## Algebraic & Geometric Topology

### Polynomial $6j$–symbols and states sums

#### Abstract

For a given $2r$–th root of unity $ξ$, we give explicit formulas of a family of $3$–variable Laurent polynomials $Ji,j,k$ with coefficients in $ℤ[ξ]$ that encode the $6j$–symbols associated with nilpotent representations of $Uξ(sl(2))$. For a given abelian group $G$, we use them to produce a state sum invariant $τr(M,L,h1,h2)$ 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 by the authors and Turaev in [Compos. Math. 145 (2009) 196–212]. For an oriented $3$–manifold $M$, the invariants are related to $τ(M,L,ϕ∈H1(M,ℂ∗))$ defined by the authors and Turaev in [arXiv:0910.1624] from the category of nilpotent representations of $Uξ(sl(2))$. They refine them as $τ(M,L,ϕ)= ∑h1τr(M,L,h1,ϕ̃)$ where $ϕ̃$ correspond to $ϕ$ with the isomorphism $H2(M,ℂ∗)≃H1(M,ℂ∗)$.

#### Article information

Source
Algebr. Geom. Topol., Volume 11, Number 3 (2011), 1821-1860.

Dates
Accepted: 8 October 2010
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715247

Digital Object Identifier
doi:10.2140/agt.2011.11.1821

Mathematical Reviews number (MathSciNet)
MR2821443

Zentralblatt MATH identifier
1232.57015

#### Citation

Geer, Nathan; Patureau-Mirand, Bertrand. Polynomial $6j$–symbols and states sums. Algebr. Geom. Topol. 11 (2011), no. 3, 1821--1860. doi:10.2140/agt.2011.11.1821. https://projecteuclid.org/euclid.agt/1513715247

#### References

• S Baseilhac, R Benedetti, Quantum hyperbolic invariants of $3$–manifolds with ${\rm PSL}(2,\mathbb C)$–characters, Topology 43 (2004) 1373–1423
• F Costantino, J Murakami, On ${\rm SL}(2,\C)$ quantum $6j$–symbols and its relation to the hyperbolic volume.
• N Geer, B Patureau-Mirand, V Turaev, Modified $6j$–symbols and $3$–manifold invariants
• N Geer, B Patureau-Mirand, V Turaev, Modified quantum dimensions and re-normalized link invariants, Compos. Math. 145 (2009) 196–212
• C Kassel, Quantum groups, Graduate Texts in Math. 155, Springer, New York (1995)
• A N Kirillov, N Y Reshetikhin, Representations of the algebra ${U}\sb q({\rm sl}(2)),\;q$–orthogonal polynomials and invariants of links, from: “Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988)”, (V G Kac, editor), Adv. Ser. Math. Phys. 7, World Sci. Publ., Teaneck, NJ (1989) 285–339
• G Masbaum, P Vogel, $3$–valent graphs and the Kauffman bracket, Pacific J. Math. 164 (1994) 361–381
• T Ohtsuki, Quantum invariants. A study of knots, $3$–manifolds, and their sets, Series on Knots and Everything 29, World Sci. Pub., River Edge, NJ (2002)
• V G Turaev, Quantum invariants of knots and $3$–manifolds, de Gruyter Studies in Math. 18, de Gruyter, Berlin (1994)
• V G Turaev, O Y Viro, State sum invariants of $3$–manifolds and quantum $6j$–symbols, Topology 31 (1992) 865–902
• O Y Viro, Quantum relatives of the Alexander polynomial, Algebra i Analiz 18 (2006) 63–157