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

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 $B^{\prime }\notin \{B,-B^{\star }\}$, then the monochromatic 3-connected (mono3c) blinks $B$ and $B^{\prime }$ 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.

26 pages, 19 figures, 23 references. version2: References for SnapPy, GAP and Sage