56 problems

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 $K$ 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…

Characterizing slopes conjecture. There exists a constant $C=C(K)$ such that any slope $p/q$ satisfying $|q|\geq 2$ and $|p|+|q|\geq C$ is characterizing for $K$.

Fix a small cut-off value $d$ for the total rank of knot Floer homology. For $x$ a volume threshold, let $f(x)$ be the fraction of knots whose total knot Floer homology rank is les…

Let $K$ be a knot, let $r(K)$ denote the total rank of its knot Floer homology, let $\operatorname{Vol}(K)$ denote its hyperbolic volume, and let $c$ denote the crossing number. As…

Let $K\subset Y$ be a knot with complement $M=Y\setminus\nu(K)$. Let $\mathit{HF}^{-}(Y,K)$ denote the decorated curves constructed from minus knot Floer homology in the marked tor…

Let $K\subset Y$ be a knot with complement $M=Y\setminus\nu(K)$. The decorated curves $\widehat{\mathit{HF}}(Y,K)$ are curves in the punctured torus $T_M^*$, and…

Let $L$ be an $n$-component link in a homology 3-sphere, and let $\boldsymbol{\mathbf{n}}$ be a choice of framing. Let $\mathit{CLI}^-(L,\boldsymbol{\mathbf{n}})$ denote the associ…

Let $L_+$, $L_-$ and $L_0$ be oriented links, with $L_0$ the resolution of a distinguished crossing, and suppose the skein exact triangle involves the module … If $L_0$ has $\ell$…

Allen's conjecture. If a connected sum of possibly several torus knots is concordant to an $L$-space knot, then it is concordant to a positive torus knot.

Let $P$ be a pattern knot in $S^1\times D^2$, let $K$ be a companion knot in $S^3$, and let $U$ denote the unknot in $S^3$. Satellite rank inequality. There is an inequality … This…

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…

A finitely generated mixed complex is a mixed complex of the form used in the paper with finite generation understood in the source, and a bordered-mixed complex is a tuple…

Suppose $K\subset S^3$ is a quasi-alternating knot. Let $\underline{\operatorname{KHI}}(-S^3,K)$ be the instanton knot Floer homology group, and let $d_+$ and $d_-$ be maps on it.…

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…