Primitive core, collar, and particle-star words

Choose the common collar basepoint

in sector zero, in the identity reference chart. The collar meridian $\rho =15/4$, traversed in increasing angle, has chronological word $I,I,J_0$.

The core meridian is $\rho =29/10$. Since

it lies in $V_i$ throughout the window. Its chosen connecting path from $b_s$ follows the sector-zero bisector, on the graph, with no seam crossing.

For $s>0$, the star meridian $\rho =s$ lies in the new star, because $h-k\le 2s/3<s$. For its connecting path, first move within the sector-zero collar to the ray with $\xi :\zeta =2:1$, then follow that ray inward to $(\xi ,\zeta )=(2s/3,s/3)$. This path has no seam crossing and ends in $U_0^{-}$. It avoids the bisector crease except at no required endpoint.

To supply a finite presentation, truncate the apex by a small positive polygonal meridian, intersect the realized faces with the resulting compact annulus, triangulate its simple polygonal pieces, and refine with the necessary chart paths and boundary paths. Mark this full annular cell structure by its realization. Its fundamental group is the marked meridian group $\mathbb{Z}$. At $s=0$, use the regular limiting realization. No claim about an incomplete interior dual complex is needed.