On transverse Hopf links

We classify transverse Hopf links in the standard contact 3-space up to transverse isotopy in terms of their components’ self-linking number. 1 The statement of result A contact structure on a 3-manifold is a completely non-integrable 2-plane field on it. Let ξ0 be the standard contact structure on 3-space R 3 = {(x, y, z)}, that is, a 2-plane field on R defined by the kernel of the 1-form dz − ydx. A transverse knot/link is an oriented knot/link which is everywhere transverse to the contact structure. Throughout, we always assume a transverse knot/link is positively oriented to ξ0. Let T be a transverse knot in the standard contact 3-space (R, ξ0) and v be a non-singular vector field in ξ0 on R . Then the self-linking number (aka. Bennequin number) sl(T ) of T is defined to be the linking number of T and a knot slightly pushed off T along v. Note that sl(T ) is independent of choices of v and an orientation of T , sl(T ) is invariant under transverse isotopy of T and that sl(T ) equals the writhe in a generic projection of T to xz-plane. In [1], D. Bennequin showed that sl(T ) is bounded by the negative Euler characteristic of a Seifert surface for T . In [3], Y. Eliashberg classified topologically trivial transverse knots in (R, ξ0) up to transverse isotopy in terms of their self-linking number. Therefore transverse knots featured in Figure 1 give a complete list of transverse unknots in (R, ξ0), where the diagrams are drawn on xz-plane and the self-linking number is a negative odd integer. In [2], J. Birman–M. Wrinkle classified topologically trivial transverse links in (R, ξ0) up to transverse isotopy in terms of their components’ self-linking number.