2-knots with solvable group

TL;DR

This paper classifies certain 2-knots with solvable, torsion-free groups, distinguishing their reflexivity, symmetry properties, and providing explicit group normal forms, advancing the understanding of their topological and algebraic structures.

Contribution

It completes the classification of 2-knots with torsion-free, solvable groups by analyzing fibred knots with specific fiber manifolds and providing explicit normal forms for their groups.

Findings

Fibred 2-knots with Hantzsche-Wendt fiber are not reflexive.

Fibred 2-knots with $ ext{N}il^3$ fiber are reflexive.

Identified conditions for amphicheirality and invertibility.

Abstract

We complete the TOP classification of 2-knots with torsion-free, solvable knot group by showing that fibred 2-knots with closed fibre the Hantzsche-Wendt flat 3-manifold $HW$ are not reflexive, while every fibred 2-knot with closed fibre a $\mathbb{N}il^3$-manifold with base orbifold $S^2(3,3,3)$ is reflexive, and by giving explicit normal forms for the strict weight orbits of normal generators for the groups of all knots in either class. We also determine when the knots are amphicheiral or invertible, and show that the only non-trivial doubly null-concordant knots with such groups are those with group $\pi\tau_29_{46}$.