Non-orientable Lagrangian fillings of Legendrian knots

Abstract: We investigate when a Legendrian knot in standard contact $\mathbb{R}^3$ has a non-orientable exact Lagrangian filling. We prove analogs of several results in the orientable setting, develop new combinatorial obstructions to fillability, and determine when several families of knots have such fillings. In particular, we determine completely when an alternating knot (and more generally a plus-adequate knot) is decomposably non-orientably fillable, and classify the fillability of most torus and 3-strand pretzel knots. We also describe rigidity phenomena of decomposable non-orientable fillings, including finiteness of the possible normal Euler numbers of fillings, and the minimization of crosscap numbers of fillings, obtaining results which contrast in interesting ways with the smooth setting.

• The paper establishes a novel relation linking the Thurston-Bennequin invariant to the Euler characteristic and normal Euler number for non-orientable Lagrangian fillings.
• It introduces combinatorial and polynomial obstructions using ruling polynomials, Kauffman, and HOMFLY invariants to determine fillability criteria.
• Classification results for alternating, torus, and three-strand pretzel knots reveal a finiteness theorem and highlight the interplay between diagrammatic and topological invariants.

Non-orientable Lagrangian Fillings of Legendrian Knots

Introduction and Context

This paper conducts a comprehensive study of non-orientable exact Lagrangian fillings of Legendrian knots in the standard contact three-space. While the landscape of orientable Lagrangian fillability is by now largely determined by a collection of classical, gauge-theoretic, and Floer-theoretic obstructions, the non-orientable case remains less understood. The authors aim to elucidate when a given smooth knot type admits a Legendrian representative that bounds a non-orientable exact Lagrangian surface in the standard symplectic ball, and to classify the topological and combinatorial restrictions entailed by this property.

Classical Invariants and Euler Number Analysis

The first principal contribution is the establishment of a formula that relates the Thurston-Bennequin invariant $\mathrm{tb}$ of a Legendrian knot, the Euler characteristic $\chi(L)$, and the normal Euler number $e(L)$ of a non-orientable Lagrangian filling $L$: $\mathrm{tb}(\Lambda) = -\chi(L) - e(L).$ While for orientable fillings the normal Euler number always vanishes, in the non-orientable setting $e(L)$ can take various values, leading to new rigidity phenomena not present in the orientable case. Importantly, the authors prove that only finitely many normal Euler numbers can be realized by decomposable non-orientable exact Lagrangian fillings of a fixed Legendrian knot, despite the fact that in the smooth category connected sums with auxiliary non-orientable surfaces can generate arbitrary Euler numbers.

Further, the paper addresses minimization properties: compelling evidence is presented that non-orientable exact Lagrangian fillings minimize crosscap number among all smooth fillings for a given normal Euler number, in analogy with the genus minimizing property for orientable Lagrangian fillings. However, the absence of a non-orientable adjunction inequality precludes a general theorem.

Combinatorial and Polynomial Obstructions

Rulings of Legendrian front diagrams are employed as a primary combinatorial tool. The canonical ruling associated to a decomposable filling encodes information about the topology of the filling. The paper advances the state of the art by defining an easily computable resolution linking number obstruction—polynomial invariants captured by Kauffman and HOMFLY polynomials serve as effective fillability obstructions.

Specifically, the paper demonstrates that:

• The existence of a non-orientable filling implies sharpness of the Kauffman polynomial bound on the maximal Thurston-Bennequin number.
• The difference between the ordinary and oriented ruling polynomials (readable off the knot's Kauffman and HOMFLY polynomials) witnesses non-fillability by a decomposable non-orientable Lagrangian surface in certain cases.

Classification Results for Alternating, Adequate, Torus, and Pretzel Knots

By leveraging the developed invariants and obstructions, the authors achieve essentially complete classifications for several large classes of knot types:

Alternating and Plus-Adequate Knots

A major outcome is the complete characterization of decomposable non-orientable Lagrangian fillability for alternating and plus-adequate knots:

• An alternating knot is decomposably non-orientably fillable if and only if it is non-positive. [Thm: alternating case]
• For plus-adequate knots, there always exists a decomposable filling; orientability of the filling is equivalent to positivity of the knot.

This dichotomy exposes a highly nontrivial contrast with orientable fillability, which is intertwined with the positivity and quasipositivity of the knot.

Torus Knots

The paper provides a nearly exhaustive classification for torus knots $T(p,q)$ with $|p| > q > 0$:

• $T(p,q)$ is orientably fillable if and only if $p>q>0$.
• $\chi(L)$0 is non-orientably fillable if $\chi(L)$1, but only these negative 2-strand torus knots admit such fillings.
• For $\chi(L)$2 and $\chi(L)$3 odd, no Lagrangian filling exists.
• If $\chi(L)$4 and $\chi(L)$5 divides $\chi(L)$6, no decomposable filling exists.

Rigorous combinatorial proofs are supplied, and explicit normal Euler numbers of canonical fillings are computed, revealing the interplay between diagrammatic and polynomial invariants.

3-strand Pretzel Knots

A parallel classification is established for three-strand pretzel knots, identifying the precise families for which decomposable non-orientable Lagrangian fillings exist (with a noted exception for one subfamily lacking a resolved maximal $\chi(L)$7 front diagram).

Rigidity and Geography of Fillable Knots

A notable structural result is the finiteness theorem: for any given Legendrian knot, there are only finitely many non-orientable decomposable exact Lagrangian fillings up to normal Euler number. However, the authors construct a family—a sequence of iterated $\chi(L)$8-twisted Whitehead doubles—where the cardinality of the set of realized Euler numbers grows arbitrarily large, showing that no universal bound exists across all knot types.

The overall landscape sketched by these results is nuanced and, in some respects, orthogonal to the orientable theory: fillable knots are neither precisely complementary to positive, negative, or quasipositive knots, exhibiting subtle intermediates.

Implications and Further Directions

The techniques and invariants developed here open clear directions for subsequent work, including:

• The pursuit of crosscap-minimizing properties for non-orientable Lagrangian fillings and potential generalizations of the adjunction inequality.
• Extensions to the geography problem for fillable smooth knot types under various combinatorial constraints.
• The search for further topological invariants or Floer-theoretic obstructions capturing non-orientable fillability.

These results firmly establish that non-orientable fillability is governed by fundamentally different mechanisms from its orientable counterpart, with deeper combinatorial and algebraic underpinnings deserving further investigation.

Conclusion

This paper provides a detailed and rigorous examination of non-orientable exact Lagrangian fillings of Legendrian knots, generating new combinatorial and topological obstructions, explicit classifications for several broad knot families, and a framework for future research directions in the study of Lagrangian fillings and Legendrian knot theory. The results clarify the delicate balance between diagrammatic, polynomial, and topological invariants in controlling the existence and uniqueness of such non-orientable fillings, highlighting critical differences from the well-studied orientable case and prompting significant questions regarding the broader topology of Lagrangian fillings.