54 problems

Let $K\subset S^3$ be a hyperbolic knot, and suppose that a surgery on $K$ yields a lens space of order $r$. Bleiler--Litherland's conjecture. The order satisfies … The paper prove…

Let $K\subset S^3$ be a hyperbolic knot of genus $g$, and suppose that a surgery on $K$ yields a lens space of order $r$. Goda--Teragaito's conjecture. The order satisfies … The co…

Let $M$ be a lens space, and call a triangulation minimal if it has the smallest number of tetrahedra among triangulations of $M$. A minimal layered-triangulation of $M$ is a layer…

$L(3,1)$ $f$-vector conjecture. The $f$-vectors of triangulations of the lens space $L(3,1)$ are characterized by

Bollobás–Riordan completeness conjecture. The Bollobás–Riordan polynomial of the Heegaard graph of lens spaces is a complete invariant for lens spaces.

Torsion-polynomial conjecture. For every prime $p\ge3$, the map $q\mapsto \tau(p,q)$ is constant precisely on the orbits $\{\pm q^{\pm1}\}$. Equivalently, $\tau(p,q)$ distinguishes…

Let $K\subset S^3$ be a lens space knot, meaning that some positive integer surgery on $K$ produces a lens space. Berge's construction gives a family of such knots. Berge conjectur…

Let $L$ be a lens space and let $K\subset L$ be a non-simple knot. The knot meridian determines a boundary slope of the knot complement, and a slope is CTF-detected if it is realiz…

Let $L$ be a lens space and let $K\subset L$ be a knot. A knot is Floer simple when its Floer-theoretic complexity is minimal, and it is simple when it belongs to the standard clas…

Let $p$ be a prime and let $k$ be an integer. Consider a dlt log-Calabi–Yau $4$-fold pair, meaning a dlt pair of dimension $4$ whose log canonical divisor is numerically trivial, a…

Let $K$ be a knot in $S^3$. A lensbordant surgery on $K$ is a positive integer surgery that is smoothly rational homology cobordant to a lens space, and let $u^+(K)$ denote the kn…

Let $M$ be a simply-connected, compact, smooth $4$-manifold with $H_2(M;\mathbb{Z})=\mathbb{Z}$, and suppose that its boundary is a connected sum … of lens spaces, where…

Generic lens-space spectral uniqueness conjecture.

Infinite core-edge counterexample conjecture. Every lens space admits infinitely many one-vertex triangulations with no core edges.

Core-edge conjecture. Every one-vertex triangulation of a lens space has a core edge.

Let $W_{p,q}(\mathbf{n})$ denote the minimal symplectic filling of $(L(p,q),\xi_{can})$ corresponding to $\mathbf{n}=(n_1,\ldots,n_k)$ in the parameterizing set…

Let $p,q\in\mathbb{N}$ with $p$ odd and $(p,q)=1$. Write $L(p,q)$ for the lens space associated to the 2-bridge knot $K(p/q)$, and let its Heegaard Floer correction terms be the va…

Matveev–Jaco–Rubinstein conjecture. For $n>3$,

Generalized cabling conjecture. If Dehn surgery on $K$ yields a connected sum of more than $n-1$ lens spaces, then there is a Seifert fibering of $Y$ where $K$ is a regular fiber.

Simple-knot detection conjecture. If

Boldt–Lauret–Schüth conjecture. If the two spin lens spaces are Dirac isospectral, then they are isometric. The claim is supported by computational evidence for spin lens spaces, w…

Let $L=L(49;1,6,15)$ and $L'=L(49;1,6,20)$ be the given lens spaces, and let $\tau$ be an irreducible representation of $\operatorname{SO}(5)$. Finiteness conjecture. There are onl…

Let $K\subset L(p,q)$ be a knot and let $m(K)\subset L(p,p-q)$ denote its mirror. Consider Legendrian representatives of these knots in the universally tight contact structures on…

Let $L(p,q)$ be a lens space with universally tight contact structure $\xi_{UT}$, and let $K\subset L(p,q)$ be a knot admitting a surgery to the 3-sphere. A Legendrian representati…

Let $\Sigma(p,q,r)$ be a Brieskorn homology sphere with its usual orientation, and let $-\Sigma(p,q,r)$ denote the same manifold with the opposite orientation. Let $K_{p,k}$ denote…