Values of the t-invariant for small Seifert manifolds

TL;DR

This paper computes the t-invariant, a specific Turaev-Viro invariant, for small Seifert manifolds with three singular fibers, revealing its dependence on manifold parameters modulo five and identifying multiple distinct values.

Contribution

It provides explicit calculations of the t-invariant for a class of Seifert manifolds and shows its relation to manifold parameters and homology.

Findings

The t-invariant values depend on parameters modulo five.

There are 12 distinct t-invariant values for these manifolds.

The t-invariant is not solely determined by the first homology group.

Abstract

The t-invariant can be considered as the Turaev-Viro invariant of order 5 computed for integer colors only. We compute all values of the t-invariant for Seifert manifolds with base sphere and three singular fibers. As a result we show that the manifolds parameters modulo five define the value of the t-invariant. Partially we show that there are 12 distinct values of the t-invariant for these manifolds. Some examples show that the t-invariant for these manifolds is not defined by the first homology group.