Polynomial 6j-Symbols and States Sums

TL;DR

This paper introduces explicit formulas for 3-variable Laurent polynomials encoding 6j-symbols of nilpotent U_qsl_2 representations, leading to new 3-manifold and link invariants via state sums at roots of unity.

Contribution

It provides explicit formulas for 6j-symbols and constructs new state sum invariants for 3-manifolds and links, refining existing invariants through skein calculus and modified dimensions.

Findings

Explicit formulas for 3-variable Laurent polynomials J_{i,j,k}

Construction of a new state sum invariant tau^r(M,L,h_1,h_2)

Relation of invariants to TV invariants and refinement via homology sums

Abstract

For q a root of unity of order 2r, we give explicit formulas of a family of 3-variable Laurent polynomials J_{i,j,k} with coefficients in Z[q] that encode the 6j-symbols associated with nilpotent representations of U_qsl_2. For a given abelian group G, we use them to produce a state sum invariant tau^r(M,L,h_1,h_2) of a quadruplet (compact 3-manifold M, link L inside M, homology class h_1\in H_1(M,Z), homology class h_2\in H_2(M,G)) with values in a ring R related to G. The formulas are established by a "skein" calculus as an application of the theory of modified dimensions introduced in [arXiv:0711.4229]. For an oriented 3-manifold M, the invariants are related to TV(M,L,f\in H^1(M,C^*)) defined in [arXiv:0910.1624] from the category of nilpotent representations of U_qsl_2. They refine them as TV(M,L,f)= Sum_h tau^r(M,L,h,f') where f' correspond to f with the isomorphism H_2(M,C^*) ~ H^1(M,C^*).