56 problems

Let $L$ and $L'$ be Legendrian knots in the standard contact 3-sphere $(S^3,\xi_{st})$. Write $L(-1)$ and $L(+1)$ for the contact manifolds obtained by Legendrian $(-1)$- and…

Let $C$ be an immersed Lagrangian cylinder between smoothly concordant Legendrian knots $K_1$ and $K_2$, constructed as in the immersion construction discussed in the paper. Let…

Let $\gamma_t$, $t\in[0,1]$, be an eversion of a smooth circular wave front with no dangerous self tangencies: a family of generic oriented wave fronts in $\mathbb{R}^2$ whose endp…

Let $\mathcal{C}=(\mathbb{Z}/2)\langle a_1,\ldots,a_n\rangle/I$ be the characteristic algebra, and let $X$ be the affine scheme in $\mathbb{A}^n$ over $\mathbb{Z}/2$ obtained from…

Let $(A,\partial)$ be the differential graded algebra of a Legendrian knot, let $\varepsilon$ be an augmentation of $A$, and let the first-order Poincaré–Chekanov polynomial be the…

Let $K$ be a Legendrian knot with maximal Thurston–Bennequin number. Let its characteristic algebra be abelianized by imposing $vw=wv$ for all elements $v,w$, and regard the result…

Contact cosmetic surgery conjecture. Any Legendrian knot in $(Y,si)$ that is not smoothly an unknot admits no cosmetic contact surgeries.

Let $\mathrm{minwr}_{\mathbb{R}^2}$ and $\mathrm{minwr}_{S^2}$ denote the minimum writhe of a max-tb Legendrian unknot front projection that is smoothly hard on the plane and spher…

Let $\Lambda \subset (\Sigma \times \mathbb{R}, \xi_\Gamma)$ be a Legendrian knot. Let $Y(\Sigma, \Gamma) \subset \Sigma \times \mathbb{R}$ be the convex sutured contact manifold a…

Let $O_n$ denote the odd graph with parameter $n\geq 2$. A graph is TB-symmetrical when its total Thurston–Bennequin number admits the symmetry described in the paper. Odd-graph TB…

Let $K$ be a null-homologous smooth knot type in a tight contact manifold $(M,\xi)$. A Legendrian representative of $K$ has a Thurston–Bennequin invariant $\tb$. Low-Thurston–Benne…

Let $K$ be a fibered knot, and let $\xi_K$ be the contact structure it supports in the sense of Giroux. Positive Thurston–Bennequin conjecture. In $\xi_K$, the knot type $K$ has Le…

Let $(M,\xi)$ be an overtwisted contact $3$-manifold, and let $\Lambda\subseteq(M,\xi)$ be an embedded Legendrian lying in some Darboux ball. Denote by the inclusion in the Legendr…

Let $(M,cxi)$ be an overtwisted contact $3$-manifold, and let the hypotheses of the principal theorem hold. Denote by the inclusion in the main inclusion the map from the space of…

Let $(\mathcal{A},\partial)$ be the Chekanov–Eliashberg algebra of a Legendrian, let $(\mathcal{L},d)$ be its LSFT algebra, and let $\epsilon:\mathcal{A}\to\Bbbk$ be an augmentatio…

Crab bucket non-destabilization conjecture. For all odd $n\geq 5$, $\beta_n$ cannot be negatively destabilized without increasing its mosaic number.

Stabilized Lagrangian isotopy conjecture. If $L_1$ and $L_2$ are smoothly ribbon isotopic, then after stabilizing both cobordisms, they become Lagrangian isotopic.

Let $\Lambda$ be a non-trivial Legendrian knot that is the Legendrian rainbow closure of a positive braid. For $k\in {\mathbb N}_+$, let $(S^{3}_{1/k}(\Lambda),\xi_{1/k}(\Lambda))$…

Let $\overrightarrow{\mathscr K}$ be a strongly invertible oriented Legendrian knot of type $8_5$, and let $\operatorname{tb}(\overrightarrow{\mathscr K})$ and…

Let $\mu(\overrightarrow{\mathscr K})$ denote the Legendrian mirror of an oriented Legendrian knot $\overrightarrow{\mathscr K}$, and let $-\mu(\mathscr K)$ denote its reversed Leg…

For a strongly invertible Legendrian knot, let its Thurston–Bennequin number be denoted by $\operatorname{tb}$, let $\overline{\operatorname{tb}}(K)$ be the maximal Thurston–Benneq…

Non-loose realization conjecture. If the hat Legendrian invariant of a Legendrian realization is non-zero, then the construction produces a non-loose realization in an overtwisted…

Let $\mathscr L$ be a link type in $\mathbb S^3$. A rectangular diagram representing $\mathscr L$ is non-simplifiable if it cannot be reduced by the simplifying moves considered fo…

Let $\mathscr L$ be a link type in $\mathbb S^3$. A Legendrian link type is non-destabilizable if it cannot be obtained from another Legendrian link type by a destabilization. Fini…

Let $(Y,\Lambda)$ be a pair consisting of a closed contact $3$-manifold $Y$, a closed Legendrian link $\Lambda \subset Y$, and a contact form $\alpha$ that is non-degenerate for…