31 problems

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…

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…

The paper considers knots represented as closed 3-braids and finite-type invariants of knots. The conjectured examples are intended to distinguish a knot from its inverse using fin…

Let $K$ be a knot. Its braid index $br(K)$ is the minimum number of strands needed to express $K$ as a closed braid, and its braid length $l(K)$ is the minimum number of crossings…

Let an oriented $KL$ be achiral when its left and right forms are equivalent by an ambient isotopy. A reduced braid is palindromic when it is invariant under the relevant mirror an…

Let a minimum braid have $s$ strands, and let a graph tree associated with it have $s+1$ vertices. For each $n$, consider graph trees with $n$ vertices that have alternating minimu…

A knot is given as the closure of a braid on four strands, and the unknot recognition problem asks whether the resulting knot is isotopic to the unknot. Braid-index four polynomial…

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 $\mathcal{T}$ be a rainbow diagram of an $r$-component unlink of $2$-spheres, and let $\mathcal{U}_r$ be the crossingless rainbow diagram with $r$ strands. MTWS conjecture. The…

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 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…

Let $\delta_n$ be the minimum dilatation among pseudo-Anosov braids with $n$ strands, and let $\underline{\delta}_n$ denote the referenced candidate value. The remaining values of…

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 $\underline{L}$ be a pre-monotone Lagrangian link in the disc, and let $\norm{\cdot}_{\underline{L}}$ denote its Hofer pseudonorm on the associated braid group. Metric conjectu…

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…

In the classical case, the Bott–Taubes configuration space integral and the Kontsevich integral are invariants, also known respectively as the Chern–Simons invariant and the Knizhn…

Crossing-reduction conjecture. The number of crossings can be reduced to $n+2$ when $d\leq n$.

Stronger Form of Conjecture 2. For a transverse link $\mathcal{T}$ in $(S^3,\xi_{std})$, the Bennequin inequality is sharp if and only if $\mathcal{T}$ is represented by a strongly…

Stronger Form of Conjecture 1. Every transverse link $\mathcal{T}$ in $(S^3,\xi_{std})$ is represented by a braid word $w$ of braid index at most…

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