Page 40 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
P. 40
1.8 Bouchet’s covering triangulations
Theorem: If φ is a generative m -valuation then we have a triangulation of G(m) whose
triangles are given by
{( (x , i )(y , j )(z , k ) ) | (x y z ) ∈ T, i + j + k = φ(t )}.
This has the same orientability as the original triangulation.
• Clear that we get two triangles containing every (x , i )(y , j ), corresponding to the two
original triangles (w x y ) and (x y z ): values of i and j force values of h and k for third
vertices (w, h) and (z , k ).
• So just need to verify proper rotations. When we follow triangles around a vertex (x , i )
from edge (x , i )(y , j ) will end up at edge (x , i )(y , j ± φ(x )) after going around x once:
since φ(x ) generates m , we end up with all neighbours of (x , i ) after doing this m
times.
Finding a generative m -valuation
Restate question in more formal algebraic way.
• Consider m V = formal m -linear combinations of vertices in G , m -module.
• For each triangle t define t = x∈t (x , t )x ∈ m V . Define φ = t ∈T φ(t )t . φ is
generative m -valuation if coefficient of φ for vertex x is a generator of m for all x :
in that case say that φ is generative element of m V . This coefficient is just what
we called φ(x ) before: formal m -linear combinations are equivalent to m -valued
functions.
• Define submodule T generated by {t | t ∈ T }. Want to know if any generative element
in T .
• Depends on structure of diagonal graph D = D(Ψ): V (D) = V (G ), join w and z if they
are in adjacent triangles (w x y ) and (x y z ).
◦ If w z ∈ E (D) then one of w + z , w − z is in T : call it α(w, z ).
◦ If u and v are in the same component of D then one of u + v , u − v is in T : again
call it α(u , v ). (Use induction on previous statement.)
◦ So if could partition each component of D into pairs of vertices (u i , vi ), add up
all α(u i , vi ) and all coefficients ±1, so have a generative element.
Theorem: If φ is a generative m -valuation then we have a triangulation of G(m) whose
triangles are given by
{( (x , i )(y , j )(z , k ) ) | (x y z ) ∈ T, i + j + k = φ(t )}.
This has the same orientability as the original triangulation.
• Clear that we get two triangles containing every (x , i )(y , j ), corresponding to the two
original triangles (w x y ) and (x y z ): values of i and j force values of h and k for third
vertices (w, h) and (z , k ).
• So just need to verify proper rotations. When we follow triangles around a vertex (x , i )
from edge (x , i )(y , j ) will end up at edge (x , i )(y , j ± φ(x )) after going around x once:
since φ(x ) generates m , we end up with all neighbours of (x , i ) after doing this m
times.
Finding a generative m -valuation
Restate question in more formal algebraic way.
• Consider m V = formal m -linear combinations of vertices in G , m -module.
• For each triangle t define t = x∈t (x , t )x ∈ m V . Define φ = t ∈T φ(t )t . φ is
generative m -valuation if coefficient of φ for vertex x is a generator of m for all x :
in that case say that φ is generative element of m V . This coefficient is just what
we called φ(x ) before: formal m -linear combinations are equivalent to m -valued
functions.
• Define submodule T generated by {t | t ∈ T }. Want to know if any generative element
in T .
• Depends on structure of diagonal graph D = D(Ψ): V (D) = V (G ), join w and z if they
are in adjacent triangles (w x y ) and (x y z ).
◦ If w z ∈ E (D) then one of w + z , w − z is in T : call it α(w, z ).
◦ If u and v are in the same component of D then one of u + v , u − v is in T : again
call it α(u , v ). (Use induction on previous statement.)
◦ So if could partition each component of D into pairs of vertices (u i , vi ), add up
all α(u i , vi ) and all coefficients ±1, so have a generative element.
