On Transverse Knots and Branched Covers

Abstract

We study contact manifolds that arise as cyclic branched covers of transverse knots in the standard contact 3-sphere. We discuss properties of these contact manifolds and describe them in terms of open books and contact surgeries. In many cases we show that such branched covers are contactomorphic for smoothly isotopic transverse knots with the same self-linking number. These pairs of knots include most of the non-transversely simple knots of Birman-Menasco and Ng-Ozsváth-Thurston. arXiv:0712.1557v1 [math.GT]

Non-simplicity of Isocontact Embeddings in All Higher Dimensions

“…Figure 2 depicts a Legendrian handlebody presentation for the Weinstein manifolds A 2n k [11,43]. These contact open book decompositions correspond to the symplectic Milnor fibration for the A k -isolated singularities [46,56].…”

“…The main goal of this section is to construct a contact surgery presentation of the contact n-fold cyclic branched covers of (Y, ξ) along τ (Λ), (C n (τ (Λ)), ξ n (τ (Λ)). In the 3-dimensional case, the articles [46,59] discuss contact surgery presentations for contact branched covers along transverse knots, and the unpublished work [4, Section 7] contains part of the ideas we develop in this section.…”

Positivities of Knots and Links and the Defect of Bennequin Inequality

“…K ′ ). Since K ′ is a quasipositive braid, the contact 3-manifold (Σ 2 (K ′ ), ξ 2 (K ′ )) is Stein fillable [22,Theorem 1.3]. Since a negative flype preserves the contact structure on the double branched covering [22, Theorem 5.9], (Σ 2 (K), ξ 2 (K)) is Stein fillable.…”

Khovanov homology, open books, and tight contact structures

“…More precisely, a sequence of Dehn twists performed in a certain order corresponds to a sequence of surgeries on push-offs of the corresponding curves. Some care is needed to determine the linking of these push-offs; we refer the reader to [13] for details (in the case of branched covers), and only state the answer for the more general case that we need here. The procedure of "Legendrianizing" an open book is illustrated in Fig.…”

“…Considering various push-offs of these curves (shown in Fig. 14, see also [13]) step-by-step, we can obtain surgery diagrams corresponding to arbitrary monodromies. We observe that the curves α 1 , .…”