13 problems

Let $(Y,\tau,\mathbf{p})$ be a multi-based real three-manifold, let $\widetilde{\operatorname{HMR}}_*(Y,\tau,\mathbf{p})$ denote framed real monopole Floer homology, and let…

Let $(L_+,L_-,L_0)$ be an oriented skein triple, and let $\mathbf{p}$ be a set of basepoints away from the resolutions. Denote by $\widetilde{\operatorname{HMR}}_*(L,\mathbf{p})$ t…

Let $(Y,\iota)$ be a real three-manifold with a compatible real $\mathrm{spin^c}$ structure $\mathfrak{s}$. Let $\widetilde{\mathit{HMR}}$, $\widecheck{\mathit{HMR}}$,…

Let $K$ be a nonzero determinant link. Write $\widetilde{\operatorname{HMR}}_{\bullet}(K)$ for its reduced real monopole Floer homology and let $\operatorname{SWF}_R(K)$ denote the…

Trivial-torsion conjecture. The torsion group $G_{\mathrm{tor}}$ in Lemma 4 is trivial:

Weak symplectic cobordism conjecture. The map

Let $L\subset S^3$ be a link, let $\Sigma(L)$ denote its branched double cover, and let $\bar{L}$ be the mirror of $L$. Write $\widetilde{\mathrm{BN}}^3_{*,*}(\bar{L})$ for the rel…

Derived-groups conjecture. The groups $A'$ and $A”$ are isomorphic, up to a grading shift, to the groups $B'$ and $B”$, respectively.

Let $\mathcal{K}0$ and $\mathcal{K}1$ be Legendrian knots in $(Y,xi)$, and suppose that $\mathcal{K}0prec{\Sigma}\mathcal{K}1$ is a Lagrangian cobordism of arbitrary genus. Write…

Let $\mathcal{K}\subset Y$ be a Legendrian knot. Let $S{+}(\mathcal{K})$ and $S{-}(\mathcal{K})$ denote its positive and negative stabilizations, respectively, and let…

Let $\mathcal{K}\subset(Y,xi)$ be a Legendrian knot, and let $ellg(\mathcal{K})in KHM(-Y,K)otimesmathcal{R}$ denote the invariant constructed using a closure of genus $g$, for…

Let $\mathcal{S}^n=T(3,6n+1)$ and $\mathcal{T}^n=T(3,6n-1)$ for $n greater than or equal to 1$. Define the auxiliary polynomials $fn(t,q)=\sum{k=0}^{n-1}t^8q^{12}$ and…

Let $T(3,6n+1)$ and $T(3,6n-1)$ be the indicated families of torus knots, and let $E^k$ denote the pages of the link surgery spectral sequence associated to a choice of analytic da…