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Is the Jones Polynomial of a Knot Really a Polynomial?

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November 8, 2006 16:32 WSPC/134-JKTR 00491 Journal of Knot Theory and Its Ramifications Vol. 15, No. 8 (2006) 983–1000 c World Scientific Publishing Company IS THE JONES POLYNOMIAL OF A KNOT REALLY A POLYNOMIAL? STAVROS GAROUFALIDIS∗ and THANG TQ LEˆ † School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA ∗[email protected] †[email protected] Accepted 19 June 2006 Dedicated to Louis Kauffman on the occasion of his 60th birthday ABSTRACT The Jones polynomial of a knot in 3-space is a Laurent polynomial in q, with integer coefficients. Many people have pondered why this is so, and what a proper generalization of the Jones polynomial for knots in other closed 3-manifolds is. Our paper centers around this question. After reviewing several existing definitions of the Jones polynomial, we argue that the Jones polynomial is really an analytic function, in the sense of Habiro. Using this, we extend the holonomicity properties of the colored Jones function of a knot in 3-space to the case of a knot in an integer homology sphere, and we formulate an analogue of the AJ Conjecture. Our main tools are various integrality properties of topological quantum field theory invariants of links in 3-manifolds, manifested in Habiro’s work on the colored Jones function. Keywords: Knots; Jones polynomial; Kauffman bracket; Habiro ring; homology spheres; TQFT; holonomic functions; A-polynomial; AJ Conjecture. Mathematics Subject Classification 2000: Primary 57N10; Secondary 57M25 J. Knot Theory Ramifications 2006.15:983-1000. Downloaded from www.worldscientific.com 1. Introduction 1.1. The Jones polynomial of a knot In 1985 Jones discovered the celebrated Jones polynomial of a knot/link in 3-space, see [14]. The framed version of the Jones polynomial of a framed oriented knot/link may be uniquely defined by the following skein theory: by MAX-PLANCK-INSTITUTE FOR MATHEMATICS LIBRARY on 03/17/20. Re-use and distribution is strictly not permitted, except for Open Access articles. From the above definition, the Jones polynomial JL of an oriented link L lies in Z[q±1/4]. 983 November 8, 2006 16:32 WSPC/134-JKTR 00491 984 S. Garoufalidis & T. T. Lˆe An important and trivial observation is that knots may be considered to lie in closed 3-manifolds other than S3. The paper is centered around the following question: Question 1.1. What is the Jones polynomial of a knot K in a closed 3-manifold N? In particular, is the Jones polynomial of a knot in a closed 3-manifold really a polynomial? Before we reveal the answer, let us review some alternative definitions of the Jones polynomial, namely the Kauffman bracket approach, the quantum group approach, the TQFT (Witten–Reshetikhin–Turaev) approach, and the perturbative TQFT approach. Note that being polynomial is closely related to the integrality discussed in [21]. 1.2. The Kauffman bracket of a knot Soon after Jones’s discovery, Kauffman gave a reformulation of the Jones polynomial in terms of the Kauffman bracket · : Framed unoriented links in S3 → Z[A±1]. The Kauffman bracket is defined by the following skein theory (see [15]): (1.1) and the relation between the Kauffman bracket and the Jones polynomial is the following: if L is an oriented m-component link projection with writhe w(L), then m 3 −w(L) JL|q1/4=A =(−1) (−A ) L. Since the Kauffman bracket takes values in Z[A±1], it implies once again that the Jones polynomial of an oriented knot/link in 3-space takes values in Z[q±1/4]. The Kauffman bracket can be generalized to framed unoriented links in a closed 3-manifold N and the resulting Kauffman bracket skein module S(N) can be defined to be the quotient of the free Z[A±1]-module on the set of framed unoriented links J. Knot Theory Ramifications 2006.15:983-1000. Downloaded from www.worldscientific.com in N, modulo the relations (1.1): S(N)=Z[A±1]-linear combinations of framed unoriented links in N/(relations (1.1)). From this point of view, the Kauffman bracket skein module of S(S3)isafree Z[A±1] with basis the empty link. This explains why the Jones polynomial of a link in S3 is an element of Z[A±1]. A knot in a general 3-manifold N gives, by definition, an element of the Kauffman bracket skein module S(N). However, little is known about the skein by MAX-PLANCK-INSTITUTE FOR MATHEMATICS LIBRARY on 03/17/20. Re-use and distribution is strictly not permitted, except for Open Access articles. module S(N). In particular, it is not known when it is free, nor is known a basis of it. In addition, S(N) is not a ring, but rather a module over the ring Z[A±1]. Thus, the skein module S(N) does not answer our Question 1.1. November 8, 2006 16:32 WSPC/134-JKTR 00491 Is the Jones Polynomial of a Knot Really a Polynomial? 985 1.3. The quantum group approach Using quantum group technology (see [25, 27]), one may define the colored Jones polynomial of a framed oriented knot in 3-space. Let us postpone the technical details to a later Sec. 2.1 and concentrate on a main idea: the notion of color. Suppose L is a framed r-component link in S3 with ordered components. The Jones polynomial of L is a powerful invariant that takes values in the Laurent polynomial ring R := Z[q±1/4]. (1.2) An even more powerful invariant is the colored Jones function r JL : N →R, which encodes the Jones polynomial of L together with its parallels. Here N is the set of positive integers, and JL(n1,...,nr)isthequantumsl2 invariant of the link L with colors the irreducible sl2-modules of dimensions n1,...,nr.When n1 = ···= nr = 2, the polynomial JL(n1,...,nr) is the Jones polynomial. Here we use the normalization such that when U is the unknot of 0 framing, qn/2 − q−n/2 JU (n)=[n]:= . q1/2 − q−1/2 1.4. The TQFT approach Let us fix a framed oriented knot K in a 3-manifold N and a nonnegative integer ∈ N ∈ Q n . The Witten–Reshetikhin–Turaev (WRT) invariant Z(N,K),ξ,Vn (ξ)is a generalization of the colored Jones polynomial and can be defined under the framework of topological quantum field theory (TQFT in short); see [27]. Here Vn is the n-dimensional irreducible representation of sl2 and ξ is a complex root of unity of order divisible by 4. When N = S3, and with the proper normalization, we may identify the WRT invariant with the colored Jones polynomial as follows: 3 Z(S ,K),ξ,Vn =evξJK (n), J. Knot Theory Ramifications 2006.15:983-1000. Downloaded from www.worldscientific.com where ±1/4 evξ : Z[q ] → Z[ξ] is the evaluation q1/4 = ξ. From the TQFT approach, it is not clear why the Jones polynomial is even a polynomial. 1.5. The perturbative approach pert by MAX-PLANCK-INSTITUTE FOR MATHEMATICS LIBRARY on 03/17/20. Re-use and distribution is strictly not permitted, except for Open Access articles. Perturbative quantum field theory also constructs an invariant Z(N,K),Vn which takes 3 pert Q 3 values in the power series ring [[h]]. When N = S , Z(S ,K),Vn is the composition of the Kontsevich integral of a knot with the sl2 weight system, and when N is November 8, 2006 16:32 WSPC/134-JKTR 00491 986 S. Garoufalidis & T. T. Lˆe pert sl2 arbitrary, Z(N,K),Vn is the composition of the LMO invariant with the weight system, see [1, 18]. When N = S3, the main identity is: pert h K ∈ Q ev J (n)=Z(N,K),Vn [[h]], where ±1/4 evh : Z[q ] → Q[[h]] is the evaluation q = eh; see, for example [22]. Thus, from the perturbative point of view, the Jones polynomial of a knot in a homology sphere exists as a formal power series in h. 1.6. The Habiro ring Despite the apparent failure of TQFT to explain the polynomial aspect of the Jones polynomial, there is one gain. Namely let us fix (N,K) a natural number n and let Ω4 denote the set of complex roots of unity whose order is divisible by 4. Then, the WRT invariant gives a function: → C Z(N,K),Vn :Ω4 . (1.3) This function is not continuous, and does not have nice analytic properties. Habiro introduced an alternative notion of analytic functions. The latter are by definition elements of the Habiro ring, defined by: 2 n Z[q] := lim Z[q]/((1 − q)(1 − q ) ···(1 − q )). (1.4) ←n The ring Z[q] can be considered as the set of all series of the form ∞ 2 n f(q)= fn(q)(1− q)(1 − q ) ···(1 − q ), wherefn(q) ∈ Z[q], (1.5) n=0 with the warning that f(q)doesnot uniquely determine (fn(q)). It turns out that the Habiro ring Z[q] shares many properties with the ring of germs of complex analytic functions, (see [9]) and plays an important role in J. Knot Theory Ramifications 2006.15:983-1000. Downloaded from www.worldscientific.com Quantum Topology. In particular, elements of the Habiro ring (a) can be differentiated with respect to q, (b) can be evaluated at the set Ω of complex roots of unity, (c) have Taylor series expansions that uniquely determine them, (d) form an integral domain.
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