11 problems

Let $L$) be the planar Murasugi sum of special alternating links $L_1,\ldots,L_n$. Murasugi sum conjecture. Then … This conjecture was proposed in connection with the analogous pro…

Jones's braid-index uniqueness conjecture. If

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

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

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 $L$ 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