17 problems

Murasugi–Przytycki's conjecture. For every alternating diagram $D$, this inequality is exact:

A bridge-braid spectrum is a sequence of integers associated to a knot, with the entries decreasing and ending in $\{\ldots,1,0\}$. Bridge-braid spectrum realization conjecture. Fo…

Let $b(d)$ denote the entries of the bridge-braid spectrum of a knot, indexed by the relevant integer parameter $d$. Concavity conjecture. The sequence $\{b(d)\}$ is concave upward…

Symmetric braid index conjecture. If $K$ is one of these eight SU knots, then

Conjecture 1. If $K$ is an alternating knot with $3\leq b(K)$, then

Conjectural Type 3 braid-index formulas. In the first positive case, $\textbf{b}(\mathcal{L})=2+\sum\alpha_j$; in the second positive case, $\textbf{b}(\mathcal{L})=1+\sum\alpha_j$…

Positive stable braid index conjecture. If $\{K_n\}$ is non-coherent, then

Stable braid index conjecture. The stable braid index is

Let $c\geq 3$, and let $n=\left\lceil\frac{c+1}{2}\right\rceil$. For each $b$ with $2\leq b\leq n$, let $k_{c,b}$ denote the number of $2$-bridge knots with crossing number $c$ and…

Let $K$ be a positive braid knot, meaning a knot represented as the closure of a braid whose crossings are all positive. Let $b(K)$ denote its bridge index and $\beta(K)$ its braid…

Let $K$ be an L-space knot, meaning a knot admitting a Dehn surgery yielding an L-space. Let $b(K)$ denote its bridge index and $\beta(K)$ its braid index. The L-space knot index c…

Let $K$ be a knot type, with braid index $\operatorname{braid}(K)$ and bridge index $\operatorname{bridge}(K)$. Consider the equilibrium shapes of closed springy wires representing…

A BB knot is a knot whose bridge index equals its braid index. For a BB knot realized as a closed springy metal wire, the equilibrium state is conjectured to have an almost planar…

Morton–Przytycki conjecture. The MP inequality of the braid index is an equality for $D$.

For an integer $n44 2$, let $B_n$ be the braid group on $n$ strands, let $a_1,\ldots,a_{n-1}$ be its standard generators, and let … be the full twist. Let $r_n$ denote a constant s…

Let $L$ be a special alternating link. Write $c(L)$ for its crossing number, $g(L)$ for its genus, and $b(L)$ for its braid index. Special alternating link inequality. One has … Th…

Murasugi-Przytycki conjecture. One has