## Algebraic & Geometric Topology

### Hopf diagrams and quantum invariants

#### Abstract

The Reshetikhin–Turaev invariant, Turaev’s TQFT, and many related constructions rely on the encoding of certain tangles ($n$–string links, or ribbon $n$–handles) as $n$–forms on the coend of a ribbon category. We introduce the monoidal category of Hopf diagrams, and describe a universal encoding of ribbon string links as Hopf diagrams. This universal encoding is an injective monoidal functor and admits a straightforward monoidal retraction. Any Hopf diagram with $n$ legs yields a $n$–form on the coend of a ribbon category in a completely explicit way. Thus computing a quantum invariant of a $3$–manifold reduces to the purely formal computation of the associated Hopf diagram, followed by the evaluation of this diagram in a given category (using in particular the so-called Kirby elements).

#### Article information

Source
Algebr. Geom. Topol., Volume 5, Number 4 (2005), 1677-1710.

Dates
Accepted: 28 November 2005
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796495

Digital Object Identifier
doi:10.2140/agt.2005.5.1677

Mathematical Reviews number (MathSciNet)
MR2186115

Zentralblatt MATH identifier
1116.57011

#### Citation

Bruguieres, Alain; Virelizier, Alexis. Hopf diagrams and quantum invariants. Algebr. Geom. Topol. 5 (2005), no. 4, 1677--1710. doi:10.2140/agt.2005.5.1677. https://projecteuclid.org/euclid.agt/1513796495

#### References

• A Bruguières, Tresses et structure entière sur la catégorie des représentations de ${\rm SL}\sb N$ quantique, Comm. Algebra 28 (2000) 1989–2028
• A Bruguières, Double Braidings, Twists, and Tangle Invariants (2004), to appear in J. Pure Appl. Alg. \arxivmath.QA/0407217
• T Kerler, V V Lyubashenko, Non-semisimple topological quantum field theories for 3-manifolds with corners, Lecture Notes in Mathematics 1765, Springer-Verlag, Berlin (2001)
• R Kirby, A calculus for framed links in $S\sp{3}$, Invent. Math. 45 (1978) 35–56
• W B R Lickorish, An introduction to knot theory, Graduate Texts in Mathematics 175, Springer-Verlag, New York (1997)
• V Lyubashenko, Modular transformations for tensor categories, J. Pure Appl. Algebra 98 (1995) 279–327
• V V Lyubashenko, Invariants of $3$-manifolds and projective representations of mapping class groups via quantum groups at roots of unity, Comm. Math. Phys. 172 (1995) 467–516
• S Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer-Verlag, New York (1998)
• S Majid, Braided groups, J. Pure Appl. Algebra 86 (1993) 187–221
• A Markoff, Foundations of the algebraic theory of tresses, Trav. Inst. Math. Stekloff 16 (1945) 53
• M H A Newman, On theories with a combinatorial definition of “equivalence.”, Ann. of Math. (2) 43 (1942) 223–243
• N Reshetikhin, V G Turaev, Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991) 547–597
• V G Turaev, Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics 18, Walter de Gruyter & Co., Berlin (1994)
• A Virelizier, Kirby elements and quantum invariants, to appear in Proc. London Math. Soc. \arxivmath.GT/0312337