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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 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 invariantCategories: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 surgeryCategories: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 surgeryCategories:57M25, 06F15

4. 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 surgeryCategories:57M25, 06F15

5. 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 surgeryCategories:57M25, 06F15

6. 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 surgeryCategory:57M25

7. 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 surfaceCategory:57M25

8. 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. Category:57M25

9. 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 surfaceCategory:57M25

10. 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, MonodromyCategories:57Q10, 57M27, 57M25

11. 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

12. 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 decompositionCategories:57M25, 57M12

13. 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 groupsCategories:57M05, 57M25, 11R04, 11R27

14. 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 groupsCategories:57M05, 57M25, 11R04, 11R27

15. 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 versionCategory:57M25

16. 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

17. 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

18. 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

19. 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

20. 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

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