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

Abstract

A 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 [14]. Switching black and gray in a blink B, giving −B, reverses the manifold orientation. The dual of the blink B in the sphere S 2 is denoted by B . Blinks B and −B induce the same 3-manifold. The paper reinforces the Conjecture that if B / ∈ {B, −B }, 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 . We also present a reformulation of the combinatorial algorithm to obtain the WRT-invariants of [13] using only the blink.