The paper under review considers transverse knots \(K\) for the standard contact structure \(\xi_0\) on the three-sphere. A trivialization of \(\xi_0\) gives a push-off of \(K\) whose linking number with \(K\) is the (well-defined) self-linking number of \(K\) with respect to \(\xi_0\). It was shown by Y. Eliashberg [“Contact 3-manifolds twenty years since J. Martinet’s work”, Ann. Inst. Fourier 42, No.1-2, 165–192 (1992; Zbl 0756.53017)] that even for the unknot the self-linking number can take all odd integer values up to \(2g-1\), where \(g\) is the Seifert genus of the knot. In particular one can not relate the self-linking number to purely topological invariants of the knot. The authors show however that the self-linking number can be related to symplectic invariants. Namely they look at the three-sphere as the boundary of the 4-ball with its standard symplectic form and consider symplectic disks which are bounded by the transverse knot \(K\). Then they prove that the self-linking number of \(K\) equals \(2t-1\), where \(t\) is the tangential index of the symplectic disk (the number of self-intersections counted with signs and multiplicities). The result applies in particular to periodic orbits of the Reeb vector field. The authors actually prove the same formula for the more general case of periodic Reeb orbits on the boundary of any bounded star-shaped domain in \(\mathbb R^4\).
For the entire collection see [Zbl 1226.57002].