The 2x3 and 3x4 chessboard complexes (form a square grid of vertices and make a simplex for any set of vertices no two of which are in the same row or column) are a 6-cycle and a triangulated torus with 24 triangles, respectively. The 4x5 chessboard complex is only a pseudomanifold — each vertex has the 3x4 torus as its link, rather than a spherical link that a proper manifold would have — but if you delete its 20 vertices you get a bona-fide cusped hyperbolic 3-manifold, triangulated by 120 regular ideal tetrahedra. It sort of looks like the kind of manifold you might get as the complement of a 20-component link in Euclidean space. Is it a link complement? And if so, which link is it the complement of?


Edit: here are a couple of general references on chessboard complexes.


Ziegler, G. M. (1994). Shellability of chessboard complexes. [Israel J. Math. 87: 97–110](http://dx.doi.org/10.1007/BF02772986).


Björner, A.; Lovász, L.; Vrecica, S. T.; Zivaljevic, R. T. (1994). Chessboard complexes and matching complexes. [J. London Math. Soc. 49: 25–39](http://web.cs.elte.hu/~lovasz/morepapers/chess.pdf).


They also have important applications to the proof of colored Tverberg theorems in discrete geometry: see, e.g.


Pavle V. M. Blagojević, Benjamin Matschke, Günter M. Ziegler (2009). Optimal bounds for the colored Tverberg problem. [arXiv:0910.4987](http://arxiv.org/abs/0910.4987).



Is the 4x5 chessboard complex a link complement?
I met this manifold before. It is a normal cover of the orbifold $\mathbb{H}^3/\mathrm{PSL}(2,\mathbb{Z}[\zeta])$ where $\zeta=e^{\pi i/3}$.


I suspect that it actually is $\mathbb{H}^3/\ker\left(\mathrm{PSL}(2,\mathbb{Z}[\zeta])\rightarrow \mathrm{PSL}(2,\mathbb{Z}[\zeta]/I)\right)$ where $I$ is the ideal $\langle 2+2\zeta\rangle$.


For $I=\langle 2+\zeta\rangle$, the manifold is a link complement, see Ian Agol's paper on the Thurston Congruence Link for a drawing.


For $I=\langle 3\rangle$, I could show that it is a link complement, but failed at constructing the link.


