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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.

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  • $\begingroup$ What kind of relationship are you looking for? For example, the stabilized page and monodromies can be written down in terms of the original one. $\endgroup$ – Kyle Hayden Feb 1 '16 at 22:03
  • $\begingroup$ I edited the problem, that's what I want to know, how to write down the two monodrmies. $\endgroup$ – nikita Feb 1 '16 at 22:08
  • $\begingroup$ I believe this is done in Section 3 of Ozbagci's "A Note on Contact Surgery Diagrams" (home.ku.edu.tr/~bozbagci/diagram.pdf). I know I've seen it elsewhere, though. $\endgroup$ – Kyle Hayden Feb 1 '16 at 22:46
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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.

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  • $\begingroup$ Note that if you do two stabilisations, one "on the left of $k$" and one "on the right of $k$", you arrive at a single open book on which each of $k_L$ and $k_R$ can be Legendrian realised. $\endgroup$ – magicker72 Feb 3 '16 at 18:05
  • $\begingroup$ That is trickier than you make it sound, though; what I understand from your wording is that you have a double stabilisation in the page. $\endgroup$ – Marco Golla Feb 3 '16 at 21:34
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    $\begingroup$ Indeed, you can Legendrian realise a double stabilisation on the page. However, if you push $k$ over only one of the two handles, you can realise a single stabilisation (and which one depends on which handle you push over). $\endgroup$ – magicker72 Feb 3 '16 at 23:22

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