27 problems

Let $\Sigma$ be a closed foam in $\R^{3}$, and let $S$ be its underlying closed pre-foam. Let $\Jsharp(\Sigma)$ denote the instanton-homological evaluation of $\Sigma$, and let…

Cazassus–Herald–Kirk–Kotelskiy conjecture. The deformed Floer homology

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 knot, let $\mathcal I$ be the finite intersection set generating the tangle Lagrangian Floer complex, and let $x\in\mathcal I$ be the distinguished generator independe…

Let $T$ be a 2-tangle, let $R_\pi(T)$ be its holonomy-perturbed traceless character variety, and let $P^*$ be the pillowcase with its singular points removed. A bounding cochain is…

Let $L$ be a link decomposed into tangles along a Conway sphere, and let R_\u03c0(T) denote the perturbed traceless $\mathrm{SU(2)}$ character variety of a tangle $T$, viewed as…

Let $T$ be an oriented 1-tangle diagram in the annulus, let $(C_0,\partial_0)$ be its associated cochain complex, and let $L_{T^0}$ be the corresponding link in $S^2 \times S^1$. W…

Sutured instanton–Heegaard Floer equivalence conjecture. For every balanced sutured manifold $(M,\gamma)$, there is an isomorphism

Let $T$ be a pointed 2-tangle and let $P^*\cong\partial(D^3,T)$ be the pillowcase. Write $I(T)$, $I^\natural(T)$ and $I^\sharp(T)$ for the proposed instanton-theoretic immersed cur…

Let $T$ be a 2-tangle, let $R_\pi(T)\looparrowright P^*$ be its holonomy-perturbed traceless character variety, and let $b\in \mathit{CF}(R_\pi(T),R_\pi(T))$. Let…

Instanton ideal conjecture. Based on preliminary calculations,

Let $Y$ be a rational homology sphere, let $E$ be the bundle used to define the equivariant instanton complex, let $R$ be the coefficient ring, and let $\sigma$ be a choice of sign…

Let $K$ be a web embedded in $\mathbb{R}^{3}$, and let $\Jsharp(K)$ be its instanton homology over the field $\mathbb{F}=\mathbb{Z}/2$. A Tait coloring of $K$ is a Tait coloring of…

Let $N^g$ be the moduli space considered above, and let $\beta\in H^4(N^g)$ denote the first Pontryagin class of its $SO(3)$ fibration. Newstead's nilpotency conjecture. Newstead c…

Spectral sequence conjecture. There is a spectral sequence with $\mathbb{Z}_2$ coefficients whose starting page is

Let $K$ be a planar web, meaning that $K$ is contained in $\mathbb{R}^{2}\subset\mathbb{R}^{3}$, and let $\mathcal{J}^{\sharp}(K)$ denote its $\mathcal{J}^{\sharp}$ instanton homol…

Let $S$ be a closed pre-foam, namely an abstract finite $2$-dimensional cell complex with foam local structure, possibly with tetrahedral points and dots. Define $\Jflat(S)\in\F$ b…

Let $K$ be a web in $\R^{3}$, meaning an embedded trivalent graph, and let $\Jsharp(K)$ be the associated finite-dimensional vector space over the field $\F$ of two elements. A Tai…

Let $K=T_{4,7}$, and write the instanton chain complex in the form … where $a$ and $b$ denote cyclic reordering parameters for the relative grading data. The (4,7)-torus-knot gradi…

Let $K$ be a torus knot and let $\Delta_K$ be its Alexander polynomial. The torus-knot instanton rank conjecture asserts that the reduced singular instanton homology has rank … whe…

Let $K$ be a $2$-bridge knot, let $r_+^\epsilon$ be the distinguished generator in the pillowcase Floer complex, let $\sigma(K)$ be the knot signature, and let $I^\natural(S^3,K)$…

Let $K\subset S^3$ be a knot decomposed as in Equation, with $(Y_1,K_1)$ a trivial tangle. Let $R(Y_0,T_0)$ be non-degenerate in the sense of Section 7 of Hedden, Herald, and Kirk,…

In the table under discussion, the parameter $b$ records one of the grading data associated with the computed knot invariants. Mod-four conjecture for $b$. The parameter satisfies…

Let $K$ be a torus knot, let $CI^\natural(K)$ be its singular instanton knot chain complex, and let $I^\natural(K)$ be the corresponding singular instanton knot homology. Let…

Let $K$ be a knot, let $CI^\natural(K)$ denote its singular instanton knot chain complex, and let $\alpha$ be its generator. Let $\sigma(K)$ denote the signature of $K$. Signature-…