Rational Mixed Tate Motivic Graphs

TL;DR

This paper explores the combinatorics of a subcomplex of the Bloch-Kriz cycle complex related to mixed Tate motives, introducing a graphical approach to analyze algebraic cycles and their realizations.

Contribution

It introduces a graphical criterion for admissibility and connects sums of bivalent graphs to coboundary elements in the cycle complex, advancing the combinatorial understanding of mixed Tate motives.

Findings

Graphical criterion for admissibility of algebraic cycles

Sums of bivalent graphs correspond to coboundary elements

Computed Hodge realizations for new algebraic cycles

Abstract

In this paper, we study the combinatorics of a subcomplex of the Bloch-Kriz cycle complex [4] used to construct the category of mixed Tate motives. The algebraic cycles we consider properly contain the subalgebra of cycles that correspond to multiple logarithms (as defined in [12]). We associate an algebra of graphs to our subalgebra of algebraic cycles. We give a purely graphical criterion for admissibilty. We show that sums of bivalent graphs correspond to coboundary elements of the algebraic cycle complex. Finally, we compute the Hodge realization for an infinite family of algebraic cycles represented by sums of graphs that are not describable in the combinatorial language of [12].