A tougher challenge to 3-manifold topologists and group algebraists

TL;DR

This paper addresses unresolved questions in 3-manifold topology, specifically focusing on the homomorphism problem for certain 3-manifolds, reporting solutions for two previously uncertain cases and providing tools for further analysis.

Contribution

It provides solutions to two open cases in the homomorphism problem for specific 3-manifolds and offers an appendix for importing links into SnapPy for further research.

Findings

Solved two of the eleven remaining cases in L. Lins' thesis.

Provided an appendix for importing links into SnapPy.

Contributed to understanding the homomorphism problem in 3-manifold topology.

Abstract

This paper poses some basic questions about instances (hard to find) of a special problem in 3-manifold topology. "Important though the general concepts and propositions may be with the modern industrious passion for axiomatizing and generalizing has presented us ... nevertheless I am convinced that the special problems in all their complexity constitute the stock and the core of mathematics; and to master their difficulty requires on the whole the harder labor." Hermann Weyl 1885-1955, cited in the preface of the first edition (1939) of A. N. Whitehead's book {\em The classical groups: their invariants and representations} \cite{whitehead1997}. In this paper I focus on new uncertainties left unanswered in L. Lins thesis \cite{lins2007blink} on the homemorphism problem of eleven concrete pairs of closed orientable 3-manifolds induced by 3-connected monochromatic {\em blinks} (\cite{kauffman1994tlr}). The eleven HG8QI-classes are the only doubts left in the thesis, but the first two of them were solved few days ago and in this work I report on their solutions. We also include an appendix which can be used to import all the links of this paper into SnapPy. The appendix was obtained by drawing the links in SnapPy, work performed by C. Nascimento.