Kirby Problem 2.21

Definitions and mathematical statement

Let $S_g^b$ be a compact connected oriented surface of genus $g\geq1$ with $b\geq0$ boundary components. Its mapping class group fixes the boundary pointwise. For a simple closed curve $c$, write $T_c$ for the right-handed Dehn twist. Choose curves $\delta_1,\ldots,\delta_b$ parallel to the boundary components and put $\Delta_{g,b}=T_{\delta_1}\cdots T_{\delta_b}$; for $b=0$ this product is the identity.

An admissible positive factorization is an equality in this mapping class group \[ \Delta_{g,b}=T_{c_1}\cdots T_{c_\ell}, \] where every $c_i$ is essential and not boundary-parallel: it bounds neither a disk nor an annulus with a boundary component. Separating curves are allowed, and every factor is a single positive twist. Define $m_{g,b}$ as the least possible length $\ell$, with value $+\infty$ when no such factorization exists. In the closed case require $\ell>0$, excluding the empty identity factorization.

Determine $m_{g,b}$ and construct factorizations attaining each finite minimum. Both genus and number of boundary components are parameters, so a complete answer also identifies the pairs for which no factorization exists. Boundary twists are allowed on the left but not as factors on the right.

Short English statement

Find the shortest nontrivial positive factorization of the boundary multitwist using twists about essential nonboundary curves, for every genus and number of boundary circles.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2769 (KP-2.21), 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 2.21. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.