2009/11/30 by Nathan Geer, Bertrand Patureau-Mirand
Mathematics · #math.GT #math.QA #msc:57M25 #msc:57M27 #msc:17B37
paper · pdf · doi:10.2140/agt.2011.11.1821
published as Algebr. Geom. Topol. 11 (2011) 1821-1860 · 31 pages, 14 figures. Minor changes and reference added in version 2
arxiv created 2013/07/11 · arxiv updated 2014/09/30
For q a root of unity of order 2r, we give explicit formulas of a family of 3-variable Laurent polynomials Ji,j,k with coefficients in Z[q] that encode the 6j-symbols associated with nilpotent representations of Uqsl2. For a given abelian group G, we use them to produce a state sum invariant taur(M,L,h1,h2) of a quadruplet (compact 3-manifold M, link L inside M, homology class h1∈ H1(M,Z), homology class h2∈ H2(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∈ H1(M,C^*)) defined in [arXiv:0910.1624] from the category of nilpotent representations of Uqsl2. They refine them as TV(M,L,f)= Sumh taur(M,L,h,f') where f' correspond to f with the isomorphism H2(M,C^*) ~ H1(M,C^*).