Feynman motives of banana graphs(7)

sinceforthebananagraphsthehypersurfacesXΓnarerational.

Remark3.16.ThepreviousremarkexplainstheappearanceofatermnTn 2intheexpression(3.14).Theremainingtermsareanalternatingsumoftori.Thistermcanbeviewedas

(3.15)Tn ( 1)n

T+1,

forg(T)=Tn.AccordingtoLemma3.4,thisistheclassofthehyperplanesectionofthecomplementofthealgebraicsimplexΣn+1inPn.However,howgeometricallyonecanassociateaPntoagraphhypersurfaceXΓn Pn 1isunclear,sothatasatisfactory

conceptualexplanationoftheoccurrenceof(3.15)in(3.14)isstillmissing.

Forcompletenesswealsogivetheexplicitformulaoftheclass(3.14)writtenintermsofclasses[Pr].

Corollary3.17.Intermsofclassesofprojectivespacestheclass[Pn 1 XΓn]isgiven

by

n 1

(3.16)[Pn 1 XΓ

k=0 n+1+2 nn 1 kPk

n]=( 1)[]+n 2

k=0 n 1k+1 ( 1)n 2 k

k[Pk].

Proof.Theformula(3.16)isobtainedeasilyusingthetransformationrulesofLemma3.3togofromexpressionsinTtoexpressionsin[Pr],sothat

(Tn ( 1)n)/(T+1)→ (P 1)(P 1)n ( 1)n