Naturality of Legendrian LOSS invariant under positive contact surgery 259 views

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.

PHD (Doctor of Philosophy)

Geometric topology, Contact topology, Heegaard Floer homology

English

All rights reserved (no additional license for public reuse)

2024-04-29