196 problems

Jones's braid-index uniqueness conjecture. If

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

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

Chisini's conjecture. Given two generic coverings $f:S\longrightarrow\mathbb{P}^2_{\mathbb{C}}$ and $f':S'\longrightarrow\mathbb{P}^2_{\mathbb{C}}$, both of degree at least $5$, wi…

Let $V$ be a finite-dimensional complex vector space, and let $R$ be a unitary solution to the Yang–Baxter equation on $V^{\otimes 2}$. Suppose that $R$ has finite projective order…

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…

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…

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…