A topological proof of the non-degeneracy of harmonic structures on Sierpinski Gaskets

TL;DR

This paper provides a direct proof that harmonic structures on Sierpinski gaskets are non-degenerate for all levels, confirming a longstanding conjecture and offering an alternative to recent proofs using Tutte's spring theorem.

Contribution

It introduces a new direct proof method for the non-degeneracy of harmonic structures on Sierpinski gaskets, bypassing the need for Tutte's spring theorem.

Findings

Confirmed non-degeneracy of harmonic structures for all levels n≥2

Provided a direct proof approach

Validated the conjecture by Hino

Abstract

We present a direct proof of the non-degeneracy of the harmonic structures on the level-$n$ Sierpinski gaskets for any $n\geq 2$, which was conjectured by Hino in [H1,H2] and confirmed to be true by Tsougkas [T] very recently using Tutte's spring theorem.