(a1) – +(0,-1) node\[below=-2.5pt\][[5]{}]{}; (a1) – ++(1,0) coordinate (a2) – +(0,1) node\[above=-2.5pt\][[2]{}]{}; (a2) – ++(1,0) coordinate (a3) – +(-45:1) node\[below right=-5pt\][[4]{}]{}; (a3) – +(45:1) node\[above right=-5pt\][[3]{}]{};

;

;

on the right we show one of its edges

For ${\rm{PT}}(1,2,3,4)$, it’s mapped to ${\mathcal{K}}_1$ .

For ${\rm{PT}}(1,2,3,4,5)$, see , it’s mapped to ${\mathcal{K}}_2$, see figure \[H5a\]. Any two adjacent vertices ,whose triangulations differs by a flip , share a common pole represented as an edge.

For ${\rm{PT}}(1,2,3,4,5,6)$, it’s mapped to ${\mathcal{K}}_3$,see figure \[H6a\]. Any two adjacent vertices ,whose triangulations differs by a flip, share two common poles represented as edge. Any adjacent two edges share a common pole represented as a face. Those Feynman diagrams sharing a common pole sit on the same face. Note that a pentagon corresponds to a two-particle pole and a square corresponds to a three-particle pole.

\(4) at (0.03,-5.83)

; (1) at (1.62,-6.35)

; (7) at (3.76,-5.28)

; (9) at (1.37,-4.49)

; (6) at (0.30,-5.02)

; (3) at (0.03,-1.83)

; (13) at (0.85,-2.08)

; (14) at (1.65,-3.67)

; (12) at (4.82,-2.09)

; (8) at (4.80,-3.40)

; (11) at (4.03,-.50)

; (2) at (2.46,-.05)

; (5) at (0.30,-1.03)

; (10) at (2.46,-2.64)

; (1)–(4)–(3)–(13)–(14)–(1); (1)–(7)–(8)–(12)–(14)–(1); (1)–(4)–(6)–(9)–(7)–(1); (14)–(12)–(11)–(13)–(14); 