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Search: MSC category 57M25 ( Knots and links in $S^3$ {For higher dimensions, see 57Q45} )

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1. CMB Online first

Hong, Kyungpyo; Oh, Seungsang
Bounds on multiple self-avoiding polygons
A self-avoiding polygon is a lattice polygon consisting of a closed self-avoiding walk on a square lattice. Surprisingly little is known rigorously about the enumeration of self-avoiding 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 self-avoiding 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 self-avoiding polygons in the $m \times n$ rectangular grid on the square lattice. For $m=2$, $p_{2 \times n} = 2^{n-1}-1$. And, for integers $m,n \geq 3$, $$2^{m+n-3} \left(\tfrac{17}{10} \right)^{(m-2)(n-2)} \ \leq \ p_{m \times n} \ \leq \ 2^{m+n-3} \left(\tfrac{31}{16} \right)^{(m-2)(n-2)}.$$

Keywords:ring polymer, self-avoiding polygon
Categories:57M25, 82B20, 82B41, 82D60

2. CMB 2017 (vol 60 pp. 830)

Motegi, Kimihiko; Teragaito, Masakazu
Generalized Torsion Elements and Bi-orderability of 3-manifold Groups
It is known that a bi-orderable 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 non-hyperbolic, 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, bi-ordering, 3-manifold group
Categories:57M25, 57M05, 06F15, 20F05

3. CMB 2016 (vol 59 pp. 472)

Clay, Adam; Desmarais, Colin; Naylor, Patrick
Testing Bi-orderability of Knot Groups
We investigate the bi-orderability of two-bridge 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 bi-orderability 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, 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

6. 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

7. 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

8. CMB 2013 (vol 57 pp. 310)

Hakamata, Ryoto; Teragaito, Masakazu
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

9. CMB 2012 (vol 56 pp. 850)

Teragaito, Masakazu
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

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 $m-1$ and $m+1$ are toroidal. This gives an affirmative answer to a question asked by Boyer and Zhang.

Keywords:cyclic surgery, toroidal surgery

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

12. CMB 2007 (vol 50 pp. 390)

Hebda, James J.; Hsieh, Chun-Chung; Tsau, Chichen M.
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.


13. CMB 2006 (vol 49 pp. 624)

Teragaito, Masakazu
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

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 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

15. CMB 2005 (vol 48 pp. 32)

Dąbkowski, Mieczysław K.; Przytycki, Józef H.; Togha, Amir A.
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

16. CMB 2003 (vol 46 pp. 356)

Ishiwata, Makiko; Przytycki, Józef H.; Yasuhara, Akira
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

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), 440--451.

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 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

19. CMB 2000 (vol 43 pp. 145)

Chang, Jae-Ho; Lee, Sang Youl; Park, Chan-Young
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

20. CMB 2000 (vol 43 pp. 79)

König, Steffen
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

21. CMB 1999 (vol 42 pp. 257)

Austin, David; Rolfsen, Dale
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

22. CMB 1999 (vol 42 pp. 149)

Boyer, S.; Zhang, X.
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

23. CMB 1998 (vol 41 pp. 140)

Bullock, Doug; Frohman, Charles; Kania-Bartoszyńska, Joanna
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

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.

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