An error formula for simplicial complexes of non-pure dimension
Kellan Steele and Dr. Julie Beier
Mathematics; Earlham College
Abstract
Imagine we have three lines, each with points on the ends; we connect each of
these lines by gluing them together, to form a triangle. With a similar method
imagine gluing four triangles together at their points to form a tetrahedron.
Now, what would happen if we took one of the triangles and glued a point
onto one of its lines? The mathematician, Richard Stanley, studied shapes,
such as two tetrahedrons glued together, and the effects of gluing an extra
point, line, etc. onto the shape. He devised ways to mathematically describe
the shapes and gluings with polynomials. We studied the uses and effects of
changing the type of shapes that are glued together. We found that Stanley’s
polynomial did not extend when we had, for example, a tetrahedron attached
to a triangle with an extra point, as there is an amount in the polynomial,
the error, missing.
Definitions
•Simplex: A convex face (point, line, triangle, etc.) formed by using d points
•Complex: One or more simplices glued together nicely
•f-polynomial: Is a polynomial that keeps track of the number of faces of
each dimension in a complex
•h-polynomial: Is a re-centering of the f-polynomial
•Stanley’s definition of the local h-polynomial: Let be a pure
d-dimensional complex and 0 be with subdivisions, then
h( 0, x) = X
F2
l( 0)h(lk )
In our case, h(lk ) = 1, so we omit this piece of the equation
•Simple subdivisions: Denoted , is a subdivision where an
n-dimensional face is added on an (n + 1)-dimensional face, and occurs only
at the vertices of
Previous work
Let be a non-pure complex, with highest dimension d, where n is the di-mension
of the face we are adding, and c is the dimension of the face we are
subdividing. We have previously shown that:
•the difference between the h-polynomial of and 0 is
(−1)d−n−1xn+1(x − 1)d−n−1
•when there is one subdivision, the error of that subdivision is
(−1)nxn+1[(−1)d−1(x − 1)d−n−1 + (−1)c(x − 1)c−n−1]
Corollary 0.1
Let be a simplicial complex of highest dimension d with components joined only at their vertices. Let 0 = kP
i=1
( + i)
be with at least one simple subdivision. Then, as a result of h( + ) − h( ) = (−1)d−n−1xn+1(x − 1)d−n−1 we
have that h( 0) − h( ) is kX
i=1
[h( + i) − h( )] = (−1)d−ni−1xni+1(x − 1)d−ni−1.
Theorem 0.2
Let be a non-pure simplicial complex of highest dimension d and let 0 be with multiple simple subdivisions. Let
h( 0)−h( ) follow from Corollary 0.1 and let E( , + i) = (−1)nixni+1[(−1)d−1(x−1)d−ni−1+(−1)ci(x−1)ci−ni−1].
Then,
E( , 0) = kX
i=1
E( , + i).
Example
We can see that the complex with all of the subdivisions 1, 2, 3 is the complex 0. So if we calculate the error for
each of the complexes, E( , + i) where i = 1, 2, 3, and add them together we will get E( , 0), by Theorem 0.2.
Conclusion
•We found that the error of a complex with multiple subdivisions is equivalent to the sum of the error of each of the
individual subdivisions.
•It is interesting to consider what would happen if we could glue at places other than points. For example, what would
happen if we glued a tetrahedron to a triangle but instead of gluing at a point, we glued at a line.