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Several knots like unknot, $4_1$, $3_1$ are known to be Legendrian simple, i.e., Thurston-Bennequin number and rotation number determine Legendrian type completely.

How about the same notion for link cases of more than two components? In this cases, of course, we may consider those numbers in a component-wise manner.

Even for the most simplest link, i.e., Hopf link cases, I couldn't find a literature for Legendrian simplicity for that.

Is there a reason not to consider Legendrian simple links?

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The question for torus links has been studied by Jennifer Dalton in her PhD thesis, Legendrian torus links, in which she proves that, in fact, positive torus links are Legendrian simple, and all torus links are Legendrian simple if you allow reshuffling of the components.

Similar questions have also been studied by Lenny Ng and his student, Wutichai Chongchitmate: they started the Legendrian link atlas. I would assume that they use Heegaard Floer invariants coming from grid diagrams to establish non-destabilisability of Legendrian links, but I can't find a reference to that in there...

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