196 problems

Non-termination conjecture. The LCS of ${\mathbf{B}}_{2,m}({\mathbb{P}^2})$ does not stop.

Circuit-free permutation conjecture. The permutation $w$ is circuit-free, and thus achieves the lower bound in Theorem, if and only if at least one of the following conditions is s…

Let $L$ be a positive integer, let $a_1,\ldots,a_{2L}$ satisfy $a_i\geq 2$, and set $D=a_1+\cdots+a_{2L}$. For the braid word $x_1^{a_1}x_2^{a_2}\dots x_1^{a_{2L-1}}x_2^{a_{2L}}$,…

Conjecture. The groups $B$ and $P$ are linear and satisfy all the properties enumerated in that theorem whenever $W$ is an irreducible complex reflection group.

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…

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 $\beta$ 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…

Let $\beta\in B_n$ be an $n$-braid, let $\widehat{\beta}$ be its closure, and let $V(t)$ denote the Jones polynomial of $\widehat{\beta}$. For a complex number $t$, choose a square…

Alternation conjecture. Braids with maximal entropy are alternated. The conjecture concerns the observed structure of entropy-maximizing braids among those considered by word lengt…

Strand-number conjecture. Braids of maximal entropy belong to $\operatorname{Br}(3)$ or $\operatorname{Br}(4)$. The paper presents this as an empirical conjecture obtained from its…

Let $f$ be a singular function, let its bifurcation diagram be the relevant discriminant in the parameter space of a versal unfolding, and let the braid monodromy group be the imag…

Let $f$ be a singular function with Milnor number $4mu$, and let a Dynkin diagram of $f$ have vertex cardinality $4mu$. Write $t_1,\cdots,t_4mu$ for generators indexed by its vertic…

Let $f$ be a singular function and let a Dynkin diagram encode its algebraic monodromy. Consider the braid subgroup associated to such a Dynkin diagram, allowing conjugation in the…

Let $f$ be a singular function, with braid monodromy group acting by the Artin representation on its algebraic monodromy homomorphism. Stabilizer conjecture. The braid monodromy gr…

Let ${\bf w}$ be an $F$-root of ${\boldsymbol\pi}$, let $w=\beta({\bf w})$, let $B(w)$ be the braid group of $C_W(wF)$, and let ${\mathcal H}(w)$ be a cyclotomic Hecke algebra for…

Let ${\bf w}$ be an $F$-root of ${\boldsymbol\pi}$, put $w=\beta({\bf w})$, and let $C_W(wF)$ be the centralizer of the corresponding regular element of $WF$. Let $B(w)$ be the bra…

Let $B$ be the braid group, let $B^+$ be its positive braid monoid, let $F$ act on $B$ by the Frobenius diagram automorphism, and let ${\boldsymbol\pi}$ be the positive generator o…

Let $B_n$ be the braid group on $n$ strands and let $K_n=[B_n,B_n]$ be its commutator subgroup. A subgroup has index $r$ if its coset action gives a transitive representation…

Let $L_n$ be the quadratic Lie algebra associated with the pure cactus group, let $\mathbf{L}_n$ be the corresponding braid Lie algebra, and let $\xi_n:L_n\to\mathbf{L}_n$ be defin…

Let $B$ be the braid group associated with a well-generated complex reflection group, let $\mathcal{R}$ be its set of braid reflections, and let $\delta$ be the braid-group analogu…

Let $B_n$ be the braid group on $n$ strands, and let $a_k(B_n)$ denote the number of subgroups of index $k$ in $B_n$. For $n$ sufficiently large, write $c(k/n)$ for a constant depe…

Let $B$ be the associated braid group, let $R$ be its set of braid reflections, and let $B_+$ be the submonoid of $B$ generated by $R$. Define a relation $\preccurlyeq$ on $B_+$ by…

In either of the two settings, let $B$ be the associated braid group, let $W$ be the corresponding Coxeter or complex reflection group, let $p:B\twoheadrightarrow W$ be the natural…

The $G_{34}$ braid-group conjecture. The braid group associated with $G_{34}$ admits the presentation