Topological invariants of some chemical reaction networks

TL;DR

This paper explores the topological invariants of certain chemical reaction networks through the lens of toric varieties and their associated homotopy groups, linking algebraic topology with physical chemistry.

Contribution

It introduces a novel connection between toric dynamical systems, homotopy groups of symplectic spectra, and noncommutative algebraic structures in chemical networks.

Findings

Identification of homotopy group elements associated with smooth toric varieties.

Establishment of links between noncommutative Hopf algebroids and formal diffeomorphisms.

Conjectural framework for a noncommutative Helmholtz free energy in complex systems.

Abstract

Certain toric dynamical systems studied in physical chemistry have associated toric varieties which, when smooth, represent elements in the homotopy groups $M\xi_*B\T$ of a symplectic variant of the $A_\infty$ Baker-Richter spectrum $M\xi$. The noncommutative Hopf algebroid $(M\xi_*,M\xi_*M\xi)$ has interesting connections to the group of formal diffeomorphism of the noncommutative line, conjecturally defining a noncommutative analog Helmholtz free energy for systems of complex symplectic (completely integrable) Hamiltonian toric manifolds