13 problems

Morton's conjecture. For every Seifert surface $S(L)$ of $L$,

Let $L \subset \Sigma \times I$ be a link, where $\Sigma$ is a surface. An essential surface for $L$ is a surface of smallest genus in which $L$ can be embedded in the thickened su…

Specialization conjecture.

Let $K\subset S^{3}$ be a generic knot, and let $\mathcal{M}_{g,d}(K)$ denote the moduli space of properly immersed minimal surfaces in $\operatorname{H}^{4}$ asymptotic to $K$ at…

Fibered w.s.a.p. coefficient conjecture. The unique non-zero coefficient $c_{i,1-\chi(L)}$ is

HOMFLY homology realization conjecture. It has been conjectured that $H(K)$ can be taken as $\mathcal{H}(K)$.

Finite-N and large-N refined index conjecture. The refined index is computed by refined Chern–Simons theory: for $SL_N$ homologies, by finite-$N$ refined Chern–Simons theory, and f…

HOMFLY–BPS correspondence conjecture. The knot homology ${\cal H}^{i,j,k}(K)$ is the same, up to re-grading, as ${\cal H}_{BPS}^{s_1,s_2,Q}(L_K)$.

Fundamental-representation superpolynomial conjecture. The invariant ${\cal P}_{\tableau{1}}(K)({\bf q},{\bf t},{\bf a})$ computes the superpolynomial of the HOMFLY knot homology.

Let $C$ be a curve, let $C^{[\star]}$ denote the disjoint union of its Hilbert schemes of points, let $l$ be the colength grading, and let $m$ be the minimal number of generators o…

Let $C$ be a locally planar curve and let $C^{[\star]}$ denote the disjoint union of its Hilbert schemes of points, with $l$ the colength grading. For each singular point $p_i$, le…

Let ${\cal L}$ be an unoriented link with $L$ components, colored by representations $R_1,\ldots,R_L$. Let $\hat f_{R_1,\ldots,R_L}(t,\nu)$ denote the reformulated colored HOMFLY i…

Let ${\cal K}$ be a knot, let $R$ be a representation, and let $\hat f_R(t,\nu)$ be the reformulated colored HOMFLY invariant obtained by applying the inverse matrix $M^{-1}$ to th…