Transverse universal links

Abstract

We show that there exists a transverse link in the standard contact structures on the 3-sphere such that all contact 3-manifolds are contact branched covers over this transverse link.

Obstructing Lagrangian concordance for closures of 3-braids

“…By Lemma 2.1, there is a symplectic approximation of C, call it φ(C) = C with transverse boundary ∂C = U where U is the transverse unknot with self linking number −1. The p-fold cyclic branched cover of S 3 branched over U is the standard S 3 (see Lemma 2.4 of [CE19]).…”

The existence of a transverse universal knot

“…Recently, R. Casals and J. Etnyre [3] proved that there is a transverse universal link. They also asked if it is possible to find one which is connected, a.k.a a transverse universal knot.…”

“…To prove this theorem, we will show that it is possible to find a contact branch covering ϕ : (S 3 , ξ std ) → (S 3 , ξ std ) branched along some knot K such that ϕ −1 (K) contains a sublink that is contact isotopic to the universal transverse link L given in [3]. Now, by composing ϕ with contact branch coverings ϕ : M → S 3 along L, we obtain all contact manifolds as contact branch coverings along K; therefore, K becomes universal.…”

“…The algorithm is long, and it will create a knot with a lot of crossings. If we want to obtain a knot with fewer crossings, a possible way is to simplify the algorithm presented here for the specific case of the link L given in [3].…”