The $\kappa_r$-version of the WRT$_r$-invariants, monochromatic 3-connected blinks and evidence for a conjecture on their induced 3-manifolds

TL;DR

This paper investigates the relationship between monochromatic 3-connected blinks and the 3-manifolds they induce, providing evidence for a conjecture that distinct blinks induce distinct manifolds, and classifies 708 such blinks using advanced invariants.

Contribution

It offers new evidence supporting the conjecture that different blinks induce different 3-manifolds and refines the computational approach to WRT-invariants using blinks alone.

Findings

Classified 708 mono3c blinks into equivalence classes

Provided evidence supporting the conjecture on distinctness of manifolds from different blinks

Reformulated the algorithm for computing WRT-invariants using blinks

Abstract

A {\em blink} is a plane graph with a bipartition (black, gray) of its edges. Subtle classes of blinks are in 1-1 correspondence with closed, oriented and connected 3-manifolds up to orientation preserving homeomorphisms \cite{lins2013B}. Switching black and gray in a blink $B$, giving $-B$, reverses the manifold orientation. The dual of the blink $B$ in the sphere $\mathbb{S}^2$ is denoted by $B^ \star$. Blinks $B$ and $-B^\star$ induce the same 3-manifold. The paper reinforces the Conjecture that if $B' \notin \{B,-B^\star\}$, then the monochromatic 3-connected (mono3c) blinks $B$ and $B'$ induce distinct 3-manifolds. Using homology of covers and length spectra, we conclude the topological classification of 708 mono3c blinks that were organized in equivalence classes by WRT-invariants in \cite{lins2007blink}. We also present a reformulation of the combinatorial algorithm to obtain the WRT-invariants of \cite{lins1995gca} using only the blink.