When Knots Break the Rules: Non-Orientable Lagrangian Fillings
This lightning talk explores a breakthrough in understanding when Legendrian knots admit non-orientable exact Lagrangian fillings. Moving beyond the well-studied orientable case, the research develops new combinatorial obstructions and classifies fillability for key knot families including alternating and torus knots. The talk reveals surprising differences between orientable and non-orientable fillings, establishes rigidity phenomena, and shows how classical knot invariants control the geometry of these exotic surfaces.Script
In three-dimensional contact geometry, some knots can be filled by surfaces that defy our usual sense of inside and outside. This paper cracks open a fundamental question: which Legendrian knots admit non-orientable exact Lagrangian fillings, surfaces with no consistent choice of orientation?
The authors prove that classical invariants like the Thurston-Bennequin number and normal Euler number act as gatekeepers. For any given Legendrian knot, only finitely many normal Euler numbers are possible among its non-orientable fillings, creating a discrete landscape of allowed geometries.
Their most striking result is a complete classification for alternating knots. Using a new obstruction called the resolution linking number, they determine exactly which alternating knots are non-orientably fillable, revealing structure invisible in the orientable theory.
The geography of non-orientable fillings differs dramatically from the orientable case. Exact non-orientable fillings minimize crosscap number for a fixed normal Euler number, and decomposable fillings exhibit unexpected rigidity, constraints with no orientable analog.
Beyond alternating knots, the authors classify torus knots and most pretzel knots, building a zoo of examples where combinatorial techniques succeed. Yet open cases remain, marking the boundary where current obstruction methods lose their grip.
This work opens a new chapter in knot fillability, where non-orientable surfaces reveal hidden structure in contact geometry. To dive deeper into how classical invariants constrain exotic fillings, visit EmergentMind.com and create your own exploration of the mathematics.
