93 problems

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 a surgery on a knot in a three-manifold produce a lens space. In the context above, non-torus knots admit at most two such surgeries, and these slopes must be successive intege…

Let $K$ be a knot in a lens space $Y$, and suppose that $K$ satisfies the relevant Floer-simple equality … Here $KHI(Y,K)$ denotes the instanton knot Floer homology and…

Let $L(p,q)$ be a lens space, let $G$ be a simply connected compact simple Lie group with complexified Lie algebra $\mathfrak g$, and let $\Lambda^{R}$ be the coroot lattice. The C…

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

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…

Matveev's lens-space complexity conjecture. The Matveev complexity of $L(2n+1,n)$ is

Let $p$ be a prime, let $n_1,\ldots,n_k$ be positive integers, and let $L_p^{2n_i-1}$ denote the corresponding lens spaces. Allday's conjecture. The free $p$-rank of their product…

For coprime integers $(p,k)$, let $K_{p,k}$ be the dual knot of a simple $(1,1)$-knot in the lens space with lens-surgery parameter $(p,k)$, and let $Y_{p,k}$ be the homology spher…

The $L(4,1)$ surgery conjecture. The lens space $L(4,1)$ cannot be obtained from a non-trivial knot in $S^3$ by Dehn surgery.

Fiberedness conjecture for hyperbolic knots yielding lens spaces. If a hyperbolic knot admits Dehn surgery yielding a lens space, then the knot is fibered.

Let $p,q$ be coprime integers with $p>q>1$. Denote by ${\cal R}_q$ the set of integers $u$ with $0\leq u\leq 2q$ and $g.c.d.(u,q)=1$. Arithmetic conjecture. [The remainder of the c…

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 $L_{p,q}$ be a lens space, and let $T_{p,q}$ denote its candidate minimal triangulation. Matveev's minimal triangulation conjecture. The triangulation $T_{p,q}$ is minimal for…

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…

Let $L(\alpha,\beta)$ be a lens space, and consider its null-homologous fibered knots. The minimal genus is the least genus among such knots in $L(\alpha,\beta)$. Existence conject…

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

Let $p>q\geq 1$ be coprime integers, let $L(p,q)$ be the corresponding lens space with contact structure $\overline\xi_{\rm st}$, and let $\mathbf Z_{p,q}$ denote the set indexing…

Let $L(p,q)$ be a three-dimensional lens space with arbitrary $p$ and $q$, and let $I_k$ denote the invariant associated with the integer $k$. The relevant values are indexed by…

Let $X^4$ be a simply connected $4$-manifold, let $L(p,q)$ be a $3$-dimensional homology lens space, and let $g_{X^4}(L(p,q))$ denote the $X^4$-genus of $L(p,q)$. Finiteness conjec…

Supersymmetric explanation conjecture. The observed behavior of the torsion may be understandable using some form of supersymmetry.

Let $M$ be a compact, orientable, simply-connected $4$-manifold with $H_1(M)=0$, $H_2(M)\cong\mathbb{Z}$, and boundary a union of spherical $3$-manifolds $Y_1,\dots,Y_n$. For an em…

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…