24 problems

Lück's homotopy-invariance conjecture. $L^2$-Reidemeister torsion is a homotopy invariant.

Asymptotic expansion conjecture. As $N\to\infty$ and $k\to\infty$ with $u$ fixed,

Let $K\subset{\mathbb S}^3$ be a hyperbolic knot, let $\gamma$ be a simple closed curve on $\partial\nu(K)$, and let $\rho_0$ be the discrete faithful representation associated wit…

Let $M$ be a closed $3$-manifold, let $\varphi$ be a smooth transitive Anosov flow on $M$, and let $\rho$ be an acyclic unitary representation of $\pi_1(M)$. Define the twisted Rue…

Higher Whitehead torsion and Reidemeister trace compatibility conjecture. The following diagram commutes up to natural homotopy:

Let $M$ be a smooth closed manifold, let $\mathcal{H}(M)$ be its stable $h$-cobordism space, and suppose the higher Reidemeister trace constructions define a map … Let…

Let $K$ be a hyperbolic knot. For a parameter $u$, let $\rho_u\colon\pi_1(S^3\setminus K)\to\operatorname{SL}(2;\mathbb{C})$ be the irreducible representation specified by the meri…

Let $\mathscr{H}\subset S^3$ be a hyperbolic knot. Let $\operatorname{cv}(\mathscr{H})$ be its complex volume, and let $T(\mathscr{H})$ be the adjoint cohomological Reidemeister to…

Let $\mathfrak{g}$ be the Lie algebra of a semisimple complex Lie group $G$, let $M$ be a connected closed oriented $3$-manifold, and let $R^{\rm irr}_G(M)$ be the set of conjugacy…

Let $M$ be a cusped hyperbolic 3-manifold, let $\mathcal{T}$ be an ideal triangulation, and choose an epimorphism $\alpha:\pi_1(M)\twoheadrightarrow\mathbb{Z}$, equivalently an inf…

Let $M$ be an oriented compact hyperbolic $3$-manifold with a torus boundary. A norm curve is a one-dimensional component of the character variety on which every trace function is…

Let $M$ be a compact $3$-manifold with a torus boundary whose interior admits a hyperbolic structure. Suppose that the character variety of irreducible…

Let $K$ be a knot in $S^3$, let $X_K$ be its exterior, and let $\phi\in H^1(X_K;\mathbb{Z})$ be a generator. A pair $(\rho,\phi)$ is assumed to be admissible, and…

Let $M$ be a compact oriented $3$-manifold with torus boundary, and let $\mathcal{X}(M)$ be the character variety of irreducible representations…

Let $N$ be a $3$-manifold with non-empty boundary and let $\mathcal T$ be an ideal triangulation with edge set $E$. Let $\{\mathbf b^{(r)}\}$ be a sequence of colorings of…

Let $M$ be a closed oriented $3$-manifold and let $L$ be a framed hyperbolic link in $M$ with $|L|$ components. Let \\{\mathbf a^{(r)}\} be a sequence of colorings of the compon…

Sign conjecture. For every $f\in R^{\rm zd}(\Sigma(m,p,q))$, the quantity $\varepsilon_f$ is negative. Consequently,

Let $M$ be an oriented compact $3$-manifold with torus boundary, and let $X^\mathrm{irr}(M)$ be the character variety of irreducible $\mathrm{SL}_{2}(\mathbb{C})$-representations.…

Let $M$ be a compact 3-manifold with a torus boundary, and suppose that every irreducible component of $X^\mathrm{irr}(M)$ has dimension $1$. For a slope…

Let $\Sigma=S^1\times S^1$ and let $\omega\in H^1(S^1\times S^1;{\mathbb Z}/2{\mathbb Z})$. For a graded vector space, write $\dim_s$ for the sum over $k\in{\mathbb Z}$ of $s^k$ ti…

Let $K$ be a hyperbolic knot, let $u\ne0$ be small, and let $S(u)$ be the asymptotic function defined by … Let $\mathbb{T}^{K}_{\mu}(u)$ be the twisted Reidemeister torsion of the…

Let $M$ be a cusped hyperbolic $3$-manifold, let $\tau_M$ be the topological invariant obtained from the 1-loop invariant on the Epstein–Penner component, and let…

Let $K$ be a knot with exterior $E_K$ and peripheral torus $\Sigma$. Let ${\mathcal M}^s(\Sigma)$ and ${\mathcal M}^s(E_K)$ denote the smooth loci of the corresponding character mo…

Cappell–Miller conjecture. The Cappell–Miller torsion satisfies