8 problems

Let $M_1$ and $M_2$ be manifolds with torus boundary, and let $\phi:\partial M_1\to\partial M_2$ be an orientation-reversing gluing map. Let $\mathit{HF}^{-}(M_i)$ denote the decor…

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…

Arnold's conjecture. There exists a sequence of invariants $\{E_i\}_{i\geq 0}$ with $E_0=J^+ + 2 St$ such that

Extended conjecture. If $\bar{L}_{i,n}>\bar{D}_{i,n}$ for $i=1,2$, then a sphere $\mathbb{S}^k$ is attached to $\mathcal{L}^{+\kappa_0}_{-\kappa_0}(\pmb{I},\pmb{Q})$, where…

Let a generic immersion be a map $S^{1}\rightarrow\mathbb{R}^{2}$ with only generic singularities, and let Reidemeister moves 1, 2, and 3 be the usual planar moves on immersed curv…

Let $T$ be an immersed curve in the plane, and let $St$, $J^+$, and $J^-$ denote its associated invariants. An inscribed square of a curve is a square whose vertices lie on the cur…

Gauss's conjecture. The interlacement structure of a Gauss code determines whether it has a planar diagram.