A function of several variables is called holonomic if, roughly speaking, it is determined from finitely many of its values via finitely many linear recursion relations with polynomial coefficients. Zeilberger was the first to notice that the abstract notion of holonomicity can be applied to verify, in a systematic and computerized way, combinatorial identities among special functions. Using a general state sum definition of the colored Jones function of a link in 3-space, we prove from first principles that the colored Jones function is a multisum of a q -proper-hypergeometric function, and thus it is q -holonomic. We demonstrate our results by computer calculations.

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Nahm sums are q-series of a special hypergeometric type that appear in character formulas in the conformal field theory, and give rise to elements of the Bloch group, and have interesting modularity properties. In our paper, we show how Nahm sums arise naturally in the quantum knot theory -we prove the stability of the coefficients of the colored Jones polynomial of an alternating link and present a Nahm sum formula for the resulting power series, defined in terms of a reduced diagram of the alternating link. The Nahm sum formula comes with a computer implementation, illustrated in numerous examples of proven or conjectural identities among q-series. MSC: Primary 57N10; Secondary 57M25.Keywords: Nahm sums; Colored Jones polynomial; Links; Stability; Modular forms; Mock-modular forms; q-holonomic sequence; q-series; Conformal field theory; Thin-thick decomposition BackgroundThe colored Jones polynomial of a link is a sequence of Laurent polynomials in one variable with integer coefficients. We prove in full a conjecture concerning the stability of the colored Jones polynomial for all alternating links.A weaker form of stability (zero stability, defined below) for the colored Jones polynomial of an alternating knot was conjectured by Dasbach and Lin. The zero stability is also proven independently by Armond for all adequate links [1], which include alternating links and closures of positive braids, see also [2]. The advantage of our approach is that it proves stability to all orders and gives explicit formulas (in the form of generalized Nahm sums) for the limiting series, which in particular implies convergence in the open unit disk in the q-plane and allow for the study of their redial asymptotics.Stability was observed in some examples by Zagier, and conjectured by the first author to hold for all knots, assuming that we restrict the sequence of colored Jones polynomials to suitable arithmetic progressions, dictated by the quasi-polynomial nature of its q-degree [3,4]. Zagier asked about modular and asymptotic properties of the limiting q-series. In a similar direction, Habiro asked about zero stability of the cyclotomic function of alternating links in [5].Our generalized Nahm sum formula comes with a computer implementation (using as input a planar diagram of a link), and allows the study of its asymptotics when q

Using elementary equalities between various cables of the unknot and the Hopf link, we prove the Wheels and Wheeling conjectures of [5,9], which give, respectively, the exact Kontsevich integral of the unknot and a map intertwining two natural products on a space of diagrams. It turns out that the Wheeling map is given by the Kontsevich integral of a cut Hopf link (a bead on a wire), and its intertwining property is analogous to the computation of 1 + 1 = 2 on an abacus. The Wheels conjecture is proved from the fact that the k -fold connected cover of the unknot is the unknot for all k .Along the way, we find a formula for the invariant of the general (k, l) cable of a knot. Our results can also be interpreted as a new proof of the multiplicativity of the Duflo-Kirillov map S(g) → U (g) for metrized Lie (super-)algebras g.

We state and prove a quantum generalization of MacMahon's celebrated Master Theorem and relate it to a quantum generalization of the boson-fermion correspondence of physics. In this article we state and prove a quantum generalization of MacMahon's celebrated Master Theorem conjectured by S.G. and T.T.Q.L. Our result was motivated by quantum topology. In addition to its potential importance in knot theory and quantum topology (explained in brief in ref. [1]There are several equivalent reformulations of MacMahon's Master Theorem (see, for example, ref. 3 and references therein). Let us mention one of these studies, which is of importance to physics. Given a matrix A ϭ (a ij ) of size r with commuting entries that lie in a ring R, and a nonnegative integer n, we can consider its symmetric and exterior powers S n (A) and ⌳ n (A), and their traces tr S n (A), and tr ⌳ n (A), respectively. Becausethe following identity] is equivalent to Eq. 1. In physics, Eq. 2 is called the boson-fermion correspondence, where bosons (fermions) are commuting (skew-commuting) particles corresponding to symmetric (exterior) powers.Quantum Algebra, Right-Quantum Matrices, and Quantum Determinants In r-dimensional quantum algebra we have r indeterminate variables x i (1 Յ i Յ r), satisfying the commutation relations x j x i ϭ qx i x j for all 1 Յ i Ͻ j Յ r. We also consider matrices A ϭ (a ij ) of r 2 indeterminates a ij , 1 Յ i, j Յ r, which commute with the x i and such that for any 2-by-2 minor of (a ij ), consisting of rows i and iЈ, and columns j and jЈ (where 1 Յ i Ͻ iЈ Յ r, and We will call such matrices A right-quantum matrices. The quantum determinant, (first introduced in ref.3) of any (not necessarily right-quantum) r by r matrix B ϭ (b ij ) may be defined bywhere the sum ranges over the set of permutations, S r , of {1, . . . , r}, and for any of its members, , inv() denotes the number of pairs 1 Յ i Ͻ j Յ r for which i Ͼ j . A q-Version of MacMahon's Master TheoremWe are now ready to state our quantum version of MacMahon's Master Theorem. The above result is not only interesting from the combinatorial point of view, but it is also a key ingredient in a finite noncom-

Objective-Previous studies have shown that oxidized products of PAPC (Ox-PAPC) regulate cell transcription of interleukin-8, LDL receptor, and tissue factor. This upregulation takes place in part through the activation of sterol regulatory element-binding protein (SREBP) and Erk 1/2. The present studies identify vascular endothelial growth factor receptor 2 (VEGFR2) as a major regulator in the activation of SREBP and Erk 1/2 in endothelial cells activated by Ox-PAPC. Methods and Results-Ox-PAPC induced the phosphorylation of VEGFR2 at Tyr 1175 in human aortic endothelial cells. Inhibitors and siRNA for VEGFR2 decreased the transcription of interleukin-8, LDL receptor, and tissue factor in response to Ox-PAPC and the activation of SREBP and Erk 1/2, which mediate this transcription. We provide evidence that the activation of VEGFR2 is rapid, sustained, and c-Src-dependent. Conclusions-These data point to a major role of VEGFR2 in endothelial regulation by oxidized phospholipids which accumulate in atherosclerotic lesions and apoptotic cells.

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