# Braid loops with infinite monodromy on the Legendrian contact DGA

@article{Casals2022BraidLW, title={Braid loops with infinite monodromy on the Legendrian contact DGA}, author={Roger Casals and Lenhard L. Ng}, journal={Journal of Topology}, year={2022} }

We present the first examples of elements in the fundamental group of the space of Legendrian links in (S, ξst) whose action on the Legendrian contact DGA is of infinite order. This allows us to construct the first families of Legendrian links that can be shown to admit infinitely many Lagrangian fillings by Floer-theoretic techniques. These families include the first known Legendrian links with infinitely many fillings that are not rainbow closures of positive braids, and the smallest…

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## References

SHOWING 1-10 OF 63 REFERENCES

### pages Art ID 14874

- 29,
- 2006

### A note on coherent orientations for exact Lagrangian cobordisms

- MathematicsQuantum Topology
- 2019

Let $L \subset \mathbb R \times J^1(M)$ be a spin, exact Lagrangian cobordism in the symplectization of the 1-jet space of a smooth manifold $M$. Assume that $L$ has cylindrical Legendrian ends…

### Infinitely many Lagrangian fillings

- MathematicsAnnals of Mathematics
- 2022

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### Rational symplectic field theory for Legendrian knots

- Mathematics
- 2010

We construct a combinatorial invariant of Legendrian knots in standard contact three-space. This invariant, which encodes rational relative Symplectic Field Theory and extends contact homology,…

### Algebraic Weaves and Braid Varieties

- Mathematics
- 2020

In this manuscript we study braid varieties, a class of affine algebraic varieties associated to positive braids. Several geometric constructions are presented, including certain torus actions on…

### Lagrangian skeleta and plane curve singularities

- MathematicsJournal of Fixed Point Theory and Applications
- 2022

We construct closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities. This yields closed Lagrangian skeleta for Weinstein pairs $$(\mathbb {C}^2,\Lambda )$$ ( C 2…

### Positive Braid Links with Infinitely Many Fillings

- Mathematics
- 2020

We prove that any positive braid Legendrian link not isotopic to a standard finite type link admits infinitely many exact Lagrangian fillings.