# facets -- the facets of a simplicial complex

## Synopsis

• Usage:
facets D
• Inputs:
• Optional inputs:
• useFaceClass => ..., default value false, Option to return faces in the class Face
• Outputs:
• , with one row, whose entries are squarefree monomials representing the facets (maximal faces) of D

## Description

In Macaulay2, every simplicial complex is equipped with a polynomial ring, and the resulting matrix of facets is defined over this ring.

The 3-dimensional sphere has a unique minimal nonface which corresponds to the interior.

 i1 : R = ZZ[a..e]; i2 : sphere = simplicialComplex monomialIdeal(a*b*c*d*e) o2 = | bcde acde abde abce abcd | o2 : SimplicialComplex i3 : facets sphere o3 = | bcde acde abde abce abcd | 1 5 o3 : Matrix R <--- R
The following faces generate a simplicial complex consisting of a triangle (on vertices a,b,c), two edges connecting c to d and b to d, and an isolated vertex e.
 i4 : D = simplicialComplex {e, c*d, b*d, a*b*c, a*b, c} o4 = | e cd bd abc | o4 : SimplicialComplex i5 : facets D o5 = | e cd bd abc | 1 4 o5 : Matrix R <--- R
There are four facets of D.

Note that no computatation is performed by this routine; all the computation was done while constructing the simplicial complex.

A simplicial complex is displayed by listing its facets, and so this function is frequently unnecessary.