DOCSLIB.ORG

Volume Conjecture and Asymptotic Expansion of Q-Series Kazuhiro Hikami

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Volume Conjecture and Asymptotic Expansion of Q-Series Kazuhiro Hikami Volume Conjecture and Asymptotic Expansion of q-Series Kazuhiro Hikami CONTENTS We consider the “volume conjecture,” which states that an 1. Introduction asymptotic limit of Kashaev’s invariant (or, the colored Jones type 2. Kashaev’s Invariant and Volume Conjecture invariant) of knot K gives the hyperbolic volume of the com- 3. Torus Knots and q-Series plement of knot K. In the first part, we analytically study an 4. Hyperbolic Knots asymptotic behavior of the invariant for the torus knot, and pro- Acknowledgments pose identities concerning an asymptotic expansion of q-series q N References which reduces to the invariant with being the -th root of unity. This is a generalization of an identity recently studied by Zagier. In the second part, we show that “volume conjecture” is numerically supported for hyperbolic knots and links (knots up to 6-crossing, Whitehead link, and Borromean rings). 1. INTRODUCTION In [Kashaev 95, Kashaev 97], Kashaev defined an invari- ant KN for knot K using a quantum dilogarithm func- tion at the N-th root of unity, and proposed the stimulat- ing conjecture that for a hyperbolic knot K an asymptotic limit N →∞of the invariant KN gives a hyperbolic volume of a knot complement, 2 π 3 lim log KN = v3 · S \K , (2–8) N→∞ N where v3 is the hyperbolic volume of the regular ideal tetrahedron, and · denotes the Gromov norm. It was later proved [Murakami and Murakami 01] that Kashaev’s invariant coincides with a specific value of the colored Jones polynomial. In several attempts since then, a geometrical aspect to relate Kashaev’s R-matrix with an ideal octahedron in the three-dimensional hyperbolic space has been clarified (see, e.g., [Thurston 99, Yokota 00, Hikami 01]). Furthermore, a relationship with the Chern–Simons invariant was pointed out [Murakami et al. 02]. In this paper, we are interested in an explicit form 2000 AMS Subject Classification: Primary 52M27, 11B65; Secondary 57M50, 11Mxx of Kashaev’s invariant for the knot K. In general, this invariant can be regarded as a reduction of certain q- Keywords: Jones polynomial, Rogers-Ramanujan identity, hyperbolic volume series. In [Zagier 01], Zagier derived a strange identity c A K Peters, Ltd. 1058-6458/2003 $ 0.50 per page Experimental Mathematics 12:3, page 319 320 Experimental Mathematics, Vol. 12 (2003), No. 3 (3,2) ∞ for a q-series, F (q)= n=0(q)n, which was origi- 2. KASHAEV’S INVARIANT nally introduced in [Stoimenow 98] as an upper bound AND VOLUME CONJECTURE of the number of linearly independent Vassiliev invari- The quantum invariant of knot K can be constructed once F (3,2) −t half- ants. He showed that (e ) is related to the we have the enhanced Yang–Baxter operator [Turaev 88], differential η of the Dedekind -function. From our view- (R, µ, α, β), satisfying point, F (3,2)(q) with q → e2πi/N is nothing but Kashaev’s (3,2) 2πi/N invariant for the trefoil, 31N = F (e ). This (R ⊗ 1)(1 ⊗ R)(R ⊗ 1) = (1 ⊗ R)(R ⊗ 1)(1 ⊗ R), (2–1) motivates us to study an asymptotic expansion of the q- µ ⊗ µ R R µ ⊗ µ , series which, when q is the N-th root of unity, reduces = (2–2) to Kashaev’s invariant for the torus knot. We introduce ±1 ±1 Tr2 R (1 ⊗ µ) = α β. (2–3) the q-series F (2m+1,2)(q) as a generalization of Zagier’s q-series and prove an identity, The operators R±1 and µ±1 are usually depicted as fol- lows; F (2m+1,2)(e2πi/N ) 2 2 3 πi − πi (2m−1) j j √ N 2 e 4 N 4(2m+1) i i 2 m +1 m−1 ij −1 ij 2 R R j 2 j +1 −Nπi (2j+1) k = k = × − m − j π 4(2m+1) ( 1) ( )sin m e × × j=0 2 +1 k k ∞ 2 (2m+1,2) n − πi (2m−1) Tn π N 4(2m+1) , +e n m N n=0 ! 4(2 +1) i (3–15) / k k −1 k and then propose a conjecture, µ = µ = k / F (2m+1,2)(e−t) ∞ 2 (2m+1,2) n (2m−1) t Tn t 8(2m+1) . =e n 3 m (3–24) K ξ n=0 ! 2 (2 +1) When the knot is given as a closure of a braid with n strands, the invariant τ1(K) is computed as for the (2m+1,2) (2m+1,2) K Here F (q), respectively T -number Tn ,are (1,1)-tangle of knot as defined in Equations (3–22), respectively (3–10). τ K α−w(ξ) β−n b ξ ⊗ µ⊗(n−1) . In Section 2, we review the volume conjecture and 1( )= Tr2,...,n R( )(1 ) explain how to construct Kashaev’s invariant from the (2–4) b ξ enhanced Yang–Baxter operator. In Section 3, we study Here we have associated the homomorphism R( )byre- σ±1 ξ R±1 w ξ analytically in detail Kashaev’s invariant for the torus placing the braid group i in with ,and ( ) knot. We consider an asymptotic expansion of the invari- denotes a writhe (a sum of exponents). ant following Kashaev-Tirkkonen [Kashaev and Tirkko- Kashaev’s invariant is originally defined by use of nen 00], and derive an asymptotic formula for q-series the quantum dilogarithm function with a deformation 2πi/N N with q → e . In Section 4, we study numerically an parameter being the -th root of unity [Fadeev and asymptotic behavior of invariants for hyperbolic knots Kashaev 94], ω π /N . and links. We use PARI/GP [PARI 00], and show that =exp(2 i ) (2–5) there is a universal logarithmic correction to invariants. The invariant is then defined as follows: We use the q- We then propose a conjecture as an extension of Equa- product, tion (2–8), n i (ω)n = (1 − ω ), 3 N 3 0 log KN ∼ v3 · S \K · + #(K) · log N + O(N ), i=1 2 π 2 (4–1) n ∗ −i where #(K) is the number of prime factors of a knot (ω)n = (1 − ω ). considered as a connected-sum of prime knots. i=1 Hikami: Volume Conjecture and Asymptotic Expansion of q-Series 321 Theorem 2.1. [Kashaev 95, Kashaev 97] (See also Conjecture 2.3. [Kashaev 97, Murakami and Murakami [Murakami and Murakami 01].) Kashaev’s invariant 01] The asymptotic behavior of Kashaev’s invariant gives KN for the knot K is defined by Equation (2–4) with the hyperbolic volume of the knot complement of knot K; R µ the following and matrices; π 2 K v · S3 \K, 1−(k−j+1)(−i) lim log N = 3 (2–8) ij Nω ij N→∞ N Rk = ∗ ∗ · θ , (ω)[−k−1] (ω) (ω)[i−j] (ω) k [j−] [k−i] where v3 is the hyperbolic volume of the regular ideal (2–6a) tetrahedron, and · denotes the Gromov norm. −1+(−i−1)(k−j) −1 ij Nω ij (R )k = ∗ ∗ · θ , A mathematically rigorous proof of this conjecture has (ω) (ω)[j−] (ω) (ω)[k−i] k [−k−1] [i−j] not been established yet (only thea case of the figure- (2–6b) eight knot was proved (see, e.g., [Murakami 00]). How- k 1 ever, several geometrical studies have been done; the µ = −δk,+1 ω 2 . (2–6c) relationship between Kashaev’s R-matrix and the ideal Here [x] ∈{0, 1,...,N − 1} modulo N,and hyperbolic octahedron has been established [Thurston 99, Yokota 00, Hikami 01], and it was found that the i ≤ k<≤ j, saddle point equation of the invariant coincides with the ij j ≤ i ≤ k<, hyperbolicity consistency condition. θ =1, if and only if k ≤ j ≤ i ≤ k (with <k), As it is well known [Neumann and Zagier 85, Yoshida k<≤ j ≤ i. 85] that the hyperbolic volume is closely related to the Chern–Simons invariant, we also propose a complexifica- tion of Conjecture 2.3. In [Murakami and Murakami 01], it was shown that this invariant coincides with a specific value of the col- Conjecture 2.4. [Murakami et al. 02, Baseilhac and ored Jones polynomial, the invariant of knot K colored Benedetti 01] by the irreducible SU(2)q-module of dimension N with a π q → π /N 2 3 parameter exp(2 i ). lim log KN = v3 · S \K +iCS(K), (2–9) N→∞ N Theorem 2.2. [Murakami and Murakami 01] Kashaev’s where CS denotes the Chern–Simons invariant, invariant KN coincides with the colored Jones polyno- 2 mial at the N-th root of unity, whose R-matrix is given CS(M)=2π cs(M), by 1 2 csM(A)= Tr A ∧ dA + A ∧ A ∧ A . min(N−1−i,j) 8 π2 3 ij i+j+n M Rk = δ,i+n δk,j−n (−1) n=0 ∗ (ω)i+n (ω)j ij+ 1 (i+j−n) × ω 2 , ∗ ∗ 3. TORUS KNOTS AND q-SERIES (ω)i (ω)j−n (ω)n (2–7a) 3.1 Invariant of the Torus Knot m, p min(N−1−j,i) We consider the ( )-torus knot, where we suppose that R−1 ij δ δ − i+j+n m and p are coprime integers. The knot is expressed in k = ,i−n k,j+n ( 1) n=0 terms of generators of Artin’s braid group as ∗ (ω)i (ω)j+n −ij− 1 (i+j−n) × ω 2 , p ∗ ξ = σ1 σ2 ···σm−1 .
Load more

Recommended publications
  • Experimental Evidence for the Volume Conjecture for the Simplest Hyperbolic Non-2-Bridge Knot Stavros Garoufalidis Yueheng Lan
    ISSN 1472-2739 (on-line) 1472-2747 (printed) 379 Algebraic & Geometric Topology Volume 5 (2005) 379–403 ATG Published: 22 May 2005 Experimental evidence for the Volume Conjecture for the simplest hyperbolic non-2-bridge knot Stavros Garoufalidis Yueheng Lan Abstract Loosely speaking, the Volume Conjecture states that the limit of the n-th colored Jones polynomial of a hyperbolic knot, evaluated at the primitive complex n-th root of unity is a sequence of complex numbers that grows exponentially. Moreover, the exponential growth rate is proportional to the hyperbolic volume of the knot. We provide an efficient formula for the colored Jones function of the simplest hyperbolic non-2-bridge knot, and using this formula, we provide numerical evidence for the Hyperbolic Volume Conjecture for the simplest hyperbolic non-2-bridge knot. AMS Classification 57N10; 57M25 Keywords Knots, q-difference equations, asymptotics, Jones polynomial, Hyperbolic Volume Conjecture, character varieties, recursion relations, Kauffman bracket, skein module, fusion, SnapPea, m082 1 Introduction 1.1 The Hyperbolic Volume Conjecture The Volume Conjecture connects two very different approaches to knot theory, namely Topological Quantum Field Theory and Riemannian (mostly Hyper- bolic) Geometry. The Volume Conjecture states that for every hyperbolic knot K in S3 2πi log |J (n)(e n )| 1 lim K = vol(K), (1) n→∞ n 2π where ± • JK (n) ∈ Z[q ] is the Jones polynomial of a knot colored with the n- dimensional irreducible representation of sl2 , normalized so that it equals to 1 for the unknot (see [J, Tu]), and c Geometry & Topology Publications 380 Stavros Garoufalidis and Yueheng Lan • vol(K) is the volume of a complete hyperbolic metric in the knot com- plement S3 − K ; see [Th].
    [Show full text]
  • Arxiv:Math-Ph/0305039V2 22 Oct 2003
    QUANTUM INVARIANT FOR TORUS LINK AND MODULAR FORMS KAZUHIRO HIKAMI ABSTRACT. We consider an asymptotic expansion of Kashaev’s invariant or of the colored Jones function for the torus link T (2, 2 m). We shall give q-series identity related to these invariants, and show that the invariant is regarded as a limit of q being N-th root of unity of the Eichler integral of a modular form of weight 3/2 which is related to the su(2)m−2 character. c 1. INTRODUCTION Recent studies reveal an intimate connection between the quantum knot invariant and “nearly modular forms” especially with half integral weight. In Ref. 9 Lawrence and Zagier studied an as- ymptotic expansion of the Witten–Reshetikhin–Turaev invariant of the Poincar´ehomology sphere, and they showed that the invariant can be regarded as the Eichler integral of a modular form of weight 3/2. In Ref. 19, Zagier further studied a “strange identity” related to the half-derivatives of the Dedekind η-function, and clarified a role of the Eichler integral with half-integral weight. From the viewpoint of the quantum invariant, Zagier’s q-series was originally connected with a gener- ating function of an upper bound of the number of linearly independent Vassiliev invariants [17], and later it was found that Zagier’s q-series with q being the N-th root of unity coincides with Kashaev’s invariant [5, 6], which was shown [14] to coincide with a specific value of the colored Jones function, for the trefoil knot. This correspondence was further investigated for the torus knot, and it was shown [3] that Kashaev’s invariant for the torus knot T (2, 2 m + 1) also has a nearly arXiv:math-ph/0305039v2 22 Oct 2003 modular property; it can be regarded as a limit q being the root of unity of the Eichler integral of the Andrews–Gordon q-series, which is theta series with weight 1/2 spanning m-dimensional space.
    [Show full text]
  • Volume Conjecture and Topological Recursion
    . Volume Conjecture and Topological Recursion .. Hiroyuki FUJI . Nagoya University Collaboration with R.H.Dijkgraaf (ITFA) and M. Manabe (Nagoya Math.) 6th April @ IPMU Papers: R.H.Dijkgraaf and H.F., Fortsch.Phys.57(2009),825-856, arXiv:0903.2084 [hep-th] R.H.Dijkgraaf, H.F. and M.Manabe, to appear. Hiroyuki FUJI Volume Conjecture and Topological Recursion . 1. Introduction Asymptotic analysis of the knot invariants is studied actively in the knot theory. Volume Conjecture [Kashaev][Murakami2] Asymptotic expansion of the colored Jones polynomial for knot K ) The geometric invariants of the knot complement S3nK. S3 Recent years the asymptotic expansion is studied to higher orders. 2~ Sk(u): Perturbative" invariants, q = e [Dimofte-Gukov-Lenells-Zagier]# X1 1 δ k Jn(K; q) = exp S0(u) + log ~ + ~ Sk+1(u) ; u = 2~n − 2πi: ~ 2 k=0 Hiroyuki FUJI Volume Conjecture and Topological Recursion Topological Open String Topological B-model on the local Calabi-Yau X∨ { } X∨ = (z; w; ep; ex) 2 C2 × C∗ × C∗jH(ep; ex) = zw : D-brane partition function ZD(ui) Calab-Yau manifold X Disk World-sheet Labrangian Embedding submanifold = D-brane Ending on Lag.submfd. Holomorphic disk Hiroyuki FUJI Volume Conjecture and Topological Recursion Topological Recursion [Eynard-Orantin] Eynard and Orantin proposed a spectral invariants for the spectral curve C C = f(x; y) 2 (C∗)2jH(y; x) = 0g: • Symplectic structure of the spectral curve C, • Riemann surface Σg,h = World-sheet. (g,h) ) Spectral invariant F (u1; ··· ; uh) ui:open string moduli Eynard-Orantin's topological recursion is applicable. x x1 xh x x x x x x 1 h x x k1 k i1 ij hj q q q = q + Σ l J g g 1 g l l Σg,h+1 Σg−1,h+2 Σℓ,k+1 Σg−ℓ,h−k Spectral invariant = D-brane free energy in top.
    [Show full text]
  • Alexander Polynomial, Finite Type Invariants and Volume of Hyperbolic
    ISSN 1472-2739 (on-line) 1472-2747 (printed) 1111 Algebraic & Geometric Topology Volume 4 (2004) 1111–1123 ATG Published: 25 November 2004 Alexander polynomial, finite type invariants and volume of hyperbolic knots Efstratia Kalfagianni Abstract We show that given n > 0, there exists a hyperbolic knot K with trivial Alexander polynomial, trivial finite type invariants of order ≤ n, and such that the volume of the complement of K is larger than n. This contrasts with the known statement that the volume of the comple- ment of a hyperbolic alternating knot is bounded above by a linear function of the coefficients of the Alexander polynomial of the knot. As a corollary to our main result we obtain that, for every m> 0, there exists a sequence of hyperbolic knots with trivial finite type invariants of order ≤ m but ar- bitrarily large volume. We discuss how our results fit within the framework of relations between the finite type invariants and the volume of hyperbolic knots, predicted by Kashaev’s hyperbolic volume conjecture. AMS Classification 57M25; 57M27, 57N16 Keywords Alexander polynomial, finite type invariants, hyperbolic knot, hyperbolic Dehn filling, volume. 1 Introduction k i Let c(K) denote the crossing number and let ∆K(t) := Pi=0 cit denote the Alexander polynomial of a knot K . If K is hyperbolic, let vol(S3 \ K) denote the volume of its complement. The determinant of K is the quantity det(K) := |∆K(−1)|. Thus, in general, we have k det(K) ≤ ||∆K (t)|| := X |ci|. (1) i=0 It is well know that the degree of the Alexander polynomial of an alternating knot equals twice the genus of the knot.
    [Show full text]
  • Modular Forms and Quantum Knot Invariants
    Modular forms and Quantum knot invariants Kazuhiro Hikami (Kyushu University), Jeremy Lovejoy (CNRS, Universite´ Paris 7), Robert Osburn (University College Dublin) March 11–16, 2018 1 Overview A modular form is an analytic object with intrinsic symmetric properties. They have en- joyed long and fruitful interactions with many areas in mathematics such as number theory, algebraic geometry, combinatorics and physics. Their Fourier coefficients contain a wealth of information, for example, about the number of points over the finite field of prime order of elliptic curves, K3 surfaces and Calabi-Yau threefolds. Most importantly, they were the key players in Wiles’ spectacular proof in 1994 of Fermat’s Last Theorem. Quantum knot invariants have their origin in the seminal work of two Fields medalists, Vaughn Jones in 1984 on von Neumann algebras and Edward Witten in 1988 on topological quantum field theory. The equivalence between vacuum expectation values of Wilson loops (in Chern- Simons gauge theory) and knot polynomial invariants (for example, the Jones polynomial, HOMFLY polynomial and Kauffman polynomial) is a major source of inspiration for the development of new knot invariants via quantum groups. Over the past two decades, there have been many tantalizing interactions between quan- tum knot invariants and modular forms. For example, quantum invariants of 3-manifolds are typically functions that are defined only at roots of unity and one can ask whether they extend to holomorphic functions on the complex disk with interesting arithmetic prop- erties. A first result in this direction was found in 1999 by Lawrence and Zagier [17] who showed that Ramanujan’s 5th order mock theta functions coincide asymptotically with Witten-Reshetikhin-Turaev (WRT) invariants of Poincare´ homology spheres.
    [Show full text]
  • [Math.GT] 18 Oct 2005 L Ihte(Omlzd Rmvnr.Ltu Tt Oepeieythis of Precisely Representations More Irreducible State the Us Using Let That, Norm
    3 2 1 COLORED JONES INVARIANTS OF LINKS IN S #kS S AND THE × VOLUME CONJECTURE FRANCESCO COSTANTINO Abstract. We extend the definition of the colored Jones polynomials to framed links and triva- 3 2 1 lent graphs in S #kS × S using a state-sum formulation based on Turaev’s shadows. Then, we prove that the natural extension of the Volume Conjecture is true for an infinite family of hyperbolic links. Contents 1. Introduction 1 2. Recalls on shadows of graphs and state sums 3 2.1. Shadows of graphs 3 2.2. Explict constructions 5 3 2 1 3. The Jones invariant for framed links and trivalent graphs in S #kS S 6 3.1. State-sum quantum invariants × 6 3.2. The Jones invariant through shadows 8 3.3. Properties and examples 9 2 1 4. The Volume Conjecture for links in #kS S 11 References × 14 1. Introduction Since its discovery in the early eighties, the Jones polynomial of links in S3 has been one of the most studied objects in low-dimensional topology. Despite this, at present, it is not yet completely clear which topological informations are carried and how are encoded by this invariant arXiv:math/0510048v2 [math.GT] 18 Oct 2005 and more in general by the larger family of Quantum Invariants. One of the most important conjectural relations between the topology of a link in a manifold and its Quantum Invariants has been given by Kashaev through his Volume Conjecture for hyperbolic links L in S3 ([9]), based on complex valued link invariants <L>d constructed by using planar (1, 1)-tangle presentations of L and constant Kashaev’s R-matrices ([10]).
    [Show full text]
  • On the Volume Conjecture for Hyperbolic Knots
    ON THE VOLUME CONJECTURE FOR HYPERBOLIC KNOTS MASTER THESIS submitted by ANNICK SCHMITGEN Department of Mathematics Under the supervision of Professor Dr. Ruth Kellerhals July 2010 ABSTRACT This thesis presents the Volume Conjecture raised by R. Kashaev and provides proofs for the figure-eight knot 41 and the knot 52. With the goal of constructing the colored Jones polynomial - one principal con- stituent of the Volume Conjecture - we first explain the main aspects of the theo- ries of knots, tangles and braids. Hyperbolic geometry, with its volume measure, forms another essential ingredient. Then we switch to the concept of Hopf algebras to derive the Yang-Baxter equa- tion, whose solutions, the so-called R-matrices, will be investigated through the formalism of quantum groups. These preliminaries culminate in a physical interpretation and their direct appli- cation to the construction of the colored Jones polynomial: consider the finite- dimensional irreducible representations of the quantum universal enveloping algebra of the Lie algebra sl2.C/, a quantum group, and decorate the .1; 1/- tangle diagram of a link according to certain assignment rules. This leads to the treasured colored Jones invariant. Parallel to this, the Kashaev invariant, based on the quantum dilogarithm, is introduced and shown to coincide with the colored Jones polynomial evaluated at the N th root of unity. Finally, the Volume Conjecture is exposed first in its original version by R. Kashaev and then in a modified formulation by H. Murakami and J. Murakami. Numerical evidence supporting the Conjecture and rigorous proofs for the cases of the knot 41 and the knot 52 are provided.
    [Show full text]
  • The ADO Invariants Are a Q-Holonomic Family
    The ADO Invariants are a q-Holonomic Family Jennifer Browna , Tudor Dimoftea , Stavros Garoufalidisb;c , and Nathan Geerd aDepartment of Mathematics and QMAP, UC Davis, 1 Shields Ave, Davis, CA 95616, USA bMax Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany cInternational Center for Mathematics, Dept. of Math., Southern Univ. of Science and Technology, Shenzhen, China dDepartment of Mathematics & Statistics, Utah State University, Logan, Utah 84322, USA [email protected] [email protected] [email protected] [email protected] May 17, 2020 Abstract We investigate the q-holonomic properties of a class of link invariants based on quantum group representations with vanishing quantum dimensions, motivated by the search for the invariants' realization in physics. Some of the best known invariants of this type, constructed from `typical' representations of the unrolled quantum group U H (sl ) at a 2 -th root of unity, ζ2r 2 r were introduced by Akutsu-Deguchi-Ohtsuki (ADO). We prove that the ADO invariants for r ≥ 2 are a q-holonomic family, implying in particular that they satisfy recursion relations that are independent of r. In the case of a knot, we prove that the q-holonomic recursion ideal of the ADO invariants is contained in the recursion ideal of the colored Jones polynomials, the subject of the celebrated AJ Conjecture. (Combined with a recent result of S. Willetts, this establishes an isomorphism of the ADO and Jones recursion ideals. Our results also confirm a recent physically-motivated conjecture of Gukov-Hsin-Nakajima-Park-Pei.) Contents 1 Introduction 2 1.1 Roots of unity, q-holonomic families, and Hamiltonian reduction.........4 1.2 Example: figure-eight knot..............................6 1.3 Acknowledgements...................................7 2 Background 7 2.1 An extension of the Drinfel'd-Jimbo algebra.....................7 H 2.1.1 Representations of U ζ2r ............................8 2.1.2 The ribbon structure on C ..........................
    [Show full text]
  • Quantum Field Theory and the Volume Conjecture
    Contemporary Mathematics Volume 541, 2011 Quantum Field Theory and the Volume Conjecture Thdor Dimofte and Sergei Gukov ABSTRACT. The volume conjecture states that for a hyperbolic knot K in the three-sphere S 3 the asymptotic growth of the colored Jones polynomial of K is governed by the hyperbolic volume of the knot complement S3 \K. The conjecture relates two topological invariants, one combinatorial and one geometric, in a very nonobvious, nontrivial manner. The goal of the present lectures is to review the original statement of the volume conjecture and its recent extensions and generalizations, and to show how, in the most general context, the conjecture can be understood in terms of topological quantum field theory. In particular, we consider: a) generalization of the volume conjecture to families of incomplete hyperbolic metrics; b) generalization that involves not only the leading (volume) term, but the entire asymptotic expansion in 1/N; c) generalization to quantum group invariants for groups of higher rank; and d) generalization to arbitrary links in arbitrary three-manifolds. 1. Preliminaries 3 Let K be an oriented knot (or link) in the three-sphere S . The original volume conjecture [21, 26] relates theN-colored Jones polynomial of K to the hyperbolic volume of the knot complement S3\K: N-colored Jones poly of K hyperbolic volume of S3 \K (1.1) (combinatorial, rep. theory) (geometric) . We begin by reviewing some of the definitions and ingredients that enter on the two sides here in order to make this statement more precise, and to serve as a precursor for its subsequent generalization.
    [Show full text]
  • Arxiv:1503.02547V4 [Math.GT] 9 Jul 2018 Perbolic Volume of the Complement of the Link
    VOLUME CONJECTURES FOR THE RESHETIKHIN-TURAEV AND THE TURAEV-VIRO INVARIANTS QINGTAO CHEN AND TIAN YANG ABSTRACT. We consider the asymptotics of the Turaev-Viro and thep Reshetikhin-Turaev invariants of a hyperbolicp 3-manifold, evaluated at the root of unity exp(2π −1=r) instead of the standard exp(π −1=r). We present evidence that, as r tends to 1, these invariants grow exponentially with growth rates respectively given by the hyperbolic and the complex volume of the manifold. This reveals an asymptotic behavior that is different from that of Witten’s Asymptotic Expansion Conjecture, which predicts polynomial growth of these invariants when evaluated at the standard root of unity. This new phenomenon suggests that the Reshetikhin-Turaev invariants may have a geometric interpretation other than the original one via SU(2) Chern-Simons gauge theory. 1. INTRODUCTION In [62], Witten provided a new interpretation of the Jones polynomial based on Chern-Simons gauge theory, and expanded on this idea to construct a sequence of complex valued 3-manifold invariants. This approach was formalized though the representation theory of quantum groups by Reshetikhin and Turaev [53, 54], who generalized the Jones polynomial to a sequence of polyno- mial invariants of a link, later called the colored Jones polynomials of that link. They also defined a sequence of 3-manifold invariants corresponding to Witten’s invariants. The Reshetikhin-Turaev construction of 3-manifold invariants starts from a surgery description [33] of the manifold, and evaluates the colored Jones polynomials of the surgery data at certain roots of unity. A different approach was developed by Turaev and Viro [60] who, from a triangulation of a closed 3-manifold, constructed real valued invariants of the manifold by using quantum 6j- symbols [35]; these Turaev-Viro invariants turned out to be equal to the square of the norm of the Reshetikhin-Turaev invariants [55, 59, 61].
    [Show full text]
  • On the Asymptotic Expansion of the Kashaev Invariant of the 52 Knot
    On the asymptotic expansion of the Kashaev invariant of the 52 knot Tomotada Ohtsuki Abstract We give a presentation of the asymptotic expansion of the Kashaev invariant of the 52 knot. As the volume conjecture states, the leading term of the expansion presents the hyperbolic volume and the Chern-Simons invariant of the complement of the 52 knot. Further, we obtain a method to compute the full Poincare asymptotics to all orders of the Kashaev invariant of the 52 knot. 1 Introduction h i 2 C ··· In [16, 17] Kashaev defined the Kashaev invariant L N of a link L for N = 2; 3; by using the quantum dilogarithm. In [18] he conjectured that, for any hyperbolic link L, log h L i 2π · lim N = vol(S3 − L); N!1 N where \vol" denotes the hyperbolic volume, and gave evidence for the conjecture for the figure-eight knot, the 52 knot and the 61 knot, by formal calculations. In 1999, H. h i Murakami and J. Murakami [24] proved that the Kashaevp invariant L N of anyp link L 2π −1=N 2π −1=N is equal to the N-colored Jones polynomial JN (L; e ) of L evaluated at e , where JN (L; q) denotes the invariant obtained as the quantum invariant of links associated with the N-dimensional irreducible representation of the quantum group Uq(sl2). Further, as an extension of Kashaev's conjecture, they conjectured that, for any knot K, p log jJ (K; e2π −1=N )j 2π · lim N = vol(S3 − K); N!1 N where \vol" in this formula denotes the simplicial volume (normalized by multiplying by the hyperbolic volume of the regular ideal tetrahedron).
    [Show full text]
  • Recent Progress of Various Volume Conjectures for Links As Well As 3-Manifolds
    Recent progress of various volume conjectures for links as well as 3-manifolds Qingtao Chen ETHZ (Swiss Federal Institute of Technology in Zuerich) Joint work with Kefeng Liu and Shengmao Zhu Joint work with Joergen Andersen Joint work with Tian Yang Joint work with Jun Murakami Workshop on Algebraic and Geoemtric Topology Emeishan, China July, 2018 Qingtao Chen (ETHZ) Volume Conjectures Emeishan,China,July, 2018 1/ 66 Sketch of my talk Theme of my talk: Intertwining of Mathematics and Physics Mathematics (Volume Conjecture for colored Jones) inspired by physics (Chern-Simons gauge theory, string theory and largeN duality) Possible new physics interpretation indicated by new discovery in mathematics (Volume Conjectures for Turaev-Viro invariants and Reshetikhin-Turaev invariants as well as asymptotics of quantum 6j symbols) Qingtao Chen (ETHZ) Volume Conjectures Emeishan,China,July, 2018 2/ 66 Sketch of my talk Theme of my talk: Intertwining of Mathematics and Physics Mathematics (Volume Conjecture for colored Jones) inspired by physics (Chern-Simons gauge theory, string theory and largeN duality) Possible new physics interpretation indicated by new discovery in mathematics (Volume Conjectures for Turaev-Viro invariants and Reshetikhin-Turaev invariants as well as asymptotics of quantum 6j symbols) Qingtao Chen (ETHZ) Volume Conjectures Emeishan,China,July, 2018 2/ 66 Sketch of my talk Theme of my talk: Intertwining of Mathematics and Physics Mathematics (Volume Conjecture for colored Jones) inspired by physics (Chern-Simons gauge theory, string theory and largeN duality) Possible new physics interpretation indicated by new discovery in mathematics (Volume Conjectures for Turaev-Viro invariants and Reshetikhin-Turaev invariants as well as asymptotics of quantum 6j symbols) Qingtao Chen (ETHZ) Volume Conjectures Emeishan,China,July, 2018 2/ 66 Part I History and Basic Setups of Quantum Topology Qingtao Chen (ETHZ) Volume Conjectures Emeishan,China,July, 2018 3/ 66 Introduction Definition A knot is a simple closed curve embedded in R3.
    [Show full text]