuma, M.

On the homology of finite abelian coverings of links
Let $A$ be a finite abelian group and $M$ be a
branched cover of an homology $3$sphere, branched over a link $L$,
with covering group $A$. We show that $H_1(M;Z[1/A])$ is determined
as a $Z[1/A][A]$module by the Alexander ideals of $L$ and certain
ideal class invariants.
Keywords:Alexander ideal, branched covering, Dedekind domain,, knot, link. Category:57M25 
