Naturality of Legendrian LOSS invariant under positive contact surgery

Abstract

Ozsvath and Stipsicz showed that the LOSS invariant is natural under +1 contact surgery. We extend their result and prove the naturality of the LOSS invariant of a Legendrian L under any positive integer contact surgery along another Legendrian 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 (without conjugation ambiguity). In particular this implies that contact invariant of the +n contact surgery along a rationally null-homologous Legendrian S depends only on the classical invariants of S. The additional generalityprovided 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.

Figures (18)

• Figure 1: Example when there are two parallel arcs (the blue and red arcs are $e_1$ and $e_2$, and the dotted circle represents a Darboux ball). On the left is part of $L$ inside a standard Darboux ball. After doing the twist we get the right diagram which is still inside the Darboux ball and is part of the new knot $L_\sigma$
• Figure 2: Schematic Whitney triangle for $(\Sigma, \alpha,\beta, \gamma)$
• Figure 3: The left diagram describes the open book with Legendrian $L$ lie on it and parallel to some binding, and the right one is the stabilization of the left one (we do right hand twist along $k$), where $L^-$ now is parallel to some binding $B$. Then after we do $n-1$ right hand twists along $L^-$ and $1$ left hand twist along $L$ we obtain an open book $(P',\phi')$ for $(Y_n(L), \xi_n^-(L))$.
• Figure 4: The left diagram is an adapted triple $(L,B,\bf\{a_i\})$ and the right one is not; one can transform the $\{a_i\}$ from one to the other by arcslides.
• Figure 5: Since all of what we care are on the $P_{+1}$ page, we can just draw things on $P_{+1}$ to capture all the information instead of drawing the whole doubly pointed Heegaard triple. (The black circles are binding, red curves are $a_i$ (parts of the $\alpha_i$), blue curves are $b_i$ (parts of the $\beta_i$), green curves are $c_i$ (parts of the $\gamma_i$). The $Spin^c$ structure $\mathfrak{s}$ is represented by the small shaded triangle. There might be genus but it's not shown on the picture.)

Theorems & Definitions (42)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Proposition 1.4
• Remark 1.5
• Corollary 1.6
• Remark 1.7
• Theorem 1.8
• Corollary 1.9
• Theorem 2.1: LOSS