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

Abstract

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...

On $1$-bridge braids, satellite knots, the manifold $v2503$ and non-left-orderable surgeries and fillings

“…Clay and Watson introduced [7] the concept of r-decayed knots. Then they proved that the (p, q)-cable of an r-decayed knot is pq-decayed if q p > r, and that sufficiently positive surgeries on decayed knots yield manifolds with non-left-orderable fundamental groups.…”

Left-orderable fundamental groups and Dehn surgery on genus one 2–bridge knots

“…Since the resulting manifold under Dehn surgery is either a Seifert fibered manifold or a connected sum of two lens spaces, this is also equivalent to 1 .K.r // being not left-orderable. See Boyer, Gordon and Watson [3], and Clay and Watson [8].…”

Left-orderable, non-$L$-space surgeries on knots

“…Example 1.6 (torus knots). For a nontrivial torus knot T p,q (p > q ≥ 2), the argument in the proof of [10,Theorem 1.4]…”

“…[10]). Let K be a satellite knot with a pattern knot k. If K(r) is irreducible and r ∈ S LO (k), then r ∈ S LO (K).Composite knots K with S LO (K) = Q and S L (K) = ∅.…”