Turaev-Viro and Epsilon invariants

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


 

[1] V. Turaev - O. Viro, State sum invariants of 3-manifolds and quantum 6j-symbols, Topology (4) 31 (1992), 865-902.

[2] S. Matveev - M. Ovchinnikov - M. Sokolov}, On a simple invariant of Turaev-Viro type, J. Math. Sci. (New York) (2) 94 (1999), 1226-1229.

[3] Matveev S. Algoritghmic topology and classification of 3-manifolds (Algorithms and Computation in Mathematics). Springer Berlin Heidelberg, 2007.492 p.