162 problems

Let a Brieskorn homology sphere be a 3-dimensional link $\Sigma(a_1,\ldots,a_n)$ of an isolated complete intersection singularity as described above, and call it nontrivial when it…

A tight fibered knot is a fibered knot supporting the tight contact structure. A knot is prime if it cannot be expressed as a nontrivial connected sum. The smooth knot concordance…

Let $(V,\xi)$ be a closed contact manifold, where $\xi$ is a cooriented contact structure, and let $\operatorname{Cont}_0(V,\xi)$ denote the identity component of its contactomorph…

Casals' conjecture. The standard Legendrian Hopf Link should have exactly two Lagrangian fillings.

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…

Honda's conjecture. These contact structures are not symplectically semi-fillable, and hence are not symplectically fillable.

Ghiggini's conjecture. The contact invariant vanishes:

Let $L$ be a link in $S^3$, and let its fiber surface be the page of a fibration of its complement over $S^1$. A Hopf plumbing or deplumbing is the operation of adding or removing…

Fuchs–Tabachnikov conjecture. Two transversal knots of $\mathbb{R}^3$ with the same knot type and the same Bennequin number are contact isotopic.

ECH–knot Floer correspondence conjecture. There is an isomorphism

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 $S$ be a compact, oriented surface with boundary, let $\operatorname{Map}(S,\partial S)$ be the group of orientation-preserving diffeomorphisms of $S$ that restrict to the iden…

A contact 3-manifold $(Y,\xi)$ has a contact invariant $c(\xi)\in\widehat{HF}(-Y)$, and Giroux torsion means that the contact structure contains a positive amount of Giroux torsion…

Let $\beta$ be a positive braid, let $L_\beta$ be its braid-positive link closure, and let the corresponding Legendrian closure be represented by the front diagram $f_\beta$. A Leg…

Giroux-torsion vanishing conjecture. If

Amorós–Bogomolov–Katzarkov–Pantev conjecture. Every right-veering mapping class is a product of positive Dehn twists:

Let $Y$ be a $3$-manifold and let $\xi$ be a contact structure on $Y$. Let ${\rm Tor}(Y,\xi)$ denote the torsion, defined as the supremum of the integers $n\geq 1$ for which there…

Let $Y$ be a $3$-manifold and let $\xi$ be a contact structure on $Y$. The torsion of $(Y,\xi)$, denoted by … is the supremum of the integers $n\geq 1$ for which there exists a con…

Reduced branched-cover contact-invariant conjecture. The invariant $psi(L)$ is mapped to the Ozsváth–Szabó contact invariant $c(xi_L)$ under this spectral sequence.

Let $L$ be a transverse representative of an alternating smooth link. Write $\psi(L)\in\widetilde{Kh}(L)$ for the transverse-link invariant in reduced Khovanov homology, let…

K-contact energy-minimization conjecture. The curl eigenform $\eta$ defined by a $K$-contact structure is always energy-minimizing on $\Xi_\eta$.

K-contact tight-minimizer conjecture. K-contact structures on $S^1$-bundles always define tight minimizers.

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…

ECH contact invariant conjecture. The homology class $c(\lambda)$ depends only on the contact structure $\xi$.

ECH–Floer correspondence conjecture. Up to a grading shift,