# Kauffman brackets, character varieties, and triangulations of surfaces

@article{Bonahon2010KauffmanBC, title={Kauffman brackets, character varieties, and triangulations of surfaces}, author={Francis Bonahon and Helen M. Wong}, journal={arXiv: Geometric Topology}, year={2010} }

A Kauffman bracket on a surface is an invariant for framed links in the thickened surface, satisfying the Kauffman skein relation and multiplicative under superposition. This includes representations of the skein algebra of the surface. We show how an irreducible representation of the skein algebra usually specifies a point of the character variety of homomorphisms from the fundamental group of the surface to PSL_2(C), as well as certain weights associated to the punctures of the surface…

## 13 Citations

Representations of the Kauffman bracket skein algebra II: punctured surfaces

- Mathematics
- 2012

In earlier work, we constructed invariants of irreducible representations of the Kauffman skein algebra of a surface. We introduce here an inverse construction, which to a set of possible invariants…

THE SKEIN ALGEBRA OF ARCS AND LINKS AND THE DECORATED TEICHM

- Mathematics, Physics
- 2011

We define an associative algebra AS_h(S) generated by framed arcs and links over a punctured surface S which is a quantization of the Poisson algebra C(S) of arcs and curves on S. We then construct a…

Skein algebras of surfaces

- MathematicsTransactions of the American Mathematical Society
- 2018

We show that the Kauffman bracket skein algebra of any oriented surface
F
F
(possibly with marked points in its boundary) has no zero divisors and that its center is generated by knots…

Asymptotics of quantum invariants of surface diffeomorphisms I: conjecture and algebraic computations

- Mathematics
- 2021

The Kashaev-Murakami-Murakami Volume Conjecture connects the hyperbolic volume of a knot complement to the asymptotics of certain evaluations of the colored Jones polynomials of the knot. We…

Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality

- MathematicsQuantum Topology
- 2019

This is the third article in the series begun with [BonWon3, BonWon4], devoted to finite-dimensional representations of the Kauffman bracket skein algebra of an oriented surface $S$. In [BonWon3] we…

Tropical Fock-Goncharov coordinates for $\mathrm{SL}_3$-webs on surfaces II: naturality

- Mathematics
- 2020

In a companion paper [DS20] we constructed non-negative integer coordinates ΦT for a distinguished collection W3,Ŝ of SL3-webs on a finite-type punctured surface Ŝ, depending on an ideal…

The Witten-Reshetikhin-Turaev representation of the Kauffman skein algebra

- Mathematics
- 2013

For A a primitive 2N-root of unity with N odd, the Witten-Reshetikhin-Turaev topological quantum field theory provides a representation of the Kauffman skein algebra of a closed surface. We show that…

Matrix formulae and skein relations for cluster algebras from surfaces

- Mathematics
- 2011

This paper concerns cluster algebras with principal coefficients A(S,M) associated to bordered surfaces (S,M), and is a companion to a concurrent work of the authors with Schiffler [MSW2]. Given any…

Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations

- Mathematics
- 2016

We study finite-dimensional representations of the Kauffman bracket skein algebra of a surface S. In particular, we construct invariants of such irreducible representations when the underlying…

Quantum traces for representations of surface groups in SL2(ℂ)

- Mathematics
- 2010

We consider two different quantizations of the character variety consisting of all representations of surface groups in SL_2. One is the skein algebra considered by Przytycki-Sikora and Turaev. The…

## References

SHOWING 1-10 OF 47 REFERENCES

MULTIPLICATIVE STRUCTURE OF KAUFFMAN BRACKET SKEIN MODULE QUANTIZATIONS

- Mathematics
- 1999

We describe, for a few small examples, the Kauffman bracket skein algebra of a surface crossed with an interval. If the surface is a punctured torus the result is a quantization of the symmetric…

UNDERSTANDING THE KAUFFMAN BRACKET SKEIN MODULE

- Mathematics
- 1996

The Kauffman bracket skein module K(M) of a 3-manifold M is defined over formal power series in the variable h by letting A = eh/4. For a compact oriented surface F, it is shown that K(F×I) is a…

Rings of SL2(C)-characters and the Kauman bracket skein module

- Mathematics
- 1997

Let M be a compact orientable 3-manifold. The set of characters of SL2( )- representations of 1(M) forms a closed ane algebraic set. We show that its coordinate ring is isomorphic to a specialization…

Rings of SL2(
$\Bbb{C}$ )-characters and the Kauffman bracket skein module

- Mathematics
- 1997

AbstractLet M be a compact orientable 3-manifold. The set of characters of SL2(
$\Bbb {C}$)-representations of
$ \pi_1(M) $ forms a closed affine algebraic set. We show that its coordinate ring is…

Skein modules and the noncommutative torus

- Mathematics
- 1998

We prove that the Kauffman bracket skein algebra of the cylinder over a torus is a canonical subalgebra of the noncommutative torus. The proof is based on Chebyshev polynomials. As an application, we…

Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms

- Mathematics
- 2007

S of the quantum Teichmuller space of a punctured surface S . This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell decompositions of S . Our…

SKEIN SPACES AND SPIN STRUCTURES

- Mathematics, Physics
- 1999

This paper relates skein spaces based on the Kauffman bracket and
spin structures. A spin structure on an oriented 3-manifold provides an isomorphism
between the skein space for parameter A and the…

Skein Modules of 3-Manifolds

- Mathematics
- 2020

It is natural to try to place the new polynomial invariants of links in algebraic topology (e.g. to try to interpret them using homology or homotopy groups). However, one can think that these new…

Trace Coordinates on Fricke spaces of some simple hyperbolic surfaces

- Mathematics
- 2009

The conjugacy class of a generic unimodular 2 by 2 complex matrix is determined by its trace, which may be an arbitrary complex number. In the nineteenth century, it was known that a generic pair…

On skein algebras and Sl2(C)-character varieties

- Mathematics
- 1997

Let M be an oriented 3-manifold. For any commutative ring R with a speci"ed invertible element A one can assign an R-moduleS 2,= (M; R, A) called the Kau!man bracket skein module of M. This invariant…