Page 24 - 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. 24
1.3 Current graphs
Edges of derived embedding have form (f , t )(g , t a ), where f and g are faces of base
graph meeting along an edge of current a ; decide which way current applies
based on rules below.
Faces of derived embedding come from vertices of base graph. For each vertex mul-
tiply currents of incident edges together in direction of vertex to get net current.
Order r of net current in current group specifies how many times that sequence
of edges is repeated to give a face of the derived graph, so vertex of degree d yields
face of degree d r .
If net current of vertex is the identity, say vertex obeys the Kirchoff Current
Law, KCL. Then vertex of degree d yields face of degree d .
Standard tracing algorithm:
at each vertex follow natural rotation;
if an edge has vertices of different directions at its ends, cross over in the middle;
if we are leaving a clockwise vertex on an edge with face f on left, face g on right,
current is a , then in derived graph (f , t ) is joined to (g , t a ) for each t ∈ Γ (current
acts 90◦ clockwise because vertex is clockwise); for an anticlockwise vertex swap
left ↔ right (current acts 90◦ anticlockwise).
There are alternative ways to trace faces that are more convenient in some ways.
Clockwise-biased tracing algorithm:
at clockwise vertices follow natural rotation;
at anticlockwise vertices follow reversed rotation;
if going along an edge with face f on left, face g on right, current is a , then in derived
graph ( f , t ) is joined to (g , t a ) for each t ∈ Γ (currents always act 90◦ clockwise).
Advantage is that we don’t have to worry about vertex rotations until we are actually at
vertex. Also less complicated when have to deal with twisted edges, later.
Also have anticlockwise-biased tracing algorithm: swap clockwise ↔ anticlockwise, left
↔ right. Can choose whether to use clockwise-biased or anticlockwise-biased algo-
rithm depending on whether more clockwise or anticlockwise vertices.
Edges of derived embedding have form (f , t )(g , t a ), where f and g are faces of base
graph meeting along an edge of current a ; decide which way current applies
based on rules below.
Faces of derived embedding come from vertices of base graph. For each vertex mul-
tiply currents of incident edges together in direction of vertex to get net current.
Order r of net current in current group specifies how many times that sequence
of edges is repeated to give a face of the derived graph, so vertex of degree d yields
face of degree d r .
If net current of vertex is the identity, say vertex obeys the Kirchoff Current
Law, KCL. Then vertex of degree d yields face of degree d .
Standard tracing algorithm:
at each vertex follow natural rotation;
if an edge has vertices of different directions at its ends, cross over in the middle;
if we are leaving a clockwise vertex on an edge with face f on left, face g on right,
current is a , then in derived graph (f , t ) is joined to (g , t a ) for each t ∈ Γ (current
acts 90◦ clockwise because vertex is clockwise); for an anticlockwise vertex swap
left ↔ right (current acts 90◦ anticlockwise).
There are alternative ways to trace faces that are more convenient in some ways.
Clockwise-biased tracing algorithm:
at clockwise vertices follow natural rotation;
at anticlockwise vertices follow reversed rotation;
if going along an edge with face f on left, face g on right, current is a , then in derived
graph ( f , t ) is joined to (g , t a ) for each t ∈ Γ (currents always act 90◦ clockwise).
Advantage is that we don’t have to worry about vertex rotations until we are actually at
vertex. Also less complicated when have to deal with twisted edges, later.
Also have anticlockwise-biased tracing algorithm: swap clockwise ↔ anticlockwise, left
↔ right. Can choose whether to use clockwise-biased or anticlockwise-biased algo-
rithm depending on whether more clockwise or anticlockwise vertices.
