Wigglyhedra

Abstract Motivated by categorical representation theory, we define the wiggly complex, whose vertices are arcs wiggling around $$n+2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> points on a line, and whose faces are sets of wiggly arcs which are pairwise pointed and non-crossing. The wiggly complex is a $$(2n-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensional pseudomanifold, whose facets are wiggly pseudotriangulations. We show that wiggly pseudotriangulations are in bijection with wiggly permutations, which are permutations of [2 n ] avoiding the patterns $$(2j-1) \cdots i \cdots (2j)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>j</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> <mml:mo>⋯</mml:mo> <mml:mi>i</mml:mi> <mml:mo>⋯</mml:mo> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>j</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for $$i &lt; 2j-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>2</mml:mn> <mml:mi>j</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and $$(2j) \cdots k \cdots (2j-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>j</mml:mi> <mml:mo>)</mml:mo> <mml:mo>⋯</mml:mo> <mml:mi>k</mml:mi> <mml:mo>⋯</mml:mo> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>j</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for $$k &gt; 2j$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>2</mml:mn> <mml:mi>j</mml:mi> </mml:mrow> </mml:math> . These permutations define the wiggly lattice, an induced sublattice of the weak order. We then prove that the wiggly complex is isomorphic to the boundary complex of the polar of the wigglyhedron, for which we give explicit and simple vertex and facet descriptions. Interestingly, we observe that any Cambrian associahedron is normally equivalent to a well-chosen face of the wigglyhedron. Finally, we recall the correspondence of wiggly arcs with objects in a category, and we develop categorical criteria for a subset of wiggly arcs to form a face of the wiggly complex. Graphical abstract