23 problems

Casals' conjecture. The standard Legendrian Hopf Link should have exactly two Lagrangian fillings.

Casals' folding conjecture. For Legendrian links of type $\mathsf{A}_{2n-1}$, $\mathsf{D}_{n+1}$, $\mathsf{E}_6$, and $\mathsf{D}_4$, Lagrangian fillings having certain…

Let $k,n,\ell$ and $d$ be as in the construction of the twist-spun torus link, and let $\Sigma_{\psi^\ell}(\Lambda(k,n-k))$ denote that twist-spun link. An orientably exact Lagrang…

Let $\lambda$ be a Legendrian link, let $\varphi$ be a symmetry used to form the twist spun $\Sigma_\varphi(\lambda)$, and consider the folded cluster algebra associated with this…

Let $\lambda$ be a Legendrian link such that $\mathcal{M}_1(\lambda)$ has a cluster structure. Let $L_1$ and $L_2$ be exact Lagrangian fillings of $\lambda$, and let…

Let $L_1$ and $L_2$ be fillings, and let $d_\mu(L_1,L_2)$ be the smallest number of mutations relating them. Let $\Lambda(L_1,L_2)$ denote their Legendrian double. Mutation-distanc…

Let $\rho$ be the twisting monodromy, let $\rho^k$ denote its $k$-fold iterate, and let $\Sigma_{\rho^k}(\lambda(2,n))$ be the corresponding twist-spun Legendrian. Let $f(k)$ be th…

A Legendrian double is a Legendrian constructed from two fillings, and the mutation distance measures the smallest number of mutations relating the fillings. Mutation-distance conj…

Let $L_{st}$ be a Lagrangian filling of the Hopf link conormal, with $s,t\in\{l,m,d\}$ and $|s|=|t|$. Let ${\mathsf{Z}}_{L_{st}}$ denote its skein-valued partition function, and le…

Let $\Lambda_\beta\subset(\mathbb{R}^3,\xi_{st})$ be the Legendrian link associated to a positive braid word $\beta$, let $T\subset\Lambda_\beta$ be a set of marked points, one per…

The examples above concern Legendrian links admitting non-orientable Lagrangian fillings, and distinguish only finitely many such fillings in each construction. Finiteness conjectu…

Classification conjecture. The only non-orientably fillable torus knots are of the form $T(p,2)$ with $p<0$.

Finite-filling conjecture. The max-tb Legendrian representative of each of these knots has finitely many exact Lagrangian fillings.

BCFG classification of Lagrangian fillings. The following symmetric cases have the stated numbers of fillings:

ADE classification of Lagrangian fillings. Exactly one of the following possibilities occurs:

Let $U$ be the standard Legendrian unknot, whose unique Reeb chord is denoted by $b$. A conical Legendrian filling $\Sigma$ of $U$ is equipped with a $1$-graded augmentation…

Finiteness conjecture. The Legendrian torus link $\Lambda(n,m)$ has finitely many Lagrangian fillings whenever

Let $L$ be a Legendrian ribbon knot in $(S^3,\xi_{\text{st}})$. A Lagrangian ribbon disk is a Lagrangian disk in $B^4$ with ribbon singularities. Finiteness conjecture. Then $L$ bo…

Exact Lagrangian fillability conjecture. All almost positive links are exact Lagrangian fillable.

Let $S$ be the boundary surface of the Lagrangian 3-manifold $L=L(\mathcal{T})$ associated to a tangle obtained by smoothing the $1$-skeleton of a foam, and let $\mathcal{M}$ be th…

Let $K$ be a smooth knot type. The HOMFLY bound on the maximum Thurston–Bennequin number of $K$ is the upper bound obtained from the HOMFLY polynomial, and $K$ is fillable if it ad…

Let $K$ be a non-split link, let $K_{-1}$ be an additional unknot, and let $N_{K\cup K_{-1}}$ be the immersed Lagrangian filling of the conormal of the split link $K\cup K_{-1}$. L…

Let $K$ be an $n$-component link, and let $P$ be a partition of its components into sublinks. Write $K(P)$ for the split union obtained by moving the sublinks far from one another.…