Fan Realizations of Type $$A$$A Subword Complexes and Multi-associahedra of Rank 3

摘要: We present complete simplicial fan realizations of any spherical subword complex type $$A_n$$An for $$n\le 3$$n≤3. This provides multi-associahedra $$\varDelta _{2k+4,k}$$Δ2k+4,k, whose facets are in correspondence with $$k$$k-triangulations a convex $$(2k+4)$$(2k+4)-gon. solves the first open case problem finding where polytopality is not known. also two previously unknown cases complexes $$A_4$$A4, namely _{9,2}$$Δ9,2 and _{11,3}$$Δ11,3.