We relate two different quantizations of the character variety consisting of all representations of surface groups in SL 2 . One is the Kauffman skein algebra considered by Bullock, Frohman and Kania-Bartoszyńska, Przytycki and Sikora, and Turaev. The other is the quantum Teichmüller space introduced by Chekhov and Fock and by Kashaev. We construct a homomorphism from the skein algebra to the quantum Teichmüller space which, when restricted to the classical case, corresponds to the equivalence between these two algebras through trace functions. 14D20, 57M25, 57R56Let S be an oriented surface of finite topological type. The goal of this paper is to establish a connection between two quantizations of the character variety R SL 2 .C/ .S/ D fr W 1 .S/ ! SL 2 .C/g = = SL 2 .C/;consisting of all group homomorphisms r from the fundamental group 1 .S / to the Lie group SL 2 .C/, considered up to conjugation by elements of SL 2 .C/. The double bar indicates here that the quotient is taken in the algebraic geometric sense of geometric invariant theory.The first quantization was introduced by D Bullock, C Frohman, J Kania-Bartoszyńska [12], J Przytycki and A Sikora [32] and V Turaev [35] and uses the Kauffman skein algebra S A .S/. This algebra is obtained by considering the vector space freely generated by all isotopy classes of framed links in S OE0; 1, and then taking the quotient of this space under the Kauffman skein relation; see Section 3.1. What makes S A .S / a quantization of R SL 2 .C/ .S/ is that, when A D 1, the skein algebra S 1 .S/ has a natural identification with the commutative algebra of regular functions on R SL 2 .C/ .S/ and that, as A tends to 1, the lack of commutativity of S A .S / is infinitesimally measured by the Goldman-Weil-Petersson Poisson structure [19; 20; 31; 36] on R SL 2 .C/ .S /; see [35]. There is a similar situation when A D C1, in which case S C1 .S/ has a natural identification with the algebra of functions on a twisted version of R SL 2 .C/ .S/; see Section 3.2. Guo and Liu [21]. This quantization takes advantage of the fact that, if one restricts to matrices with real coefficients, a large subset of R SL 2 .R/ .S/ with nonempty interior has a natural identification with the Teichmüller (or Fricke-Klein) space T .S/, consisting of isotopy classes of all complete hyperbolic metrics on S . Starting with an ideal triangulation of the surface, Thurston [33;34] introduced for the Teichmüller space T .S/ a set of coordinates, called shear coordinates, in which the Goldman-Weil-Petersson form is expressed in a particularly simple way. The quantum Teichmüller space is a quantization of T .S/ that is based on these shear coordinates. As this construction requires the existence of an ideal triangulation, the surface must have at least one puncture.A natural conjecture is that these two quantizations are "essentially equivalent".In the classical cases where q D 1 and A D˙1, the correspondence is relatively clear because of the identifications of S˙1.S/ and T 1 S with algebras of functions...

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 parameter q = e 2πi is a root of unity. The main one of these invariants is a point in the character variety consisting of group homomorphisms from the fundamental group π 1 (S) to SL 2 (C), or in a twisted version of this character variety. The proof relies on certain miraculous cancellations that occur for the quantum trace homomorphism constructed by the authors. These miraculous cancellations also play a fundamental role in subsequent work of the authors, where novel examples of representations of the skein algebra are constructed.For an oriented surface S of finite topological type and for a Lie group G, many areas of mathematics involve the character varietywhere G acts on homomorphisms by conjugation. For G = SL 2 (C), Turaev [Tu 1 ] showed that the corresponding character variety R SL 2 (C) (S) can be quantized by the Kauffman bracket skein algebra of the surface; see also [BuFK 1 , BuFK 2 , PrS]. In fact, if one follows the physical tradition that a quantization of a space X replaces the commutative algebra of functions on X by a non-commutative algebra of operators on a Hilbert space, the points of an actual quantization of the character variety R SL 2 (C) (S) should be representations of the Kauffman bracket skein algebra.This article studies finite-dimensional representations of the skein algebra of a surface. The Kauffman bracket skein algebra S A (S) depends on a parameter A = e πi ∈ C − {0}, and is defined as follows. One first considers the vector space freely generated by all isotopy classes of framed links in the thickened surface S × [0, 1], and then one takes the quotient of this space by two relations: the main one is the skein relation thatwhenever the three links K 1 , K 0 and K ∞ ⊂ S × [0, 1] differ only in a little ball where they are as represented on Figure 1; the second relation states that [O] = −(A 2 + A −2 )[∅] for the trivial framed knot O and the empty link ∅. The algebra multiplication is defined by superposition of skeins. See §2 for details.Our goal is to study representations of the skein algebra, namely algebra homomorphisms ρ : S A (S) → End(V ) where V is a finite-dimensional vector space over C. See [BoW 2 ] for an interpretation of such representations as generalizations of the Kauffman bracket Date: June 5, 2018.

How knotted proteins fold has remained controversial since the identification of deeply knotted proteins nearly two decades ago. Both computational and experimental approaches have been used to investigate protein knot formation. Motivated by the computer simulations of Bölinger et al. [Bölinger D, et al. (2010) PLoS Comput Biol 6:e1000731] for the folding of the 61-knotted α-haloacid dehalogenase (DehI) protein, we introduce a topological description of knot folding that could describe pathways for the formation of all currently known protein knot types and predicts knot types that might be identified in the future. We analyze fingerprint data from crystal structures of protein knots as evidence that particular protein knots may fold according to specific pathways from our theory. Our results confirm Taylor’s twisted hairpin theory of knot folding for the 31-knotted proteins and the 41-knotted ketol-acid reductoisomerases and present alternative folding mechanisms for the 41-knotted phytochromes and the 52- and 61-knotted proteins.

In earlier work [BoW 3 ], we constructed invariants of irreducible finite-dimensional representations of the Kauffman bracket skein algebra of a surface. We introduce here an inverse construction, which to a set of possible invariants associates an irreducible representation that realizes these invariants. The current article is restricted to surfaces with at least one puncture, a condition that is lifted in subsequent work [BoW 6 ] relying on this one. A step in the proof is of independent interest, and describes the arithmetic structure of the Thurston intersection form on the space of integer weight systems for a train track.This article is a continuation of [BoW 3 ] and is part of the program described in [BoW 2 ], devoted to the analysis and construction of finite-dimensional representations of the Kauffman bracket skein algebra of a surface.Let S be an oriented surface of finite topological type without boundary. The Kauffman bracket skein algebra S A (S) depends on a parameter A = e πi ∈ C − {0}, and is defined as follows: One first considers the vector space freely generated by all isotopy classes of framed links in the thickened surface S × [0, 1], and then one takes the quotient of this space by two relations. The first and main relation is the skein relation, which states thatwhenever the three links K 1 , K 0 and K ∞ ⊂ S × [0, 1] differ only in a little ball where they are as represented in Figure 1. The second relation is the trivial knot relation, which asserts thatwhenever O is the boundary of a disk D ⊂ K × [0, 1] disjoint from K, and is endowed with a framing transverse to D. The algebra multiplication is provided by the operation of superposition, for which the product [K] · [L] is represented by the union, 1] are respectively obtained by rescaling the framed links K ⊂ S × [0, 1] and L ′ ⊂ S × [0, 1] in the [0, 1] direction.Turaev [Tu 1 ], Bullock-Frohman-Kania-Bartoszyńska [BFK 1 , BFK 2 ] and Przytycki-Sikora [PrS] showed that the skein algebra S A (S) provides a quantization of the character variety R SL 2 (C) (S) = {group homomorphisms r: π 1 (S) → SL 2 (C)}/ /SL 2 (C) where SL 2 (C) acts on homomorphisms by conjugation, and where the double bar indicates that the quotient is to be taken in the sense of geometric invariant theory [MFK]. In fact, if

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. Conversely, we sketch a proof of the fact that each point of the character variety, endowed with appropriate puncture weights, uniquely determines a Kauffman bracket. Details will appear elsewhere.