Ordered groups, eigenvalues, knots, surgery and L-spaces

Clay, Adam; Rolfsen, Dale

doi:10.1017/s0305004111000557

Cited by 32 publications

(36 citation statements)

References 23 publications

(30 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The focus of our discussion concerning fibred knots will be the following two results. More details may be found in [85] and [22].…”

Section: Fibered Knotsmentioning

confidence: 99%

“…According to [17], among the knots of 12 or fewer crossings, 1246 of them are fibred and among those knots 485 1 have Alexander polynomials with no roots in R + , so they cannot be bi-orderable. A complete list of them can be found in [22]; the examples with up to ten crossings are: 3 1 , 5 1 , 6 3 , 7 1 , 7 7 , 8 7 , 8 10 , 8 16 , 8 19 , 8 20 , 9 1 , 9 17 , 9 22 , 9 26 , 9 28 , 9 29 , 9 31 , 9 32 , 9 44 , 9 47 , 10 5 , 10 17 , 10 44 , 10 47 , 10 48 , 10 62 , 10 69 , 10 73 , 10 79 , 10 85 , 10 89 , 10 91 , 10 99 , 10 100 , 10 104 , 10 109 , 10 118 , 10 124 , 10 125 , 10 126 , 10 132 , 10 139 , 10 140 , 10 143 , 10 145 , 10 148 , 10 151 , 10 152 , 10 153 , 10 154 , 10 156 , 10 159 , 10 161 , 10 163 .…”

Section: Knotmentioning

confidence: 99%

“…Here is an outline the proof, one may consult [22] for details. By Yi Ni [80] if surgery on K yields an L-space, K must be fibred.…”

Section: Bi-orderability and Surgerymentioning

confidence: 99%

See 2 more Smart Citations

Ordered Groups and Topology

Clay

Rolfsen

2016

Graduate Studies in Mathematics

Self Cite

View full text Add to dashboard Cite

Preface ix Chapter 1. Orderable groups and their algebraic properties 1.1. Invariant orderings 1.2. Examples 1.3. Bi-orderable groups 1.4. Positive cone 1.5. Topology and the spaces of orderings 1.6. Testing for orderability 1.7. Characterization of left-orderable groups 1.8. Group rings and zero divisors 1.9. Torsion-free groups which are not left-orderable Chapter 2. Hölder's theorem, convex subgroups and dynamics 2.1. Hölder's Theorem 2.2. Convex subgroups 2.3. Bi-orderable groups are locally indicable 2.4. The dynamic realization of a left-ordering Chapter 3. Free groups, surface groups and covering spaces 3.1. Surfaces 3.2. Ordering free groups 3.3. Ordering surface groups 3.4. A theorem of Farrell Chapter 4. Knots 4.1. Review of classical knot theory 4.2. The Wirtinger presentation 4.3. Knot groups are locally indicable 4.4. Bi-ordering certain knot groups 4.5. Crossing changes: a theorem of Smythe Chapter 5. Three-dimensional manifolds 5.1. Ordering 3-manifold groups 5.2. Surgery 5.3. Branched Coverings 5.4. Bi-orderability and surgery Chapter 6. Foliations 6.1. Examples 6.2. The leaf space vii viii CONTENTS 6.3. Seifert fibred spaces 6.4. R-covered foliations 6.5. The universal circle Chapter 7. Left-orderings of the braid groups 7.1. Orderability properties of the braid groups 7.2. The Dehornoy ordering of B n 7.3. Thurston's orderings of B n 7.4. Applications of the Dehornoy ordering to knot theory Chapter 8. Groups of Homeomorphisms 8.1. Homeomorphisms of a space 8.2. PL homeomorphisms of the cube 8.3. Proof of Theorem 8.6 8.4. Generalizations 8.5. Homeomorphisms of the cube Chapter 9. Conradian left-orderings and local indicability 9.1. The defining property of a Conradian ordering 9.2. Characterizations of Conradian left-orderability Chapter 10. Spaces of orderings 10.1. The natural actions on LO(G) 10.2. Orderings of Z n and Sikora's theorem 10.3. Examples of groups without isolated orderings 10.4. The space of orderings of a free product 10.5. Examples of groups with isolated orderings 10.6. The number of orderings of a group 10.7. Recurrent orderings and a theorem of Witte-Morris Bibliography Index PrefaceThe inspiration for this book is the remarkable interplay, expecially in the past few decades, between topology and the theory of orderable groups. Applications go in both directions. For example, orderability of the fundamental group of a 3manifold is related to the existence of certain foliations. On the other hand, one can apply topology to study the space of all orderings of a given group, providing strong algebraic applications. Many groups of special topological interest are now known to have invariant orderings, for example braid groups, knot groups, fundamental groups of (almost all) surfaces and many interesting manifolds in higher dimensions.There are several excellent books on orderable groups, and even more so for topology. The current book emphasizes the connections between these subjects, leaving out some details that are available elsewhere, although we have tried to include enough...

show abstract

“…The focus of our discussion concerning fibred knots will be the following two results. More details may be found in [85] and [22].…”

Section: Fibered Knotsmentioning

confidence: 99%

Section: Knotmentioning

confidence: 99%

See 1 more Smart Citation

Ordered Groups and Topology

Clay

Rolfsen

2016

Graduate Studies in Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is known that for nontrivial knots which are fibred [4], or which have onerelator presentations of a particular form [2], if the Alexander polynomial has no positive real roots then the knot group is not bi-orderable. This begs the following question.…”

Section: Satellites and Sumsmentioning

confidence: 99%

Generalized Torsion in Knot Groups

Naylor¹,

Rolfsen²

2016

Can. math. bull.

Self Cite

View full text Add to dashboard Cite

Abstract. In a group, a nonidentity element is called a generalized torsion element if some product of its conjugates equals the identity. We show that for many classical knots one can find generalized torsion in the fundamental group of its complement, commonly called the knot group. It follows that such a group is not bi-orderable. Examples include all torus knots, the (hyperbolic) knot 5 2 and algebraic knots in the sense of Milnor.

show abstract

“…(see [1,5,6,16,21]). Recall that an L-space is a rational homology sphere with Heegaard Floer homology that is as simple as possible, in the sense that rk HF(Y ) = |H 1 (Y ; Z)| (see [15]).…”

Section: Definition 1 a Group G Is Left-orderable If There Exists A mentioning

confidence: 99%

On cabled knots, Dehn surgery, and left-orderable fundamental groups

Clay

Watson

2011

Mathematical Research Letters

Self Cite

View full text Add to dashboard Cite

Abstract. Previous work of the authors establishes a criterion on the fundamental group of a knot complement that determines when Dehn surgery on the knot will have a fundamental group that is not left-orderable [6]. We provide a refinement of this criterion by introducing the notion of a decayed knot; it is shown that Dehn surgery on decayed knots produces surgery manifolds that have non-left-orderable fundamental group for all sufficiently positive surgeries. As an application, we prove that sufficiently positive cables of decayed knots are always decayed knots. These results mirror properties of L-space surgeries in the context of Heegaard Floer homology. Definition 1. A group G is left-orderable if there exists a partition of the group elementssatisfying P · P ⊆ P and P = ∅. The subset P is called a positive cone. This is equivalent to G admitting a left-invariant strict total ordering. For background on left-orderable groups relevant to this paper see [2,6]; a standard reference for the theory of left-orderable groups is [12]. As established by Boyer, Rolfsen and Wiest [2] (compare [11]), the fundamental group π 1 (K) of the complement of a knot K in S 3 is always leftorderable. Indeed, this follows from the fact that any compact, connected, irreducible, orientable 3-manifold with positive first Betti number has left-orderable fundamental group [2, Theorem 1.1]. However, the question of left-orderability for fundamental groups of rational homology 3-spheres is considerably more subtle (see [2,6]) and seems closely tied to certain codimension one structures on the 3-manifold (see [2,3,17]). Continuing along the lines of [6] this paper focuses on Dehn surgery, an operation on knots that produces rational homology 3-spheres. We recall this construction in order to fix notation and conventions.For any knot K in S 3 there is a preferred generating set for the peripheral subgroup Z⊕Z ⊂ π 1 (K) provided by the knot meridian µ and the Seifert longitude λ. The latter is uniquely determined (up to orientation) by the existence of a Seifert surface for K. We orient µ so that it links positively with K, and orient λ so that µ · λ = 1. For any rational number r with reduced form p q we denote the peripheral element µ p λ q by α r . At the level of the fundamental group, the result of Dehn surgery along α r is summarized by the short exact sequenceDate: March 11, 2011. Both authors partially supported by NSERC postdoctoral fellowships. Here α r denotes the normal closure of α r , and S 3 r (K) is the 3-manifold obtained by attaching a solid torus to the boundary of S 3 ν(K), sending the meridian of the torus to a simple closed curve representing the classWe will blur the distinction between α r as an element of the fundamental group or as a primitive class in the (projective) first homology of the boundary, and refer to these peripheral elements as slopes.While many examples of rational homology 3-spheres have left-orderable fundamental group [2], there exist infinite families of knots for which sufficiently positive Deh...

show abstract

Ordered groups, eigenvalues, knots, surgery and L-spaces

Cited by 32 publications

References 23 publications

Ordered Groups and Topology

Ordered Groups and Topology

Generalized Torsion in Knot Groups

On cabled knots, Dehn surgery, and left-orderable fundamental groups

Contact Info

Product

Resources

About