19 problems

Let $g_{n,k}$ denote the number of curve diagrams of genus $n$ and complexity $k$, as in the paper. Let $n\geq 2$ be an integer. Polynomial-growth conjecture. There exist positive…

For an integer $n\ge 1$, an $n$-bridge position of a link in $S^3$ is a bridge presentation with $n$ bridges, considered up to bridge isotopy. Fixed-level bridge finiteness conject…

Let $b$ be a braid with a fixed number of strands, let $\widehat{b}$ denote its closure, and let the number of crossings of $b$ be the input size. Przytycki–Silvero conjecture. Com…

Let $K$ be a knot. A knot is nice when it admits a braid representative equipped with an inversion datum, as defined in the paper. It is fibered when its complement fibers over the…

Let $L$ be a fibered link in $S^3$. A braided open book is an open book in $S^3$ whose binding is braided relative to the standard open book with unknotted binding; equivalently, i…

Let two rainbow diagrams represent isotopic surfaces. A rainbow moves conjecture. Any two rainbow diagrams representing isotopic surfaces are related by a sequence of rainbow moves…

Let $b$ be a braid word, and consider its even plat closure or doubly even plat closure. For even plats, connect consecutive endpoint pairs at the top and shifted consecutive pairs…

Let a knot or link have a plat presentation of width greater than or equal to $3$, and suppose the presentation is $3$-highly twisted. Plat rigidity conjecture. The statement of Th…

Let $\underline{\delta}_9$ be the stated candidate value for the minimum dilatation of pseudo-Anosov braids with 9 strands, and let $x$ denote the variable in its defining polynomi…

For integers $0\leq d_1\leq d_2\leq\cdots\leq d_n$, define the Coxeter braid … where $\operatorname{JM}_k$ is a Jucys–Murphy element on the first $k$ strands. A braid is parity whe…

Let $\mathbf{G}$ be the countably infinite set of regular maps on the torus, where each $\mathcal{G}\in\mathbf{G}$ is an embedded graph in the torus that is arc-transitive. For eac…

Minimum-genus Bennequin surface conjecture. Every transverse link $\mathcal{T}$ in $(S^3,\xi_{std})$ is represented by a braid word $w$ whose Bennequin surface $F_w$ contains…

Split-link characterization. $B$ represents an $n$-split link if and only if

Prasad–Saeki braid conjecture. The Khovanov homology of a closed $3$-braid can have only $\mathbb{Z}_2$-torsion. The Khovanov homology of a closed $4$-braid cannot have odd torsion…

Let $B_i$ be the braids occurring in the paper's construction, and let $K_{2,1,3,7}$ and $K_{4,1,3,3}$ denote the corresponding knots. A thin position is an embedding realizing min…

Let $g_{n,k}$ denote the number of curve diagrams of genus $n$ and complexity $k$, as in the paper. Let $n\geq 2$ be an integer. Periodic-limit conjecture. There exists a positive…

Let $L$ be a link of braid index $b$, and let $B_1$ and $B_2$ be braid diagrams of $L$, each with $b$ strands. The writhe of a braid diagram is the signed crossing number. Jones' c…

Quasi-isomorphism conjecture. The cochain maps from Theorem 1 are quasi-isomorphisms for $n>3$. This conjecture asserts that the diagram complexes calculate the cohomology of space…

Jp-speciality conjecture. All minimal positive braids are Jp-special.