31 problems

Let $K$ be a knot in a lens space $Y$, and suppose that $K$ satisfies the relevant Floer-simple equality … Here $KHI(Y,K)$ denotes the instanton knot Floer homology and…

Let $K_i\subset Y_i$ for $i=1,2$ be rationally null-homologous knots. Write $K_1\# K_2$ for their connected sum, and let $B^\pm(K)$ denote the bent complexes associated to a knot…

Let $Y$ be a connected, closed, oriented $3$-manifold, and let $I^\sharp(Y;\mathbb{F}_2)$ denote its framed instanton homology over the field with two elements. Instanton Poincaré…

Heegaard–instanton Floer correspondence conjecture. It is conjectured that

Let $G$ be a reductive group and $M$ a closed, connected, oriented $3$-manifold. Let $\Loc^{\ff}_{G}(M)$ be the representation variety of $G$-local systems, let…

Instanton L-space surgery conjecture. Under these assumptions, $S^3_{p/q}(K)$ must be an instanton L-space, meaning

Instanton knot Floer correspondence. The two knot homology theories should be isomorphic:

Instanton–Heegaard Floer correspondence. The two theories should be isomorphic:

Let $K$ be a knot in $S^3$. Let $q_M(K)$ and $q_M^\dagger(K)$ be the invariants defined from equivariant singular instanton Floer theory, let $q_\tau(K)$ be Hendricks–Lipshitz–Sark…

Let $K$ be a non-local null-homologous knot in $\mathbb{RP}^3$. Let $\widetilde{\operatorname{Kh}}(K;\mathbb{Z}/2)$ denote reduced Khovanov homology over $\mathbb{Z}/2$, and let…

Let $K$ be a knot, and let $\tau(K)$, $\tau^\flat(K)$, and $tau^ atural(K)$ denote the knot concordance invariants appearing in the source, with $tau^ atural$ the instanton tau-inv…

Let $K$ be a knot in $S^3$. Write $h(S^3_1(K))$ for the instanton Frøyshov invariant of $S^3_1(K)$ and $V_0(K)$ for the Heegaard Floer knot invariant. Frøyshov–Heegaard Floer corre…

Suppose $\hat{\mu}=q\mu+p\lambda$ with $q>0$, and suppose $\hat{\lambda}=q_0\mu+p_0\lambda$ is chosen as in the setup. For $m\in\mathbb{Z}$ such that $\hat{\lambda}-m\hat{\mu}$ is…

Let $\widehat{K}$ be a rationally null-homologous knot in $\widehat{Y}$, and let $\hat{\mu}$ and $\hat{\lambda}$ be the slope data used in the construction. Given $m\in\mathbb{Z}$…

The bypass-map conjecture. The displayed square commutes for every $s\in[-g(K),g(K)]$. This predicts that the bypass-map summand is identified with the corresponding basic-class su…

Let $Y$ be a closed three-manifold, let $K\subset Y$ be a knot, and let $Y_n(K)$ denote the result of $n$-surgery on $K$. Let $W_n^\prime(K)$ be the cobordism defined above, and le…

Let $L$ be the four-component link L8n8, whose linking diagram is a cycle, and let $\operatorname{I}^\natural(L)$ denote its instanton Floer homology. L8n8 conjecture. The link L8n…

Let $K\subset S^3$ be a knot, let $S^3_{-n}(K)$ be obtained by $(-n)$-surgery, and let $W$ be the natural cobordism from $S^3$ to $S^3_{-n}(K)$. Let $t_i:H_2(W_{-n})\to\mathbb{Z}$…

Let $(\widehat{Y},\widehat{K})$ satisfy the hypotheses of the source's torsion spin$^c$ decomposition theorem, and let $\widehat{K}'\subset\widehat{Y}$ be another knot satisfying t…

Let $(M,\gamma)$ be a balanced sutured manifold. Write $SHI$ for sutured instanton Floer homology, $SFH$ for sutured Heegaard Floer homology, $I^{\sharp}$ for framed instanton Floe…

Tau-sharp equality conjecture.

Floer-rank mutation conjecture. The ranks of $I^\#$ and $\widehat{\mathit{HF}}$ are preserved by genus-2 mutation.

Let $K\subset S^3$ be a knot, and let $S^3(K)$ denote its knot complement. Consider the irreducible homomorphisms … that map a chosen meridian $m$ to $\mathbf{i}\in SU(2)$. Assume…

Let $K$ be a knot in the 3-sphere. Denote its sutured instanton knot homology by $KHI(K)$ and its knot Floer homology by $\widehat{HFK}(K)$. Isomorphism conjecture. These knot inva…

Let $(Y,L)$ be decomposed along a separating sphere as … and let $I^\natural(Y,L)$ denote the $\mathbb{Z}/4$-graded instanton Floer homology defined using a sufficiently small holo…