Quasi-right-veering braids and nonloose links

Abstract

We introduce a notion of "quasi-right-veering" for closed braids, which plays an analogous role to "right-veering" for open books. We show that a transverse link K in a contact 3-manifold (M, ξ) is non-loose if and only if every braid representative of K with respect to every open book decomposition that supports (M, ξ) is quasi-right-veering. We also show that several definitions of "right-veering" closed braids are equivalent.

The defect of Bennequin-Eliashberg inequality and Bennequin surfaces

“…The FDTC c(φ, K, C) is well-defined; namely, if braids K 1 and K 2 are braid isotopic and p(K i ∩ S 0 ) ⊂ ν(C) for both i = 1, 2 then c(φ, K 1 , C) = c(φ, K 2 , C). In fact, a stronger statement can be found in [27,Proposition 2.4].…”

On the fractional Dehn twist coefficients of branched coverings

“…In this section we briefly review the definition of the FDTC for closed braids in open books. For detail we refer the reader to [13].…”

“…Construction of ϕ L from ϕ and L is given in Section 2 of [13], where only the weaker condition (2.2) is essentially used as mentioned in [13,Remark 2.8]. To compare the distinguished monodromies of equivalent closed braids, we use the following: Definition 2.3 (Point-changing isomorphism).…”

“…Although tight contact 3-manifolds are supported by right-veering open books [8], non-loose transverse links are not necessarily represented by right-veering braids. In [13] a new property of closed braids called quasi-right-veering is introduced and it is shown that quasi-right-veering property characterizes non-loose transverse links. This poses a natural question: What is the property of transverse links that corresponds to right-veering-ness of closed braids?…”

Positive factorizations of symmetric mapping classes

“…The set Dehn + (S) ⊂ Mod(S) is a monoid generated by positive Dehn twists, Tight(S) ⊂ Mod(S) is a monoid consisting of monodromies supporting tight contact structures, and Veer + (S) ⊂ Mod(S) is a monoid consisting of right-veering mapping classes. One can see that Ψ(b) is right-veering if and only if b is right-veering (see [20,Section 3] for the definition(s) of right-veering braids).…”