398 problems

Brady–McCammond conjecture. The metric space

Let $B_{(W,S)}$ be the generalized braid group associated with the Coxeter system $(W,S)$, and let $2$-${\mathcal B}r$ be the monoidal category generated by the Rouquier complexes…

Przytycki–Sazdanović conjecture. The Khovanov homology of a closed $3$-braid can only have torsion of order $2$.

Let $B_4$ be the braid group on four strands, and let $\mathrm{X}^{(2)}_{\gamma,\delta}$ be the set on which the induced braid-group action is defined. Assume Assumption. Transitiv…

Knight Move Conjecture. The Khovanov homology of the closure of every $3$-strand braid over $\mathbb{Q}$ is the direct sum of one pawn move piece and several knight move pieces.

Categorical braid group action conjecture. The functor $\cal{T}_{i,+1}'$ extends to an autoequivalence of $\textsf{Com}(\dot{\cal{U}}_Q(\mathfrak{g}))$ such that the induced automo…

Let $W=(s_i\mid i\in I)$ be the Weyl group of $\mathfrak{g}$, let $\tau$ be the diagram involution, and let $\overline{I}_\tau$ denote the indices whose $\tau$-orbits are of finite…

Birman's injectivity conjecture. The monoid of singular braids maps injectively into the group algebra of the braid group.

Let $(\mathbf{U},\mathbf{U}^{\imath})$ be an iquantum group of finite type, where $\mathbf{U}^{\imath}$ is the coideal subalgebra of a Drinfeld–Jimbo quantum group $\mathbf{U}$ ass…

Let $K$ be a knot or link type, let $\chi(K)$ be its maximal Euler characteristic, and let $\operatorname{SL}(K)$ be the maximum self-linking number of its braid representatives. D…

Naidu–Rowell's conjecture. The category $\mathbf{C}$ has property F precisely when $\operatorname{FPdim}(\mathbf{C})$ is an integer.

Let $M$ be a connected aspherical $2$-dimensional orbifold, meaning that its universal orbifold cover $\widetilde M$ is a contractible orbifold, equivalently that…

Let $\beta$ denote the Burau representation of the braid group on $n$ strands, let $q$ be a $d$-th root of unity, and let $\sigma$ denote the relevant braid-group generator. Write…

Let $B_n$ be the Artin braid group and let $\Gamma_{g,k}$ be the mapping class group of an orientable surface $S_{g,k}$ of genus $g$ with $k$ boundary components. Let $B_{\infty}$…

Let $\mathcal{C}$ be a braided fusion category and let $X$ be a simple object of $\mathcal{C}$. Denote by $\rho_X$ the braid-group representations on…

Polynomial bound conjecture. For each fixed $n$, the size of ${\rm SSS}(x)$ is bounded by a polynomial in the canonical length of $x$.

Let $M$ be the underlying vector space, and let $\tau=\triangle^\rho$ and $\rho=\triangle^\rho$ be the coproduct and product data determined by…

Let ${\cal B}_n^t$ be the quotient of the braid algebra ${\cal B}_n$ with parameters $t_{ij}$, and let $\mathbb{Q}[t]=\mathbb{Q}[t_{ij},\ 1\leq i<j\leq n]$. Finite-generation conje…

Let $\Gamma$ be a graph and let $N$ be a positive integer. Write ${\cal C}^N(\Gamma)$ for the configuration space of $N$ points on $\Gamma$. Torus-gluing conjecture. The configurat…

Full-twist odd-superbridge conjecture. For $n\ge3$ and $k\ge0$, the closure of this braid is a $(2n-1)$-superbridge knot.

Let $s$ and $k$ be positive integers, let $\varepsilon_i\in\{\pm1\}$ for $1\leq i\leq s-1$, and consider the closure of the braid … A generalized rosette knot is this closed braid.…

Let ${\cal W}$ be an arbitrary element of the braid group $B_n$, and let $W$ be a word in the band generators representing it. The authors' algorithm also generalizes to the conjug…

Trivial-kernel conjecture. The kernel of $\eta$ is trivial; equivalently, no nontrivial singular braid in $SB_n$ maps to zero in $\mathbb{C}B_n$. This conjecture asks whether the n…

Completeness conjecture. These are all the braid representatives in each conjugacy class that are in flype position and have minimal letter length.

Existence conjecture. There exists an action of the semi-direct product $B_n\ltimes\mathbb{Z}^n$ on ${\rm U}'_q(\mathfrak{sp}_{2n})$ analogous to the $B_N$-action on…