Kirby Problem 2.9

Definitions and mathematical statement

Let $D$ be an oriented disk with $n\geq2$ distinguished interior points $P$. The braid group $B_n$ is the mapping class group of $D$ preserving $P$ setwise and fixing $\partial D$ pointwise. For each isotopy class of embedded arc $a$ joining two points of $P$, with interior disjoint from $P$, let $h_a$ be the positive half-twist supported near $a$. It interchanges the endpoints by a positive half-turn. These elements are precisely the conjugates of a standard positive Artin generator.

Seek a useful presentation of $B_n$ whose generators are all the $h_a$, with relations expressed by simple, explicit rules for configurations of arcs. In particular, the source asks for homogeneous positive relations: each should be an equality \[ h_{a_1}\cdots h_{a_r}=h_{b_1}\cdots h_{b_r} \] without inverse letters and with the same number of factors on the two sides. The rules must present the whole braid group, so both validity and completeness of the relations are required. Ideally a finite collection of local geometric rules describes this infinite generating set. Merely giving the usual Artin presentation on adjacent half-twists does not specify relations among all these generators. Likewise, restricting to a particular finite family of band generators changes the requested presentation.

Short English statement

Find a transparent presentation of the braid group using every positive elementary half-twist, preferably with a small set of homogeneous geometric relation rules.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2757 (KP-2.9), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 2.9. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.