The precise description of these invariants can be found in papers V. Turaev and O. Viro [1], S. Matveev,M. Ovchinnikov, M. Sokolov [2] and in chapter 8 of S. Matveev's book [3].
Epsilon-invariant is homologically trivial part of Turaev-viro invariant. Epsilon-invariant always computes more faster than Turaev-Viro invariant of the same order because only even colorings are involved. Epsilon-invariant of order 5 is indentic to &Epsilon-invariant described by Sergey Matveev in independent terms. See [3] chapter 8 for details.
The Recognizer can compute Turaev-Viro or Epsilon invariants order from 3 upto 16. The computation can be interrupted by clicking Stop button on toolbar or by pressing Ctrl+Break. But such stopping may take afew seconds.
To understand using notation, recall that Turaev-Viro invariants of order r are Laurent polynomials in q, where q is a 2r-th root of unity (such that its square is a primitive root of unity of degree r). The symbol s_k here stands for the sum
If r=5 then σ1 =½ (1 +√2). It is the biggest root of equation ε2=ε+1 - the value using in constructing of Matveev's ε-invariant.
Below you can see the result of calculation of the Turaev-Viro and the epsilon invariants up to order 11 for the 3-manifold (S^2; (2,1),(3,1),(9,1),(1,-1)) presented by this spine.
Turaev-Viro Invariants : order value T_3 1 T_4 1 T_5 s_1 +1 T_6 3 T_7 s_2 +s_1 +4 T_8 2s_2 +3 T_9 7s_2 +13s_1 +18 T_10 3s_2 +2 T_11 5s_4 +5s_3 +12s_2 +8s_1 +19
Homologically trivial part of Turaev-Viro Invariants: order value E_3 1 E_4 1 E_5 s_1 +1 E_6 3 E_7 s_2 +s_1 +4 E_8 2s_2 +3 E_9 7s_2 +13s_1 +18 E_10 3s_2 +2 E_11 5s_4 +5s_3 +12s_2 +8s_1 +19