12 problems

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 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 $\rho$ 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…

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…

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

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…

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…

Hayden–Sabloff's conjecture. A knot is exact Lagrangian fillable if and only if it is quasipositive and the HOMFLYPT bound on the maximal Thurston–Bennequin number of $K$ is sharp.

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

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…