95 problems

Let $K$ be an $L$-space knot, and let $g(K)$ denote the minimal genus of a Seifert surface for $K$. Lidman–Sivek conjecture. For every $L$-space knot $K$, … This conjecture would i…

Let $K subseteq S^3$ be a knot, and let $theta$ denote the Legendrian surgery coefficient associated to a Legendrian representative of $K$. Lidman and Sivek showed that when…

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…

Ungraded complete ruling invariant conjecture. When $\rho=1$, the multiset $\Theta_1(K)$ depends only on the topological type and Thurston--Bennequin number of $K$.

Let $K$ be a Legendrian knot. Its Chekanov number $ch(K)$ is the number of distinct Chekanov polynomials $P_\varepsilon(t)$ obtained from augmentations $\varepsilon$ of the differe…

Etnyre–Ng conjecture. $\Gamma_{-1,1}$ is the only non-loose Legendrian unknot that can be found in any overtwisted contact structure on $S^3$; equivalently, every non-loose Legendr…

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…

Let $(A,\partial)$ be the differential graded algebra associated to the Legendrian object, let $\mathcal{C}=A/I$ be its characteristic algebra, and let the degree distribution of…

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…

Legendrian Low conjecture. If the skies of two events are Legendrian linked, then the events are causally related.

Casals–Etnyre–Kegel conjecture. If the contact $(+1)$-surgeries and contact $(-1)$-surgeries on two Legendrian knots are contactomorphic, then the knots are Legendrian isotopic.

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 $K$ be a knot type in $(S^3,\xi_{std})$, and let $w(K)$ be its width and $\tbb(K)$ its maximal Thurston–Bennequin invariant. Width conjecture. … The conjecture proposes a compl…

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 $L$ be a Legendrian knot in $S^3$ with its standard tight contact structure. Two contact surgeries on a Legendrian knot are cosmetic if the resulting contact manifolds are cont…

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.