Relation between the number of facets and of free faces
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First, give the definition.
- A facet is any simplex in a complex that is not a face of any larger simplex. (aka maximal face)
- A simplex $tau$ is called a free face if it is the face of only one facet in a simplicial complex.
Here is an example. Suppose we have a simplicial complex {{1,2,3}, {3,4}}. {1,2,3} and {3,4} are all facets and {4}, {1,2}, {2,3}, {1,3} are free faces.
Here is my question: is there any research about the maximal number of free faces given an arbitrary simplicial complex which has n 0-simplex, which can be regarded vertices? Is there any relationship between the number of free faces and of the facets for an arbitrary simplicial complex?
simplicial-complex
asked Dec 17 '18 at 9:36
SoonerSooner
197
$endgroup$
add a comment |
$begingroup$
First, give the definition.
- A facet is any simplex in a complex that is not a face of any larger simplex. (aka maximal face)
- A simplex $tau$ is called a free face if it is the face of only one facet in a simplicial complex.
Here is an example. Suppose we have a simplicial complex {{1,2,3}, {3,4}}. {1,2,3} and {3,4} are all facets and {4}, {1,2}, {2,3}, {1,3} are free faces.
Here is my question: is there any research about the maximal number of free faces given an arbitrary simplicial complex which has n 0-simplex, which can be regarded vertices? Is there any relationship between the number of free faces and of the facets for an arbitrary simplicial complex?
simplicial-complex
asked Dec 17 '18 at 9:36
SoonerSooner
197
$endgroup$
$begingroup$
Your definition of free face is not the usual one. Normally a free face is required to be a codimension one face of a facet (and not contained in any other facets), so ${1}$ and ${2}$ are not free faces in your example.
$endgroup$
– Eric Wofsey
Dec 17 '18 at 15:46
$begingroup$
Thanks for your mention! Do you have any idea that given n 0-simplex, how to construct a simplicial complex which has the maximum number of free faces?
$endgroup$
– Sooner
Dec 18 '18 at 2:11
add a comment |
$begingroup$
First, give the definition.
- A facet is any simplex in a complex that is not a face of any larger simplex. (aka maximal face)
- A simplex $tau$ is called a free face if it is the face of only one facet in a simplicial complex.
Here is an example. Suppose we have a simplicial complex {{1,2,3}, {3,4}}. {1,2,3} and {3,4} are all facets and {4}, {1,2}, {2,3}, {1,3} are free faces.
Here is my question: is there any research about the maximal number of free faces given an arbitrary simplicial complex which has n 0-simplex, which can be regarded vertices? Is there any relationship between the number of free faces and of the facets for an arbitrary simplicial complex?
simplicial-complex
asked Dec 17 '18 at 9:36
SoonerSooner
197
$endgroup$
First, give the definition.
- A facet is any simplex in a complex that is not a face of any larger simplex. (aka maximal face)
- A simplex $tau$ is called a free face if it is the face of only one facet in a simplicial complex.
Here is an example. Suppose we have a simplicial complex {{1,2,3}, {3,4}}. {1,2,3} and {3,4} are all facets and {4}, {1,2}, {2,3}, {1,3} are free faces.
Here is my question: is there any research about the maximal number of free faces given an arbitrary simplicial complex which has n 0-simplex, which can be regarded vertices? Is there any relationship between the number of free faces and of the facets for an arbitrary simplicial complex?
simplicial-complex
simplicial-complex
asked Dec 17 '18 at 9:36
SoonerSooner
197
asked Dec 17 '18 at 9:36
SoonerSooner
197
edited Dec 18 '18 at 5:54
Sooner
asked Dec 17 '18 at 9:36
SoonerSooner
197
asked Dec 17 '18 at 9:36
SoonerSooner
197
asked Dec 17 '18 at 9:36
SoonerSooner
197
197
$begingroup$
Your definition of free face is not the usual one. Normally a free face is required to be a codimension one face of a facet (and not contained in any other facets), so ${1}$ and ${2}$ are not free faces in your example.
$endgroup$
– Eric Wofsey
Dec 17 '18 at 15:46
$begingroup$
Thanks for your mention! Do you have any idea that given n 0-simplex, how to construct a simplicial complex which has the maximum number of free faces?
$endgroup$
– Sooner
Dec 18 '18 at 2:11
add a comment |
$begingroup$
Your definition of free face is not the usual one. Normally a free face is required to be a codimension one face of a facet (and not contained in any other facets), so ${1}$ and ${2}$ are not free faces in your example.
$endgroup$
– Eric Wofsey
Dec 17 '18 at 15:46
$begingroup$
Thanks for your mention! Do you have any idea that given n 0-simplex, how to construct a simplicial complex which has the maximum number of free faces?
$endgroup$
– Sooner
Dec 18 '18 at 2:11
$begingroup$
Your definition of free face is not the usual one. Normally a free face is required to be a codimension one face of a facet (and not contained in any other facets), so ${1}$ and ${2}$ are not free faces in your example.
$endgroup$
– Eric Wofsey
Dec 17 '18 at 15:46
$begingroup$
Your definition of free face is not the usual one. Normally a free face is required to be a codimension one face of a facet (and not contained in any other facets), so ${1}$ and ${2}$ are not free faces in your example.
$endgroup$
– Eric Wofsey
Dec 17 '18 at 15:46
$begingroup$
Thanks for your mention! Do you have any idea that given n 0-simplex, how to construct a simplicial complex which has the maximum number of free faces?
$endgroup$
– Sooner
Dec 18 '18 at 2:11
$begingroup$
Thanks for your mention! Do you have any idea that given n 0-simplex, how to construct a simplicial complex which has the maximum number of free faces?
$endgroup$
– Sooner
Dec 18 '18 at 2:11
add a comment |
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$begingroup$
Your definition of free face is not the usual one. Normally a free face is required to be a codimension one face of a facet (and not contained in any other facets), so ${1}$ and ${2}$ are not free faces in your example.
$endgroup$
– Eric Wofsey
Dec 17 '18 at 15:46
$begingroup$
Thanks for your mention! Do you have any idea that given n 0-simplex, how to construct a simplicial complex which has the maximum number of free faces?
$endgroup$
– Sooner
Dec 18 '18 at 2:11