Expand all Collapse all | Results 1 - 20 of 20 |
1. CMB 2014 (vol 57 pp. 431)
The Rasmussen Invariant, Four-genus and Three-genus 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, four-genus, Rasmussen invariant Categories:57M27, 57M25 |
2. CMB Online first
Left-orderable 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 two-bridge knot corresponding to the rational number $3/7$, has left-orderable
fundamental group if the slope $r$ satisfies $0\le r \le 4$.
Keywords:left-ordering, Dehn surgery Categories:57M25, 06F15 |
3. CMB Online first
Left-orderable 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 two-bridge knot corresponding to the rational number $3/7$, has left-orderable
fundamental group if the slope $r$ satisfies $0\le r \le 4$.
Keywords:left-ordering, Dehn surgery Categories:57M25, 06F15 |
4. CMB 2013 (vol 57 pp. 310)
Left-orderable 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 two-bridge knot corresponding to the rational number $3/7$, has left-orderable
fundamental group if the slope $r$ satisfies $0\le r \le 4$.
Keywords:left-ordering, Dehn surgery Categories:57M25, 06F15 |
5. CMB 2012 (vol 56 pp. 850)
Left-orderability and Exceptional Dehn Surgery on Twist Knots We show that any exceptional non-trivial Dehn surgery on a twist knot, except the trefoil,
yields a $3$-manifold whose fundamental group is left-orderable.
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:left-ordering, twist knot, Dehn surgery Categories:57M25, 06F15 |
6. CMB 2010 (vol 54 pp. 556)
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 $m-1$
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 |
7. CMB 2009 (vol 52 pp. 257)
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 |
8. CMB 2007 (vol 50 pp. 390)
Linking Number of Singular Links and the Seifert Matrix We extend the notion of linking number of an
ordinary link of two components to that of a singular link
with transverse intersections in which case the linking
number is a half-integer. We then apply it to simplify
the construction of the Seifert matrix, and therefore
the Alexander polynomial, in a natural way.
Category:57M25 |
9. CMB 2006 (vol 49 pp. 624)
On Non-Integral Dehn Surgeries Creating Non-Orientable Surfaces For a non-trivial 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 non-integral surgery on a hyperbolic knot
can create a closed non-orientable surface of any genus greater than two.
Keywords:knot, Dehn surgery, non-orientable surface Category:57M25 |
10. CMB 2006 (vol 49 pp. 55)
Non Abelian Twisted Reidemeister Torsion for Fibered Knots In this article, we give an explicit formula to compute the
non abelian twisted sign-deter\-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 |
11. CMB 2005 (vol 48 pp. 32)
Non-Left-Orderable 3-Manifold Groups We show that several torsion free 3-manifold groups
are not left-orderable.
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 left-orderable.
Many other examples of non-orderable groups are obtained by taking
3-fold branched covers of $S^3$ branched along various hyperbolic
2-bridge knots.
%with various hyperbolic 2-bridge 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
3-manifold with the smallest volume.
Categories:57M25, 57M12, 20F60 |
12. CMB 2003 (vol 46 pp. 356)
Branched Covers of Tangles in Three-balls We give an algorithm for a surgery description of a $p$-fold cyclic branched
cover of $B^3$ branched along a tangle. We generalize constructions of
Montesinos and Akbulut-Kirby.
Keywords:tangle, branched cover, surgery, Heegaard decomposition Categories:57M25, 57M12 |
13. CMB 2002 (vol 45 pp. 231)
Erratum:~~The Lehmer Polynomial and Pretzel Links Erratum to {\it The Lehmer Polynomial and Pretzel Links},
Canad. J. Math. {\bf 44}(2001), 440--451.
Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups Categories:57M05, 57M25, 11R04, 11R27 |
14. CMB 2001 (vol 44 pp. 440)
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 well-known 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 |
15. CMB 2000 (vol 43 pp. 145)
On the 2-Parallel 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, 2-parallel version Category:57M25 |
16. CMB 2000 (vol 43 pp. 79)
Cyclotomic Schur Algebras and Blocks of Cyclic Defect An explicit classification is given of blocks of cyclic defect of
cyclotomic Schur algebras and of cyclotomic Hecke algebras, over
discrete valuation rings.
Categories:20G05, 20C20, 16G30, 17B37, 57M25 |
17. CMB 1999 (vol 42 pp. 257)
Homotopy of Knots and the Alexander Polynomial Any knot in a 3-dimensional homology sphere is homotopic to a knot
with trivial Alexander polynomial.
Categories:57N10, 57M05, 57M25, 57N65 |
18. CMB 1999 (vol 42 pp. 149)
A Note on Finite Dehn Fillings Let $M$ be a compact, connected, orientable 3-manifold 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
Culler-Shalen norm of a non-zero 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 |
19. CMB 1998 (vol 41 pp. 140)
Skein homology A new class of homology groups associated to a 3-manifold is defined.
The theories measure the syzygies between skein relations in a skein
module. We investigate some of the properties of the homology theory
associated to the Kauffman bracket.
Categories:57M25, 57M99 |
20. CMB 1997 (vol 40 pp. 309)
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 |