Geodesic
$\Delta$-graphs of polytopes in Bruns and Gubeladze $K$-theory
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 6 (2013), pp. 19-24

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W. Bruns and J. Gubeladze introduced a new variant of algebraic $K$-theory, where \linebreak $K$-groups are additionally parametrized by polytopes of some type. In this paper we propose a notion of stable $E$-equivalence which can be used to calculate $K$-groups for high-dimensional polytopes. Polytopes which are stable $E$-equivalent have similar inner structures and isomorphic $K$-groups. In addition, for each polytope we define a $\Delta$-graph which is an oriented graph being invariant under a stable $E$-equivalence. 
@article{VMUMM_2013_6_a3,
     author = {M. V. Prikhod'ko},
     title = {$\Delta$-graphs of polytopes in {Bruns} and {Gubeladze} $K$-theory},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {19--24},
     publisher = {mathdoc},
     number = {6},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2013_6_a3/}
}
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M. V. Prikhod'ko. $\Delta$-graphs of polytopes in Bruns and Gubeladze $K$-theory. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 6 (2013), pp. 19-24. http://geodesic.mathdoc.fr/item/VMUMM_2013_6_a3/