[Paper Review] Symmetrical laws of structure of helicoidally-like biopolymers in the framework of algebraic topology. I. Root lattice E8 and the closed sequence of algebraic polytopes

M. I. Samoylovich, А. Л. Талис|arXiv (Cornell University)|Oct 12, 2012

Point processes and geometric inequalities8 references3 citations

TL;DR

This paper introduces a topological framework using algebraic topology to describe ordered non-crystalline structures in helicoidally-like biopolymers, centered on the E8 root lattice. It constructs a closed sequence of 4D algebraic polytopes derived from E8’s second coordination sphere, revealing a hierarchy of manifolds and helicoidal surfaces that exhibit topological stability via zero instability index and Weierstrass representation, extending beyond classical crystallography to model biopolymer symmetry.

ABSTRACT

In the framework of algebraic topology the closed sequence of 4-dimensional polyhedra(algebraic polytopes) was defined. These polytopes were determined by the second coordination sphere of 8-dimensional lattice E8. The ordered non-crystalline structure is determined by a chain of constructions of algebraic topology: an algebraic polytope, a homogeneous manifold in a 3-dimensional Euclidean space E3, locally-homogeneous manifold, locally minimal surface, one-parameter family of helicoids, bundle (cover) with a base of cell complexes, local-lattice packing of cell complexes into a substructure of E3, determined by helicoids. The formalism being developed allows one to surmount restrictions of classical crystallography and to single out a class of ordered non-crystalline structures, invariant with respect to structures determined by the lattice E8. The topological stability of such substructures is determined by their relatedness to Weierstrass' representation, as well as the condition that the instability index of the surface equals zero. Formation of such structures corresponds to lifting a configuration degeneracy, and the stability of a state - to existence of a point of bifurcation.

Motivation & Objective

• To extend the scope of structural biology beyond classical crystallography by identifying ordered non-crystalline structures invariant under E8 lattice symmetries.
• To formalize a sequence of 4-dimensional algebraic polytopes derived from the second coordination sphere of the E8 lattice.
• To establish a topological framework linking algebraic topology, minimal surfaces, and helicoidal biopolymer architectures.
• To demonstrate topological stability of such structures through the instability index and Weierstrass representation.

Proposed method

• Construct a closed sequence of 4-dimensional algebraic polytopes based on the second coordination sphere of the E8 root lattice.
• Use algebraic topology to define a chain of geometric objects: algebraic polytope → homogeneous manifold in E3 → locally-homogeneous manifold → locally minimal surface.
• Model the minimal surface as a one-parameter family of helicoids, representing the geometric core of helicoidally-like biopolymers.
• Define a bundle (cover) with a base of cell complexes, enabling local-lattice packing of these complexes into a substructure of E3.
• Apply Weierstrass representation to characterize the surface's conformal structure and ensure topological stability.
• Enforce zero instability index as a condition for structural stability, indicating bifurcation point existence.

Experimental results

Research questions

• RQ1 How can algebraic topology describe ordered non-crystalline structures in helicoidally-like biopolymers beyond classical crystallography?
• RQ2 What is the role of the E8 root lattice in generating a closed sequence of 4D algebraic polytopes?
• RQ3 How do helicoidal surfaces arise from the topological construction chain and what is their geometric significance?
• RQ4 What conditions ensure topological stability of such biopolymer-like substructures?
• RQ5 How does the Weierstrass representation and zero instability index relate to the formation and stability of these structures?

Key findings

• A closed sequence of 4-dimensional algebraic polytopes is formally defined, rooted in the second coordination sphere of the E8 lattice.
• The construction yields a one-parameter family of helicoidal surfaces as minimal surfaces embedded in E3, modeling biopolymer helicity.
• Topological stability is achieved when the instability index of the surface equals zero, indicating a bifurcation point in the configuration space.
• The structures are invariant under E8 lattice symmetries, extending the concept of order beyond periodic crystalline lattices.
• Weierstrass representation provides a conformal framework that links the surface geometry to the underlying algebraic topology.
• The model reveals that formation of such structures corresponds to lifting configuration degeneracy, stabilizing the system via topological invariance.