Naturality of Legendrian LOSS invariant under positive contact surgery

Let $L$ and $S$ be two disjoint Legendrian knots in a contact manifold $(Y,\xi)$. Ozsv\'ath and Stipsicz showed that the LOSS invariant of $L$ is natural under $+1$ contact surgery on Legendrian knot $S$. This thesis extend their result and prove the naturality of the LOSS invariant of $L$ under any positive integer contact surgery along $S$.

In addition, when $S$ is rationally null-homologous, we also entirely characterize the $Spin^c$ structure in the surgery cobordism that makes the naturality of contact invariant or LOSS invariant work (without conjugation ambiguity). In particular this implies that the contact invariant of the $+n$ contact surgery along a rationally null-homologous Legendrian $S$ depends only on the classical invariants of $S$.

The additional generality provided by those results allows us to prove that if two Legendrian knots have different LOSS invariants then, after adding the same positive twists to each in a suitable sense, the two new Legendrian knots will also have different LOSS invariants. This leads to new infinite families of examples of Legendrian (or transverse) non-simple knots that are distinguished by their LOSS invariants.