12 problems

Geodetic-number conjecture. If ${\boldsymbol{\alpha}}$ is a reduced expression, then the geodetic number of $\operatorname{\mathcal{B}}({\boldsymbol{\alpha}})$ is two. The source s…

Isometric-dimension conjecture. If $(W,S)$ is type $\Lambda$ and ${\boldsymbol{\alpha}}$ is a reduced expression, then

Odd-length braid-move conjecture. Most of the known results concerning braid graphs in Coxeter systems of type $\Lambda$ can be generalized to handle braid moves of odd length. The…

Box-product indecomposability conjecture. Braid graphs for links in all simply-laced Coxeter systems are indecomposable with respect to box product. Equivalently, if a graph $G$ is…

Unique-diametrical-pair conjecture. For a link ${\boldsymbol{\alpha}}$, there is a unique diametrical pair that determines the diameter of…

Unique-geodetic-pair conjecture. If ${\boldsymbol{\alpha}}$ is a link, then there is a unique set of cardinality two whose geodetic closure is all of $[${boldsymbol{alpha}}]$. The…

Diameter conjecture. If $(W,S)$ is type $\Lambda$ and ${\boldsymbol{\alpha}}$ is a reduced expression, then

Simply-laced partial-cube conjecture. The conclusion of Theorem holds for all simply-laced Coxeter systems, without requiring the Coxeter graph to be triangle-free.

Median braid-graph conjecture. Every such braid graph is a median graph.

Djoković–Winkler classes conjecture. The equivalence classes of the Djoković–Winkler relation are precisely the collections of edges corresponding to specific braid shadows.

$Y$-subgraph braid-graph conjecture. The graph $B({\boldsymbol{\alpha}})[Y_{\boldsymbol{\sigma}}]$ should also be the braid graph for some link derived from a link in…

Generalized support conjecture. A result analogous to Proposition should hold for $(W,S)$.