Photography principle, data transmission, and invariants of manifolds

TL;DR

This paper develops techniques to construct non-trivial invariants of 3-manifolds using the photography principle and Ptolemy relations, addressing previous limitations and proposing improvements.

Contribution

It introduces a novel approach to 3-manifold invariants based on Ptolemy relations and discusses methods to overcome triviality issues in the invariants.

Findings

Initial implementation yields trivial invariants

Proposed improvements aim to produce meaningful invariants

Framework connects photography principle with manifold invariants

Abstract

In the present paper we develop the techniques suggested in \cite{ManturovNikonov} and the photography principle \cite{ManturovWan} for constructing an invariant of 3-manifolds based on Ptolemy relation. We show that a direct implementation of the techniques leads to a trivial invariant and discuss how this approach can be improved to circumvent the difficulties encountered.