37 problems

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…

Higher-dimensional torus conjecture. There exists a $T^2$-invariant contact structure $\xi$ on $T^{2n+1}$ such that

Let $\Gamma$ be a spatial graph, and let $S$ be the boundary of a tubular neighborhood of $\Gamma$. The spatial-graph DGA is the DGA constructed from a diagram of $\Gamma$ by the e…

Let $K$ be a knot in $S^3$. Its augmentation variety is the locus in $(\mathbb{C}^*)^2$ of pairs $(\lambda_0,\mu_0)$ for which the framed knot DGA of $K$, specialized at…

Let $K$ be a codimension $2$ submanifold of a manifold $M$, let $LK$ be its unit conormal Legendrian submanifold in the contact manifold $ST^*M$ of unit covectors, and let the cord…

Let $K$ be a knot, let $\Sigma_2(K)$ be its double branched cover, and let $HC_0^{\operatorname{ab}}(K)$ denote the abelianized degree-zero contact homology, identified with the co…

Let $K$ be a knot, let $\mathcal{A}$ be the tensor algebra generated by its binormal chords, and let $\partial$ be the differential defined by counting zero-dimensional moduli spac…

Vanishing conjecture. The integer $m$ is always $0$ in Proposition. This would remove the undetermined extra free summand from the description of $HC_1^{\operatorname{lin}}(K)$ and…

O's conjecture. When $Q=\mathbb{R}^n$, the graded algebra $H^{\mathrm{string}}_*(\mathbb{R}^n,K)$ is isomorphic to the Legendrian contact homology of $\Lambda_K$ with coefficients…

Let $x$ be an isolated closed Reeb orbit, let $k$ be a positive integer, and let $\operatorname{CH}(x;{\mathbb Q})$ denote its local contact homology. Let…

Let $(Y,\xi)$ be a closed contact manifold with contact form $\alpha$. Write $\mathfrak{gap}_\sigma(Y,\alpha)$ for the spectral gap associated with a class $\sigma\in CH(Y,\xi)$. P…

Let $(Y,\xi)$ be a closed contact manifold with a periodic contact form $\alpha$ of period $T$. Let $CH(Y,\xi)$ denote contact homology, $\mathfrak{s}_\sigma(Y,\alpha)$ the spectra…

Let $(Y,\xi)$ be a closed contact manifold with a periodic contact form $\alpha$. The closed Reeb orbits form components $S$ of the orbit orbifolds, and to each component associate…

Eliashberg–Hofer conjecture. The differential satisfies $d^2=0$, and the homology $H_*(C_*,d)$ is independent of the contact form $\alpha$ for $\xi$ and of the complex structure…

Let $x=y^k$ be the $k$-th iteration of a simple Reeb orbit $y$, and let $f$ be the return map associated to $y$. The local contact homology of $x$ is denoted by $HC(x)$, and…

Let $\Lambda$ be a Legendrian in a $1$-jet space of any dimension whose front has mild singularities. Let $(\mathcal{A},\partial)$ denote its simplicial DGA, and let the Legendrian…

Let $\Lambda$ be a Legendrian knot in $\mathbb R^3$ and let $\mathbbm k$ be a field. For each integer $n\geq 1$, let $\mathcal{R}ep_n(\Lambda,\mathbbm k)$ be the $A_\infty$ categor…

Counting-factor conjecture. The factor is as above, with the remaining factors arising from extending the abstract perturbation and taking limits to the original disk with constant…

Bourgeois–Ekholm–Eliashberg cyclic surgery conjecture. Under this isomorphism, the surgery exact triangle for cyclic Legendrian contact homology is identified with the exact triang…

Let $W$ be a Liouville cobordism corresponding to attaching critical handles along a collection $\Lambda$ of Legendrian spheres, and let $L{\mathbb H}^{\text{Ho}}(\Lambda)_*$ and…

Bourgeois–Ekholm–Eliashberg surgery conjecture. These isomorphisms identify the tautological exact triangle involving $SH_*^{=0}$, $SH_*$, and $SH_*^{>0}$ for $(W,\partial^-W)$ wit…

Framed cord algebra conjecture. The framed cord algebra is the zero-th relative contact homology of $\Lambda_K$ in $ST^*M$.

Let $(M^{2n-1},\xi)$ be a co-oriented contact manifold, so that $\xi=\ker\alpha$ for a globally defined $1$-form $\alpha$. Assume that all closed orbits of the Reeb vector field as…

Let the assumptions of Lemma(c) hold, and consider the inequality in Remark. Equality conjecture. Under these assumptions, the inequality is an equality. This proposed equality wou…