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author:
- The Author
title: Brief Article
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The paper gives a complete classification of Legendrian 4-plats, that have exact orientable Lagrangian fillings.  It was already proven, that positive knots have fillable Legendrian representatives \[MR3314092\] while a knot needs to be quasipositive \[MR1836094\] to have such a filling.  As a consequence of the main theorem of the paper on sees, that among Legendrian 4-plats, only the (smoothly) positive ones are fillable. 

Legendrian 4-plats are Legendrian knots that have front projections with exactly 4 cusps (2 left and 2 right).  Smoothly these knots are 2-bridge knots, but not all Legendrian representatives of 2-bridge knots are Legendrian isotopic to Legendrian 4-plats.  Once the left cusps are moved to the left-most, and the right cusps are moved to the right most of the front projection, Legendrian 4-plats have a standard form, where alternatingly, the internal two strands cross some number of times, and then the outer two strands cross some number of times. 

The main theorem states that a Legendrian 4-plat has an exact orientable Lagrangian filling if and only if

-   none of the internal crossings are single;

-   there are no more than 2 consecutive negative crossings between the same strands. 

This statement is proved using linearised Legendrian contact homology and normal rulings for giving obstructions to fillability, and gives explicit constructions for the Lagrangian fillings in the other cases.