Knot homology via derived categories of coherent sheaves II, $\mathfrak{sl}_{m}$ case

Using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum knot polynomial. Our knot homology naturally satisfies the categorified MOY relations and is conjecturally isomorphic to Khovanov–Rozansky homology. Our construction is motivated by the geometric Satake correspondence and is related to Manolescu’s by homological mirror symmetry.     

On the colored Jones polynomial and the Kashaev invariant

We express the colored Jones polynomial as the inverse of the quantum determinant of a matrix with entries in the q-Weyl algebra of q-operators, evaluated at the trivial function (plus simple substitutions). The Kashaev invariant is proved to be equal to another special evaluation of the determinant. We also discuss the similarity between our determinant formula of the Kashaev invariant and the determinant formula of the hyperbolic volume of knot complements, hoping it will lead to a proof of the volume conjecture. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 57–78, 2005.     

On the Melvin–Morton–Rozansky conjecture

We prove a conjecture stated by Melvin and Morton (and elucidated further by Rozansky) saying that the Alexander–Conway polynomial of a knot can be read from some of the coefficients of the Jones polynomials of cables of that knot (i.e., coefficients of the “colored” Jones polynomial). We first reduce the problem to the level of weight systems using a general principle, which may be of some independent interest, and which sometimes allows to deduce equality of Vassiliev invariants from the equality of their weight systems. We then prove the conjecture combinatorially on the level of weight systems. Finally, we prove a generalization of the Melvin–Morton–Rozansky (MMR) conjecture to knot invariants coming from arbitrary semi-simple Lie algebras. As side benefits we discuss a relation between the Conway polynomial and immanants and a curious formula for the weight system of the colored Jones polynomial. Oblatum 28-VII-1994 & 5-XI-1995 & 22-XI-1995     

The sl3 Jones polynomial of the trefoil: A case study of q-holonomic sequences

The ${\mathfrak{sl}}_3$ colored Jones polynomial of the trefoil knot is a q-holonomic sequence of two variables with natural origin, namely quantum topology. The paper presents an explicit set of generators for the annihilator ideal of this q-holonomic sequence as a case study. On the one hand, our results are new and useful to quantum topology: this is the first example of a rank 2 Lie algebra computation concerning the colored Jones polynomial of a knot. On the other hand, this work illustrates the applicability and computational power of the employed computer algebra methods.     

Random Walks And The Colored Jones Function

It can be conjectured that the colored Jones function of a knot can be computed in terms of counting paths on the graph of a planar projection of a knot. On the combinatorial level, the colored Jones function can be replaced by its weight system. We give two curious formulas for the weight system of a colored Jones function: one in terms of the permanent of a matrix associated to a chord diagram, and another in terms of counting paths of intersecting chords. Electronic supplementary material to this article is available at and is accessible to authorized users. * S. G. was partially supported by an NSF and by an Israel-US BSF grant. † M. L. was partly supported by GAUK 158 grant and by the Project LN00A056 of the Czech Ministry of Education.     

On the asymptotic expansion of the colored Jones polynomial for torus knots

In the asymptotic expansion of the hyperbolic specification of the colored Jones polynomial of torus knots, we identify different geometric contributions, in particular Chern–Simons invariant and twisted Reidemeister torsion with coefficients in the adjoint representation. This work is supported in part by the Swiss National Science Foundation, the first author (J. Dubois) is also supported by the European Community with Marie Curie Intra–European Fellowship (MEIF–CT–2006–025316). While writing the paper, J. Dubois visited the CRM. He thanks the CRM for its hospitality.     

KZ equation,G-opers,quantum Drinfeld–Sokolov reduction,and quantum Cayley–Hamilton identity

The Lax operator of Gaudin-type models is a 1-form at the classical level. In virtue of the quantization scheme proposed by D. Talalaev, it is natural to treat the quantum Lax operator as a connection; this connection is a partcular case of the Knizhnik–Zamolodchikov connection. In this paper, we find a gauge trasformation that produces the “second normal form,” or the “Drinfeld–Sokolov” form. Moreover, the differential operator nurally corresponding to this form is given precisely by the quantum characteristic polynomial of the Lax operator (this operator is called the G-oper or Baxter operator). This observation allows us to relate solutions of the KZ and Baxter equations in an obvious way, and to prove that the immanent KZ equation has only meromorphic solutions. As a corollary, we obtain the quantum Cayley–Hamilton identity for Gaudin-type Lax operators (including the general case). The presented construction sheds a new light on the geometric Langlands correspondence. We also discuss the relation with the Harish-Chandra homomorphism. Bibliography: 19 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 246–259.     

Free abelian covers, short loops, stable length, and systolic inequalities

We explore the geometry of the Abel–Jacobi map f from a closed, orientable Riemannian manifold X to its Jacobi torus . Applying M. Gromov’s filling inequality to the typical fiber of f, we prove an interpolating inequality for two flavors of shortest length invariants of loops. The procedure works, provided the lift of the fiber is non-trivial in the homology of the maximal free abelian cover, , classified by f. We show that the finite-dimensionality of the rational homology of is a sufficient condition for the homological non-triviality of the fiber. When applied to nilmanifolds, our “fiberwise” inequality typically gives stronger information than the filling inequality for X itself. In dimension 3, we present a sufficient non-vanishing condition in terms of Massey products. This condition holds for certain manifolds that do not fiber over their Jacobi torus, such as 0-framed surgeries on suitable links. Our systolic inequality applies to surface bundles over the circle (provided the algebraic monodromy has 1-dimensional coinvariants), even though the Massey product invariant vanishes for some of these bundles. A. I. Suciu was supported by the National Science Foundation (grant DMS-0105342).     

A version of the volume conjecture

We propose a version of the volume conjecture that would relate a certain limit of the colored Jones polynomials of a knot to the volume function defined by a representation of the fundamental group of the knot complement to the special linear group of degree two over complex numbers. We also confirm the conjecture for the figure-eight knot and torus knots. This version is different from S. Gukov's because of a choice of polarization.     

Injectivity of real polynomial maps and Łojasiewicz exponents at infinity

In this paper we give a lower bound for the Łojasiewicz exponent at infinity of a special class of polynomial maps , s ≥ 1. As a consequence, we detect a class of polynomial maps that are global diffeomorphisms if their Jacobian determinant never vanishes. Work supported by DGICYT Grant BFM2003–02037/MATE.     

On rectifiable curves with <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-bounds on global curvature: self-avoidance,regularity, and minimizing knots

We discuss the analytic properties of curves γ whose global curvature function ρ _G [γ]⁻¹ is p-integrable. It turns out that the L ^p -norm is an appropriate model for a self-avoidance energy interpolating between “soft” knot energies in form of singular repulsive potentials and “hard” self-obstacles, such as a lower bound on the global radius of curvature introduced by Gonzalez and Maddocks. We show in particular that for all p > 1 finite -energy is necessary and sufficient for W ^2,p -regularity and embeddedness of the curve. Moreover, compactness and lower-semicontinuity theorems lead to the existence of -minimizing curves in given isotopy classes. There are obvious extensions to other variational problems for curves and nonlinearly elastic rods, where one can introduce a bound on to preclude self-intersections.     

Quantum Knizhnik-Zamolodchikov equation, totally symmetric self-complementary plane partitions, and alternating sign matrices

We present multiple-residue integral formulas for partial sums in the basis of link patterns of the polynomial solution of the level-1 quantum Knizhnik-Zamolodchikov equation at arbitrary values of the quantum parameter q. These formulas allow rewriting and generalizing a recent conjecture of Di Francesco connecting these sums to generating polynomials for weighted totally symmetric self-complementary plane partitions. We reduce the corresponding conjectures to a single integral identity, yet to be proved. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 3, pp. 387–408, March, 2008.     

The Extremal Truncated Moment Problem

For a degree 2n real d-dimensional multisequence to have a representing measure μ, it is necessary for the associated moment matrix to be positive semidefinite and for the algebraic variety associated to β, , to satisfy rank card as well as the following consistency condition: if a polynomial vanishes on , then . We prove that for the extremal case , positivity of and consistency are sufficient for the existence of a (unique, rank -atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness of . The first-named author’s research was partially supported by NSF Research Grants DMS-0099357 and DMS-0400741. The second-named author’s research was partially supported by NSF Research Grant DMS-0201430 and DMS-0457138.     

Weak Hopf Algebras Corresponding to Borcherds-Cartan Matrices

Let y be a generalized Kac-Moody algebra with an integral Borcherds-Cartan matrix. In this paper, we define a d-type weak quantum generalized Kac-Moody algebra wUq^d（y）, which is a weak Hopf algebra. We also study the highest weight module over the weak quantum algebra wUdq^d（y） and weak A-forms of wUq^d（y）.     

The colored Jones polynomial and the A-polynomial of Knots

We study relationships between the colored Jones polynomial and the A-polynomial of a knot. The AJ conjecture (of Garoufalidis) that relates the colored Jones polynomial and the A-polynomial is established for a large class of two-bridge knots, including all twist knots. We formulate a weaker conjecture and prove that it holds for all two-bridge knots. Along the way we also calculate the Kauffman bracket skein module of the complements of two-bridge knots. Some properties of the colored Jones polynomial are established.     

Tautness for riemannian foliations on non-compact manifolds

For a riemannian foliation on a closed manifold M, it is known that is taut (i.e. the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean curvature form (relatively to a suitable riemannian metric μ) is zero (cf. álvarez in Ann Global Anal Geom 10:179–194, 1992). In the transversally orientable case, tautness is equivalent to the non-vanishing of the top basic cohomology group , where (cf. Masa in Comment Math Helv 67:17–27, 1992). By the Poincaré Duality (cf. Kamber et and Tondeur in Astérisque 18:458–471, 1984) this last condition is equivalent to the non-vanishing of the basic twisted cohomology group , when M is oriented. When M is not compact, the tautness class is not even defined in general. In this work, we recover the previous study and results for a particular case of riemannian foliations on non compact manifolds: the regular part of a singular riemannian foliation on a compact manifold (CERF). J. I. Royo Prieto was partially supported by EHU06/05, by a PostGrant from the Basque Government and by the MCyT of the Spanish Government. R. Wolak was partially supported by the KBN grant 2PO3A 021 25.     

Jones多项式的零点

利用数论理论证明了纽结的Jones多项式仅有可能的有理根是O,而链环的Jones多项式仅有可能的有理根是0和-1.给出了作为Jones多项式根的所有可能单位根,以及所有可能的具有平凡Mahler测度的Jones多项式.最后指出了交叉数不超过11的纽结中,只有4_1,8_9,9_(42),K11n19的Jones多项式具有平凡的Mahler测度,从而回答了林晓松提出的关于Mahler测度的一个问题.     

On the Notion of the Bochner–Martinelli Integral for Domains with Rectifiable Boundary

In this paper we discuss the notion of the Bochner–Martinelli kernel for domains with rectifiable boundary in , by expressing the kernel in terms of the exterior normal due to Federer (see [17,18]). We shall use the above mentioned kernel in order to prove both Sokhotski–Plemelj and Plemelj–Privalov theorems for the corresponding Bochner–Martinelli integral, as well as a criterion of the holomorphic extendibility in terms of the representation with Bochner–Martinelli kernel of a continuous function of two complex variables. Explicit formula for the square of the Bochner–Martinelli integral is rediscovered for more general surfaces of integration extending the formula established first by Vasilevski and Shapiro in 1989. The proofs of all these facts are based on an intimate relation between holomorphic function theory of two complex variables and some version of quaternionic analysis. Submitted: September 6, 2006. Accepted: November 1, 2006.     

Cyclicity in the Dirichlet space

Let be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We give a new sufficient condition, not far from the known necessary condition, for a function f∈ to be cyclic, i.e. for {pf: p is a polynomial} to be dense in . The proof is based on the notion of Bergman–Smirnov exceptional set introduced by Hedenmalm and Shields. Our methods yield the first known examples of such sets that are uncountable. One of the principal ingredients of the proof is a new converse to the strong-type inequality for capacity.     

Chromatic polynomial, q-binomial counting and colored Jones function

We define a q-chromatic function and q-dichromate on graphs and compare it with existing graph functions. Then we study in more detail the class of general chordal graphs. This is partly motivated by the graph isomorphism problem. Finally we relate the q-chromatic function to the colored Jones function of knots. This leads to a curious expression of the colored Jones function of a knot diagram K as a chromatic operator applied to a power series whose coefficients are linear combinations of long chord diagrams. Chromatic operators are directly related to weight systems by the work of Chmutov, Duzhin, Lando and Noble, Welsh.