stabilization of Legendrian knots There are two ways to stabilize a Legendrian knot $k$ in standard contact sphere $(S^3,\xi_{st})$ i.e. adding right cusps or left cusps, let's call these two stabilized Legendrian knots $k_R$ and $k_{L}$ respectively. If the knot $k$ is embedded on the page of an open book $(\Sigma,\phi)$ of $(S^3,\xi_{st})$, then $k_R$ and $k_L$ can also be embedded on page of an open book obtained by positive stabilization of $(\Sigma,\phi)$. I want to know what are the monodromies of the two new open books relative to $(\Sigma,\phi)$.
I know they cannot be equivalent, since $(-1)$-contact surgery on each of the stabilized knots gives two non-homotopic contact structures (their first chern class is different), so the contact structures are not isotopic and therefore the open books cannot be equivalent.
 A: As you mentioned, $k_L$ and $k_R$ both live in a stabilisation of the open book $(\Sigma,\phi)$.
Namely, suppose you have the triple $(\Sigma,\phi,k)$, where now I think of $k$ as an embedded, nonseparating curve in $\Sigma$. Recall that $k$ is oriented (in order to make sense of "left" and "right" stabilisations), therefore it makes sense to consider a path $\gamma$ that goes from $k$ to $\partial \Sigma$ to the left of $k$ or to the right of $k$, to a given point $p\in\partial\Sigma$.
Once you've chosen such a path, stabilise $(\Sigma,\phi)$ near $p$, choosing a stabilisation arc that is boundary-parallel. There is a natural way to make $k$ go through the new handle, running along $\gamma$ and than back along $\gamma^{-1}$.
The choice of a left/right path $\gamma$ determines whether the stabilisation is a left or a right stabilisation (although the standard terminology is positive vs negative).
I'm not 100% sure that a left path corresponds to a left stabilisation in your language, but there's a 50% chance that I'm right. In any event, I have learnt this from Lisca-Ozsváth-Stipsicz-Szabó (the definition of the LOSS invariant in Heegaard Floer homology), and they credit this to John Etnyre, in his (extremely valuable) Lectures on open book decompositions and contact structures.
