87 problems

Let $K\subset Y$ be a fibered knot of genus $g$, with monodromy $\varphi$, and let $T_\varphi$ denote its mapping torus. Write $\overline{Y}$ and $\overline{K}$ for the orientation…

Rasmussen's rank conjecture. The rank of reduced Khovanov homology is always greater than or equal to the rank of knot Floer homology:

Let $L$ be a link in $S^3$. Write $coverline{H}(L)$ and $H^-(L)$ for the reduced, middle, and unreduced HOMFLY-PT homologies of $L$, and $coverline{\operatorname{HFK}}(L)$,…

Manolescu's conjecture. When $S$ is in braid position, the homology $H_{*}(C_{F}(S))$ is isomorphic to the HOMFLY-PT homology of $S$.

Manolescu's conjecture. There is an isomorphism

Let $L$ be a link in $S^3$ and let $n$ be a positive integer. Write $\overline{H}_n(L)$ and $H_n(L)$ for reduced and unreduced $\mathfrak{sl}(n)$ homology, and…

Let $L$ be a link in $S^3$. The HOMFLY-PT homologies carry gradings $\operatorname{gr}_T$ and $\operatorname{gr}_M$, and the grading-shifted knot Floer homologies…

ECH–knot Floer correspondence conjecture. There is an isomorphism

Let $K$ be a transverse knot in a contact three-manifold $(Y,\alpha)$. Denote by $\widehat{ECK}(K,Y,\alpha)$ the hat version of embedded contact knot homology, and by…

Let $K$ be a knot in $S^3$. Let $\mathcal{K}$ be an $n$-crossing braid diagram for $K$ with its outermost strand cut, and let $G_I$ for $I\in\{0,1\}^n$ be the planar trivalent grap…

Let $K$ be a knot in $S^3$ and let $R$ be a genus $g$ Seifert surface of $K$. Write $\widehat{HFK}(K,g)$ for the top Alexander-graded knot Floer homology and…

HOMFLY homology differentials conjecture. There is a homology theory $\mathcal H_*$ categorifying the HOMFLY polynomial, with differentials $\{d_N\}$ satisfying those three axioms,…

Knot Floer homology conjecture. The group $\widehat{HFK}(J)$ is supported nontrivially in exactly $g$ diagonals, shifted up in Maslov grading from $K$. Hence the corresponding exot…

Real Floer realization conjecture. There is an isomorphism of bigraded groups

Coverage conjecture. The bigraded group $\widehat{\operatorname{HFK}}(K)$ can be covered from

Let $Y$ be a $3$-manifold and $K\subset Y$ a knot, with meridian $\mu$. Let $N$ be the twisted $I$-bundle over the Klein bottle, and let $Y_N^\mu$ be the N-filling of…

Let $Y$ be a closed, oriented $3$-manifold and $K\subset Y$ a knot. Let $HFK'_2(Y,K;\mathbb{C})$ be the homology of the mapping-cone complex … and let $I^\sharp(Y,K;\mathbb{C})$ de…

Let $K$ be a strongly quasipositive knot with genus $g$, and let $\widehat{\mathit{HFK}}_m(K,g)$ denote the summand of knot Floer homology in Maslov grading $m$ and top Alexander g…

Cable stabilization conjecture. For a given normalized Alexander degree $\widetilde{\mathbf{s}}$, the dimension of…

Let $L_m$ be the $m$-fold cable link of a link $L$, and let $\overline{\mathbf{s}}$ be a normalized Alexander degree. The maps … are the connecting maps in the directed system defi…

Let $\includegraphics{mm.pdf}_{\ell}$ denote the $\ell$th knot in the Manolescu–Marengon family, and let $\operatorname{HFK}^-(\includegraphics{mm.pdf}_{\ell})$ be its minus knot F…

Let $K$ be a prime fibered positive knot of genus $g$, and let $\widehat{HFK}_{-1}(K,g-1)$ denote its knot Floer homology in Alexander grading $g-1$ and Maslov grading $-1$, with c…

Ghiggini–Spano's conjecture. More generally than the established identification

Let $T_{p,q}$ be the $(p,q)$-torus knot, where $p$ and $q$ are relatively prime integers, and let ${\rm u}^u_b(T_{p,q})$ denote its unoriented band unknotting number, the minimum n…

Diagonal-knot equivalence conjecture. Diagonal knots are conjectured to be equivalent to positive braid knots.