Representations of the Kau man Bracket Skein Algebra of a Surface

Transcription

1 Representations of the Kau man Bracket Skein Algebra of a Surface IAS Member s Seminar November 20, 2017 Helen Wong von Neumann fellow

2 Quantum topology In the early 1980s, the Jones polynomial for knots and links in 3-space was introduced. Witten followed quickly with an amplification of the Jones polynomial to 3-dimensional manifolds. Straightaway, these quantum invariants solved some long-standing problems in low-dimensional topology. However, their definition is seemingly unrelated to any of the classically known topological and geometric constructions. To this day there is still no good answer to the question: What topological or geometric properties do the Jones and Witten quantum invariants measure?

3 Quantum topology The Kau man bracket skein algebra of a surface has emerged as a likely point of connection. In particular, it generalizes the (Kau man bracket formulation of) Jones polynomial invariant for links features in the Topological Quantum Field Theory description of Witten s 3-manifold invariant relates to hyperbolic geometry of surface, particularly the SL 2 C-character variety

4 Plan for talk 1. Intro to the Kau man bracket skein algebra S q p q. 2. How S q p q is related to SL 2 C-character variety. 3. Representations of S q p q. This talk describes joint work with Francis Bonahon.

5 The Kau man bracket for links in S 3 The Kau man bracket xky is an invariant of isotopy classes of framed links in S 3. The Jones polynomial can be seen as a modification of it for unframed, oriented links.

6 The Kau man bracket for links in S 3 The Kau man bracket xky is an invariant of isotopy classes of framed links in S 3. The Jones polynomial can be seen as a modification of it for unframed, oriented links. Choose a variable q e 2 i~ P C t0u and a square root q 1 2.

7 The Kau man bracket for links in S 3 The Kau man bracket xky is an invariant of isotopy classes of framed links in S 3. The Jones polynomial can be seen as a modification of it for unframed, oriented links. Choose a variable q e 2 i~ P C t0u and a square root q 1 2. The Kau man bracket of a framed link K in the 3 sphere S 3

8 The Kau man bracket for links in S 3 The Kau man bracket xky is an invariant of isotopy classes of framed links in S 3. The Jones polynomial can be seen as a modification of it for unframed, oriented links. Choose a variable q e 2 i~ P C t0u and a square root q 1 2. The Kau man bracket of a framed link K in the 3 sphere S 3 is evaluated from a diagram using the relations: x y q 1 2 x y` q 1 2 x y x y p q q 1 qx y.

9 The Kau man bracket for links in S 3 The Kau man bracket xky is an invariant of isotopy classes of framed links in S 3. The Jones polynomial can be seen as a modification of it for unframed, oriented links. Choose a variable q e 2 i~ P C t0u and a square root q 1 2. The Kau man bracket of a framed link K in the 3 sphere S 3 is evaluated from a diagram using the relations: x y q 1 2 x y` q 1 2 x y x y p q q 1 qx y. Convention: Link diagrams are drawn with framing perpendicular to the blackboard.

10 Example x y q 1 2 x y`q 1 2 x y

11 Example x y q 1 2 x y`q 1 2 x y q 1 2 p q q 1 q 2 ` q 1 2 p q q 1 q

12 Example x y q 1 2 x y`q 1 2 x y q 1 2 p q q 1 q 2 ` q 1 2 p q q 1 q q 3 2 p q q 1 q,

13 Example x y q 1 2 x y`q 1 2 x y q 1 2 p q q 1 q 2 ` q 1 2 p q q 1 q q 3 2 p q q 1 q, x y q q 1.

14 Example x y q 1 2 x y`q 1 2 x y q 1 2 p q q 1 q 2 ` q 1 2 p q q 1 q q 3 2 p q q 1 q, x y q q 1. This shows that (as framed links).

15 Example x y q 1 2 x y`q 1 2 x y q 1 2 p q q 1 q 2 ` q 1 2 p q q 1 q q 3 2 p q q 1 q, x y q q 1. This shows that (as framed links). Observation: The Kau man bracket can be written as a linear combination of diagrams without crossings, which bound disks in S 3.ThisimpliesxKy PZrq 1 2, q 1 2 s.

16 Kau man bracket in S 3 is multiplicative Observation: x y x y x y

17 Kau man bracket in S 3 is multiplicative Observation: x y x y x y x y

18 Kau man bracket in S 3 is multiplicative Observation: x y x y x y x y Juxtaposition/superposition defines a (commutative) multiplication of the Kau man bracket in S 3 :. xk 1 y xk 2 y xk 1 next to K 2 y xk 1 over K 2 y

19 The Kau man bracket skein algebra of S 3 We consider the structure of the Kau man bracket of all possible framed links in S 3.

20 The Kau man bracket skein algebra of S 3 We consider the structure of the Kau man bracket of all possible framed links in S 3. Choose a variable q e 2 i~ P C t0u and a square root q 1 2.

21 The Kau man bracket skein algebra of S 3 We consider the structure of the Kau man bracket of all possible framed links in S 3. Choose a variable q e 2 i~ P C t0u and a square root q 1 2. Define S q ps 3 q to be the Zrq 1 2, q 1 2 s-module consisting of linear combinations of framed links K in S 3,

22 The Kau man bracket skein algebra of S 3 We consider the structure of the Kau man bracket of all possible framed links in S 3. Choose a variable q e 2 i~ P C t0u and a square root q 1 2. Define S q ps 3 q to be the Zrq 1 2, q 1 2 s-module consisting of linear combinations of framed links K in S 3, subject to the relations: x y q 1 2 x y` q 1 2 x y x y p q q 1 qx y,

23 The Kau man bracket skein algebra of S 3 We consider the structure of the Kau man bracket of all possible framed links in S 3. Choose a variable q e 2 i~ P C t0u and a square root q 1 2. Define S q ps 3 q to be the Zrq 1 2, q 1 2 s-module consisting of linear combinations of framed links K in S 3, subject to the relations: x y q 1 2 x y` q 1 2 x y x y p q q 1 qx y, with multiplication by juxtaposition/superposition of framed links.

24 Generalization to oriented 3-manifolds We consider the structure of the Kau man bracket of all possible framed links in an oriented 3-manifold M.

25 Generalization to oriented 3-manifolds We consider the structure of the Kau man bracket of all possible framed links in an oriented 3-manifold M. Choose a variable q e 2 i~ P C t0u and a square root q 1 2. Define S q pmq to be the Zrq 1 2, q 1 2 s-module consisting of linear combinations of framed links K in M, subject to the relations: x y q 1 2 x y` q 1 2 x y x y p q q 1 qx y. with multiplication by juxtaposition/superposition of framed links.

26 Kau man bracket skein algebra of a surface

27 Kau man bracket skein algebra of a surface Special case: M ˆr0, 1s and is an oriented, compact surface with finitely many punctures.

28 Kau man bracket skein algebra of a surface Special case: M ˆr0, 1s and is an oriented, compact surface with finitely many punctures. Choose a variable q e 2 i~ P C t0u and a square root q 1 2. Define S q p q to be the Zrq 1 2, q 1 2 s-module consisting of linear combinations of framed links K in M, subject to the relations: x y q 1 2 x y` q 1 2 x y x y p q q 1 qx y. with multiplication by superposition of framed links.

29 Example D loops not bounding disks ùñ many generators of S q p q. x y q 1 2 x y`q 1 2 x y

30 Example D loops not bounding disks ùñ many generators of S q p q. x y q 1 2 x y`q 1 2 x y ùñ multiplication is not commutative. x y x y x y x y x y x y

31 Kau man bracket skein algebra of a surface Remarks Every 3-manifold can be decomposed into two handlebodies along a closed surface.

32 Kau man bracket skein algebra of a surface Remarks Every 3-manifold can be decomposed into two handlebodies along a closed surface. In Witten s TQFT for q N 1, fi Ñ V q p q V q p q is a finite dim. quotient of S q pmq, wherebm. The action S q p q ü S q pmq provides an irreducible repn Witten : S q p q ÑEndpV q p qq

33 Kau man bracket skein algebra of a surface Remarks Every 3-manifold can be decomposed into two handlebodies along a closed surface. In Witten s TQFT for q N 1, fi Ñ V q p q V q p q is a finite dim. quotient of S q pmq, wherebm. The action S q p q ü S q pmq provides an irreducible repn Witten : S q p q ÑEndpV q p qq

34 Algebraic Properties of S q p q 1. S q p q is not commutative, except when: is a 2-sphere with 0, 1, or 2 punctures, disk with 1 or 2 punctures, or annulus q 1, whenq 1 2 q 1 2 and.

35 Algebraic Properties of S q p q 1. S q p q is not commutative, except when: is a 2-sphere with 0, 1, or 2 punctures, disk with 1 or 2 punctures, or annulus q 1, whenq 1 2 q 1 2 and. 2. ZpS q p qq contains puncture loops.

36 Algebraic Properties of S q p q 1. S q p q is not commutative, except when: is a 2-sphere with 0, 1, or 2 punctures, disk with 1 or 2 punctures, or annulus q 1, whenq 1 2 q 1 2 and. 2. ZpS q p qq contains puncture loops. 3. (Przytycki, Turaev, 1990) S q p q is (usually) infinitely generated as a module, by framed links whose projection to has no crossings.

37 Algebraic Properties of S q p q 1. S q p q is not commutative, except when: is a 2-sphere with 0, 1, or 2 punctures, disk with 1 or 2 punctures, or annulus q 1, whenq 1 2 q 1 2 and. 2. ZpS q p qq contains puncture loops. 3. (Przytycki, Turaev, 1990) S q p q is (usually) infinitely generated as a module, by framed links whose projection to has no crossings. 4. (Bullock, 1999) S q p q is finitely generated as an algebra.

38 Algebraic Properties of S q p q (very recently proved) 1. (Przytycki-Sikora, Le, Bonahon-W.) S q p q has no zero divisors. 2. (Przytycki-Sikora) S q p q is Noetherian. 3. (Frohman-Kania-Bartoszynska-Le) S q p q is almost Azyumaya. 4. (Bonahon-W., Le, Frohman-Kania-Bartozynska-Le) Its center ZpS q p qq is determined.

39 Part II How S q p q is related to hyperbolic geometry

40 Hyperbolic geometry Let be a surface with finite topological type.

41 Hyperbolic geometry Let be a surface with finite topological type. Its Teichmuller space T p q consists of isotopy classes of complete hyperbolic metrics on with finite area. image from geom.uiuc.edu Every metric m P T p q corresponds to a monodromy representation r m : 1 p q ÑIsom`pH 2 q PSL 2 prq, upto conjugation.

42 Hyperbolic geometry Hence consider R SL2 R tr : 1 p q ÑSL 2 Ru{SL 2 R.

43 Hyperbolic geometry Hence consider R SL2 R tr : 1 p q ÑSL 2 Ru{SL 2 R. Notice that we lifted PSL 2 R to SL 2 R

44 Hyperbolic geometry Hence consider R SL2 C tr : 1 p q ÑSL 2 Cu{SL 2 C. Notice that we lifted PSL 2 R to SL 2 R and complexify to SL 2 C.

45 Hyperbolic geometry Hence consider R SL2 C tr : 1 p q ÑSL 2 Cu{SL 2 C. Notice that we lifted PSL 2 R to SL 2 R and complexify to SL 2 C. The SL 2 C-character variety R SL2 Cp q is an algebraic variety that contains a copy of Teichmuller space T p q as a real subvariety. Note that rr 1 s rr 2 s if and only if trace r 1 trace r 2.

46 Hyperbolic geometry

47 Trace functions For every loop K in, define the trace function Tr K : R SL2 Cp q ÑC by Tr K prq traceprprksqq.

48 Trace functions For every loop K in, define the trace function Tr K : R SL2 Cp q ÑC by Tr K prq traceprprksqq. The trace functions generate the function algebra CrR SL2 Cp qs (consisting of all regular functions R SL2 Cp q ÑC).

49 Trace functions For every loop K in, define the trace function Tr K : R SL2 Cp q ÑC by Tr K prq traceprprksqq. The trace functions generate the function algebra CrR SL2 Cp qs (consisting of all regular functions R SL2 Cp q ÑC). Important observation: The trace functions Tr K satisfy the Kau man bracket skein relation when q 1 2 1, p 1q ` p 1q

50 Trace functions For every loop K in, define the trace function Tr K : R SL2 Cp q ÑC by Tr K prq traceprprksqq. The trace functions generate the function algebra CrR SL2 Cp qs (consisting of all regular functions R SL2 Cp q ÑC). Important observation: The trace functions Tr K satisfy the Kau man bracket skein relation when q 1 2 1, p 1q ` p 1q from identity TrpMqTrpNq TrpMNq`TrpMN 1 q in SL 2 C.

51 Trace functions For every loop K in, define the trace function Tr K : R SL2 Cp q ÑC by Tr K prq traceprprksqq. The trace functions generate the function algebra CrR SL2 Cp qs (consisting of all regular functions R SL2 Cp q ÑC). Important observation: The trace functions Tr K satisfy the Kau man bracket skein relation when q 1 2 1, p 1q ` p 1q. from identity TrpMqTrpNq TrpMNq`TrpMN 1 q in SL 2 C.

52 Skein algebras and hyperbolic geometry Theorem (..., Bullock-Frohman-Kania-Bartoszynska, Przytycki-Sikora) For q 1 2 1, thereisanisomorphism between the skein algebra S 1 p q and the function algebra CrR SL2 Cp qs given by rks P S 1 p q Ñ Tr K P CrR SL2 Cp qs.

53 Skein algebras and hyperbolic geometry Theorem (..., Bullock-Frohman-Kania-Bartoszynska, Przytycki-Sikora) For q 1 2 1, thereisanisomorphism between the skein algebra S 1 p q and the function algebra CrR SL2 Cp qs given by rks P S 1 p q Ñ Tr K P CrR SL2 Cp qs. Alternatively, there is an isomorphism t : S 1 p q ÑCu Ñ R SL2 Cp q.

54 Skein algebras and hyperbolic geometry Theorem (..., Bullock-Frohman-Kania-Bartoszynska, Przytycki-Sikora) For q 1 2 1, thereisanisomorphism between the skein algebra S 1 p q and the function algebra CrR SL2 Cp qs given by rks P S 1 p q Ñ Tr K P CrR SL2 Cp qs. Alternatively, there is an isomorphism t : S 1 p q ÑCu Ñ R SL2 Cp q. Notice that S 1 p q and CrR SL2 Cp qs are commutative.

55 Skein algebras and hyperbolic geometry Theorem (..., Bullock-Frohman-Kania-Bartoszynska, Przytycki-Sikora) For q 1 2 1, thereisanisomorphism between the skein algebra S 1 p q and the function algebra CrR SL2 Cp qs given by rks P S 1 p q Ñ Tr K P CrR SL2 Cp qs. Alternatively, there is an isomorphism t : S 1 p q ÑCu Ñ R SL2 Cp q. Notice that S 1 p q and CrR SL2 Cp qs are commutative. Can we think of S q p q as a non-commutative version of them?

56 Skein algebras and hyperbolic geometry Theorem (..., Bullock-Frohman-Kania-Bartoszynska, Przytycki-Sikora) For q 1 2 1, thereisanisomorphism between the skein algebra S 1 p q and the function algebra CrR SL2 Cp qs given by rks P S 1 p q Ñ Tr K P CrR SL2 Cp qs. Alternatively, there is an isomorphism t : S 1 p q ÑCu Ñ R SL2 Cp q. Notice that S 1 p q and CrR SL2 Cp qs are commutative. Can we think of S q p q as a non-commutative version of them?...yes!

57 Skein algebras and hyperbolic geometry Turaev noticed that the commutator in S q p q has the form rksrls rlsrks

58 Skein algebras and hyperbolic geometry Turaev noticed that the commutator in S q p q has the form rksrls rlsrks pq 1 2 q 1 2 q pq 1 2 q 1 2 q.

59 Skein algebras and hyperbolic geometry Turaev noticed that the commutator in S q p q has the form rksrls rlsrks pq 1 2 q 1 2 q pq 1 2 q 1 2 q. And when q e 2 i~, rksrls rlsrks 2 i~ p q ` higher order terms in ~

60 Skein algebras and hyperbolic geometry Turaev noticed that the commutator in S q p q has the form rksrls rlsrks pq 1 2 q 1 2 q pq 1 2 q 1 2 q. And when q e 2 i~, rksrls rlsrks 2 i~ p ttr K, Tr L u q ` higher order terms in ~ where t, u is the Weil-Petersson-Atiyah-Bott-Goldman Poisson bracket, using computations of Goldman.

61 Skein algebra is quantization of Teichmuller space Theorem (Turaev ) S q p q is a quantization of R SL2 Cp q with respect to the Weil-Petersson Poisson structure.

62 Skein algebra is quantization of Teichmuller space Theorem (Turaev ) S q p q is a quantization of R SL2 Cp q with respect to the Weil-Petersson Poisson structure. Alternatively, : S q p q Ñ C P R SL2 Cp q : S q p q ÑEndpC d q P quantization of R SL2 Cp q

63 Skein algebra is quantization of Teichmuller space Theorem (Turaev ) S q p q is a quantization of R SL2 Cp q with respect to the Weil-Petersson Poisson structure. Alternatively, : S q p q Ñ C P R SL2 Cp q : S q p q ÑEndpC d q P quantization of R SL2 Cp q Often, we say: S q p q is a quantization of Teichmuller space T p q.

64 Skein algebra is quantization of Teichmuller space Theorem (Turaev ) S q p q is a quantization of R SL2 Cp q with respect to the Weil-Petersson Poisson structure. Alternatively, : S q p q Ñ C P R SL2 Cp q : S q p q ÑEndpC d q P quantization of R SL2 Cp q Often, we say: S q p q is a quantization of Teichmuller space T p q. There s another quantization of Teichmuller space, related to quantum cluster algebras. (Chekhov-Fock, Kashaev, Bonahon et al.)

65 Another quantization of Teichmuller space Suppose has at least one puncture, with ideal triangulation. The shear parameters x i prq PR along edges 1,... n completely describe a hyperbolic metric r P R SL2 Cp q. X i

66 Another quantization of Teichmuller space Suppose has at least one puncture, with ideal triangulation. The shear parameters x i prq PR along edges 1,... n completely describe a hyperbolic metric r P R SL2 Cp q. X i The trace functions Tr K are polynomials in x 1 2 i. So the function algebra CrR SL2 Cp qs is also generated by commutative shear parameters x 1 2 1,...,x 1 2 n.

67 Another quantization of Teichmuller space Following the theory of quantum cluster algebras, Chekhov-Fock define the quantum Teichmuller space T q p q to be the algebra generated by non-commutative shear parameters Z1 1,...,Z 1 n Z i Z j q 1 Z j Z i if the ith and jth edges share a triangle, Z i Z j Z j Z i otherwise relations for independence from choice of ideal triangulation

68 Another quantization of Teichmuller space Following the theory of quantum cluster algebras, Chekhov-Fock define the quantum Teichmuller space T q p q to be the algebra generated by non-commutative shear parameters Z1 1,...,Z 1 n Z i Z j q 1 Z j Z i if the ith and jth edges share a triangle, Z i Z j Z j Z i otherwise relations for independence from choice of ideal triangulation Theorem (Bonahon-W. ) There exists an injective map S q p q ãñ T q p q which is compatible with S q p q as a quantization of R SL2 Cp q with respect to the Weil-Petersson Poisson structure.

69 Another quantization of Teichmuller space Following the theory of quantum cluster algebras, Chekhov-Fock define the quantum Teichmuller space T q p q to be the algebra generated by non-commutative shear parameters Z1 1,...,Z 1 n Z i Z j q 1 Z j Z i if the ith and jth edges share a triangle, Z i Z j Z j Z i otherwise relations for independence from choice of ideal triangulation Theorem (Bonahon-W. ) There exists an injective map S q p q ãñ T q p q which is compatible with S q p q as a quantization of R SL2 Cp q with respect to the Weil-Petersson Poisson structure. Great because multiplication in T q p q is easier to manipulate. Not great because the map is complicated.

70 Summary of Part II The Kau man bracket skein algebra of a surface is the quantization of Teichmuller space (really, CrR SL2 Cp qs) in multiple ways. In one : S q p q Ñ C P R SL2 Cp q : S q p q ÑEndpC d q P quantization of R SL2 Cp q Also S q p q ãñ T q p q

71 Summary of Part II The Kau man bracket skein algebra of a surface is the quantization of Teichmuller space (really, CrR SL2 Cp qs) in multiple ways. In one : S q p q Ñ C P R SL2 Cp q : S q p q ÑEndpC d q P quantization of R SL2 Cp q What are these repns? Also S q p q ãñ T q p q

72 Part III Representations of S q p q

73 Theorem (Bonahon-W.) Let q be a N-root of 1, with N odd. Let be a surface with finite topological type. 1. Every irreducible representation of S q p q uniquely determines puncture invariants p i P C for every puncture of classical shadow r P R SL2 Cp q 2. For every compatible tp i u and r P R SL2 Cp q, thereexistsan irreducible representation of S q p q with those invariants. We ll sketch the methods of construction, starting with a revisit to the algebraic structure of S q p q. This will involve Chebyshev polynomials.

74 Remarks on Theorem irred representation : S q p q ÑEndpC n q 1 1? Ñ puncture invs p i P C and classical shadow r P R SL2 Cp q

75 Remarks on Theorem irred representation : S q p q ÑEndpC n q 1 1? Ñ puncture invs p i P C and classical shadow r P R SL2 Cp q D two irrepns of S q p q ( Witten and Thm ) whose classical shadow is the trivial character r : 1 p q Ñtp qu.

76 Remarks on Theorem irred representation : S q p q ÑEndpC n q 1 1? Ñ puncture invs p i P C and classical shadow r P R SL2 Cp q D two irrepns of S q p q ( Witten and Thm ) whose classical shadow is the trivial character r : 1 p q Ñtp qu. Breaking news: In July preprint, Frohman, Kania-Bartoszynska, and Le prove 1-1 for a Zariski dense, open subset of R SL2 Cp q.

77 Chebyshev polynomials

78 Chebyshev polynomials Let T n pxq be the Chebyshev polynomials of the first kind, defined recursively as follows: T 0 pxq 2 T 1 pxq x T 2 pxq x 2 2 T 3 pxq x 3 3x. T n pxq xt n 1 pxq T n 2 pxq.

79 Chebyshev polynomials Let T n pxq be the Chebyshev polynomials of the first kind, defined recursively as follows: T 0 pxq 2 T 1 pxq x T 2 pxq x 2 2 T 3 pxq x 3 3x. T n pxq xt n 1 pxq T n 2 pxq. Note: cospn q 1 2 T np2 cos q TrpM n q T n p TrpMqq for all M P SL 2 pcq.

80 Chebyshev polynomials Let T n pxq be the Chebyshev polynomials of the first kind, defined recursively as follows: T 0 pxq 2 T 1 pxq x T 2 pxq x 2 2 T 3 pxq x 3 3x. T n pxq xt n 1 pxq T n 2 pxq. Note: cospn q 1 2 T np2 cos q TrpM n q T n p TrpMqq for all M P SL 2 pcq. Witten s TQFT uses the Chebyshevs of the second kind.

81 Threading skeins by Chebyshev polynomials

82 Threading skeins by Chebyshev polynomials Example: rks x y and T 3 x 3 3x,

83 Threading skeins by Chebyshev polynomials Example: rks x y and T 3 x 3 3x, rk T 3 s T3

84 Threading skeins by Chebyshev polynomials Example: rks x y and T 3 x 3 3x, rk T 3 s T3 3.

85 Threading skeins by Chebyshev polynomials Example: rks x y and T 3 x 3 3x, rk T 3 s T3 3. Observe: For a framed link K with n crossings, the Kau man bracket rks has 2 n resolutions, whereas rk T N s has Op2 n2 q resolutions.

86 Threading skeins by Chebyshev polynomials Theorem (Bonahon-W. ) Let q be a N-root of 1, with N odd. Framed links threaded by T N in S q p q:

87 Threading skeins by Chebyshev polynomials Theorem (Bonahon-W. ) Let q be a N-root of 1, with N odd. Framed links threaded by T N in S q p q: 1. TN TN

88 Threading skeins by Chebyshev polynomials Theorem (Bonahon-W. ) Let q be a N-root of 1, with N odd. Framed links threaded by T N in S q p q: 1. TN TN 2. TN TN p 1q TN TN ` p 1q TN TN.

89 Threading skeins by Chebyshev polynomials Theorem (Bonahon-W. ) Let q be a N-root of 1, with N odd. Framed links threaded by T N in S q p q: 1. TN TN rk T N s are in the center. 2. TN TN p 1q TN ` p 1q TN TN TN. rk T N s satisfy the skein relation, q

90 Threading skeins by Chebyshev polynomials Theorem (Bonahon-W. ) Let q be a N-root of 1, with N odd. Framed links threaded by T N in S q p q: 1. TN TN rk T N s are in the center. 2. TN TN p 1q TN ` p 1q TN TN TN. rk T N s satisfy the skein relation, q Proof: Inject S q p q ãñ T q p q. There are many miraculous cancellations when checking identities in the quantum Teichmuller space T q p q.

91 Threading skeins by Chebyshev polynomials Theorem (Bonahon-W. ) Let q be a N-root of 1, with N odd. Framed links threaded by T N in S q p q: 1. TN TN rk T N s are in the center. 2. TN TN p 1q TN ` p 1q TN TN TN. rk T N s satisfy the skein relation, q Proof: Inject S q p q ãñ T q p q. There are many miraculous cancellations when checking identities in the quantum Teichmuller space T q p q. (See also Le s skein-theoretic proof.)

92 Sketch Proof of Theorem Theorem (Bonahon-W.) Let q be a N-root of 1, with N odd. Let be a surface with finite topological type. 1. Every irreducible representation of S q p q uniquely determines puncture invariants p i P C for every puncture of classical shadow r P R SL2 Cp q 2. For compatible p i P C and r P R SL2 Cp q, thereexistsan irreducible representation of S q p q with those invariants.

93 Sketch proof of Theorem, part 1 Suppose : S q p q ÑEndpC d q is irreducible.

94 Sketch proof of Theorem, part 1 Suppose : S q p q ÑEndpC d q is irreducible. The i-th puncture loop rp i s is central. Schur s Lemma ñ : rp i s fi Ñ p i Id for p i P C puncture invs.

95 Sketch proof of Theorem, part 1 Suppose : S q p q ÑEndpC d q is irreducible. The i-th puncture loop rp i s is central. Schur s Lemma ñ : rp i s fi Ñ p i Id for p i P C puncture invs. TN AthreadedlinkrK T N sps q p q is central. TN Schur s Lemma ñ : rk T N s fi Ñ ˆ prksq Id for ˆ pkq PC.

96 Sketch proof of Theorem, part 1 Suppose : S q p q ÑEndpC d q is irreducible. The i-th puncture loop rp i s is central. Schur s Lemma ñ : rp i s fi Ñ p i Id for p i P C puncture invs. TN AthreadedlinkrK T N sps q p q is central. TN Schur s Lemma ñ : rk T N s fi Ñ ˆ prksq Id for ˆ pkq PC. rk T N s satisfy the skein relation TN TN. TN TN TN TN

97 Sketch proof of Theorem, part 1 Suppose : S q p q ÑEndpC d q is irreducible. The i-th puncture loop rp i s is central. Schur s Lemma ñ : rp i s fi Ñ p i Id for p i P C puncture invs. TN AthreadedlinkrK T N sps q p q is central. TN Schur s Lemma ñ : rk T N s fi Ñ ˆ prksq Id for ˆ pkq PC. rk T N s satisfy the skein relation TN TN ˆ : S 1 p q ÑC. So TN TN TN TN

98 Sketch proof of Theorem, part 1 Suppose : S q p q ÑEndpC d q is irreducible. The i-th puncture loop rp i s is central. Schur s Lemma ñ : rp i s fi Ñ p i Id for p i P C puncture invs. TN AthreadedlinkrK T N sps q p q is central. TN Schur s Lemma ñ : rk T N s fi Ñ ˆ prksq Id for ˆ pkq PC. rk T N s satisfy the skein relation TN TN TN TN TN TN. ˆ : S 1 p q ÑC Ñ r P R SL2 Cp q classical shadow So from the isomorphism between S q p q and CrR SL2 Cp qs.

99 Sketch proof of Theorem, part 2 Suppose we are given compatible puncture invariants p i classical shadow r P R SL2 Cp q. P C and Recall that for with at least one puncture, the quantum Teichmuller space T q p q is the algebra generated by Z1 1,...,Z 1 n with Z i Z j q 1 Z j Z i if the ith and jth edges share a triangle. Bonahon-Liu construct all representations of T q p q, whichare classified by compatible numbers p i P C and r P R SL2 Cp q. When has at least one puncture, compose : S q p q ãñ T q p q p i,r Ñ EndpC d q. When has no punctures, drill punctures and show there is an invariant subspace that is blind to punctures.

100 Further remarks In 2016 preprint, Frohman-Kania-Bartoszynska describe another method of constructing representations when has at least one puncture, which matches our construction for generic characters by July 2017 preprint of Frohman-Kania-Bartoszynska-Le. S q p q ãñ T q p q is used to prove many of the algebraic properties mentioned earlier (e.g., no zero divisors, determination of center). How can we use it to more closely tie together the Jones and Witten quantum invariants with hyperbolic geometry?

101 Thanks!