18 problems

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…

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

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 $K_i\subset Y_i$ 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…

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 $m\in\mathbb{Z}$…

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 $K\subset S^3$ 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 $(M,\Gamma)$ be a sutured submanifold of $(M',\Gamma')$, and let $\xi$ be a contact structure on $M'\setminus\operatorname{int}(M)$ with dividing set $\Gamma\cup\Gamma'$. Given…

Let $K\subset S^{3}$ be a knot. Consider the irreducible homomorphisms $c1:c0_{1}(S^{3}\setminus K)\to\operatorname{SU}(2)$ that map a chosen meridian $m$ to the element…

Let $K\subset S^{3}$ be a knot. Denote the knot homologies defined by Ozsváth–Szabó and Rasmussen by their respective knot homology groups, and denote Floer's instanton homology fo…