invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

cohomology for frobenius kernels of sl2 abstract let sl 2 r be the r th frobenius kernels of the group scheme sl 2 defined over an algebraically closed field of characteristic p 0 in this paper we give for r 1 a complete description of the cohomology groups for sl 2 r we also prove that the reduced cohomology ring h sl 2 r k red is cohen macaulay geometrically we show for each r 1 that the maximal ideal spectrum of the cohomology ring for sl 2 r is homeomorphic to the fiber product g b u r finally we adapt our calculations to obtain analogous results for the cohomology of higher frobenius lusztig kernels of quantized enveloping algebras of type sl 2

a polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed we prove that for any dimension d there exists a polynomial time algorithm for counting integral points in polyhedra in the d dimensional euclidean space previously such algorithms were known for dimensions d 1 2 3 and 4 only

second cohomology groups for frobenius kernels and related structures abstract let g be a simple simply connected affine algebraic group over an algebraically closed field k of characteristic p for an odd prime p let b be a borel subgroup of g and u be its unipotent radical in this paper we determine the second cohomology groups of b and its frobenius kernels for all simple b modules we also consider the standard induced modules obtained by inducing a simple b module to g and compute all second cohomology groups of the frobenius kernels of g for these induced modules also included is a calculation of the second ordinary lie algebra cohomology group of lie u with coefficients in k

the coloured jones function the invariantsjk k of a framed knotk coloured by the irreduciblesu 2 q module of dimensionk are studied as a function ofk by means of the universalr matrix it is shown that whenjk k is written as a power series inh withq eh the coefficient ofhd is an odd polynomial ink of degree at most 2d 1 this coefficient is a vassiliev invariant ofk in the second part of the paper it is shown that ask varies these invariants span ad dimensional subspace of the space of all vassiliev invariants of degreed for framed knots the analogous questions for unframed knots are also studied

cabled wilson loops in bf theories a generating function for cabled wilson loops in three dimensional bf theories is defined and a careful study of its behavior for vanishing cosmological constant is performed this allows an exhaustive description of the unframed knot invariants coming from the pure bf theory based on su 2 and in particular it proves a conjecture relating them to the alexander conway polynomial

extensions for frobenius kernels 1 1 let g be a simple algebraic group over an algebraically closed field k of characteristic p 0 for r 1 let gr be the rth frobenius kernel of g it is well known that the representations for g1 are equivalent to the restricted representations for lie g historically the cohomology for frobenius kernels has been best understood for large primes friedlander and parshall fp first computed the cohomology ring h g1 k for p 3 h 1 where h is the coxeter number of the underlying root system they proved that the cohomology ring can be identified with the coordinate algebra of the nullcone andersen and jantzen aj later verified this fact for p h furthermore they generalized this calculation by looking at h g1 h0 where h0 indb for p h their results had some restrictions on the type of root system involved kumar lauritzen and thomsen klt removed the restrictions on the root systems through the use of frobenius splittings the cohomology ring h g1 k modulo nilpotents can be identified in general with the coordinate algebra of the restricted nullcone n1 x lie g x p 0 for good primes nakano parshall and vella npv proved that this variety is irreducible and can be identified with the closure of some richardson orbit recently carlson lin nakano and parshall clnp have given an explicit description of n1 these recent results provide some indication that one can systematically study extensions of frobenius kernels for small primes by using general formulas which exhibit generic behavior for large primes

support varieties for infinitesimal group schemes the representation theory of a connected smooth affine group scheme over a field k of characteristic p 0 is faithfully captured by that of its family of frobenius kernels such frobenius kernels are examples of infinitesimal group schemes affine group schemes g whose coordinate hopf algebra k g is a finite dimensional local k algebra this paper presents a study of the cohomology algebra h g k of an arbitrary infinitesimal group scheme over k we provide a geometric determination of the cohomological support variety cgl spec h g k analogous to that given by d quillen for the cohomology of finite groups q we further study finite dimensional rational g modules m for arbitrary infinitesimal group schemes g over k in a manner initiated by j alperin and l evens a e and j carlson ci for finite groups we consider the variety igim c igi of the ideal im ker hev g k ext m m and provide a geometric description of this variety which is analogous to that given by g avrunin and l scott for finite dimensional modules for finite groups a s this paper is a continuation of our recent work establishing the finite generation of h g k f s and investigating the infinitesimal 1 parameter subgroups of g s f b earlier work of e friedlander and b parshall fp1 fp2 fp3 fp4 and j jantzen ji concerning the cohomology of restricted lie algebras are forerunners of the results presented here finite dimensional restricted lie algebras are in 1 1 correspondence with infinitesimal group schemes of height 1 our main theorems theorems 5 2 and 6 7 below when restricted to infinitesimal group schemes of height 1 significantly strengthen previously known cohomological information for restricted lie algebras an interesting aspect of our work is the extent to which infinitesimal 1 parameter subgroups v g a r g for infinitesimal group schemes g of height r play the role of elementary abelian p subgroups and their generalizations shifted subgroups for finite groups indeed much of our effort is dedicated to proving that cohomology classes are detected modulo nilpotence by such 1 parameter subgroups this is first done in 2 for unipotent infinitesimal group schemes using an induction argument made possible by a structure theorem presented in 1 this structure theorem is the analogue in our context of a theorem of j p serre characterizing elementary abelian p groups s

on nilpotent commuting varieties and cohomology of frobenius kernels abstract the paper studies the dimensions of irreducible components of commuting varieties of restricted nilpotent r tuples in a classical lie algebra g defined over an algebraically closed field k as applications we obtain some new results on the structure of the even cohomology ring of frobenius kernels g r for each r 1 where g is the simply connected simple algebraic group such that lie g g explicit calculations for rank two groups are also presented

three dimensional quantum gravity chern simons theory and the a polynomial we study three dimensional chern simons theory with complex gauge group sl 2 which has many interesting connections with three dimensional quantum gravity and geometry of hyperbolic 3 manifolds we show that in the presence of a single knotted wilson loop in an infinite dimensional representation of the gauge group the classical and quantum properties of such theory are described by an algebraic curve called the a polynomial of a knot using this approach we find some new and rather surprising relations between the a polynomial the colored jones polynomial and other invariants of hyperbolic 3 manifolds these relations generalize the volume conjecture and the melvin morton rozansky conjecture and suggest an intriguing connection between the sl 2 partition function and the colored jones polynomial

a unified witten reshetikhin turaev invariant for integral homology spheres we construct an invariant j m of integral homology spheres m with values in a completion widehat mathbb z q of the polynomial ring q such that the evaluation at each root of unity gives the the su 2 witten reshetikhin turaev invariant m of m at thus j m unifies all the su 2 witten reshetikhin turaev invariants of m it also follows that m as a function on behaves like an analytic function defined on the set of roots of unity

higher order terms in the melvin morton expansion of the colored jones polynomial we formulate a conjecture about the structure of upper lines in the expansion of the colored jones polynomial of a knot in powers of q 1 the melvin morton conjecture states that the bottom line in this expansion is equal to the inverse alexander polynomial of the knot we conjecture that the upper lines are rational functions whose denominators are powers of the alexander polynomial we prove this conjecture for torus knots and give experimental evidence that it is also true for other types of knots

the universalr matrix burau representation and the melvin morton expansion of the colored jones polynomial abstract p melvin and h morton 9 studied the expansion of the colored jones polynomial of a knot in powers of q 1 and color they conjectured an upper bound on the power of color versus the power of 1 they also conjectured that the bounding line in their expansion generated the inverse alexander conway polynomial these conjectures were proved by d bar natan and s garoufalidis 1 we have conjectured 12 that other lines in the melvin morton expansion are generated by rational functions with integer coefficients whose denominators are powers of the alexander conway polynomial here we prove this conjecture by using the r matrix formula for the colored jones polynomial and presenting the universal r matrix as a perturbed burau matrix

variational analysis of non lipschitz spectral functions we consider spectral functions f where f is any permutation invariant mapping from cn to r and is the eigenvalue map from the set of n n complex matrices to cn ordering the eigenvalues lexicographically for example if f is the function maximum real part then f is the spectral abscissa while if f is maximum modulus then f is the spectral radius both these spectral functions are continuous but they are neither convex nor lipschitz for our analysis we use the notion of subgradient extensively analyzed in variational analysis r t rockafellar and r j b wets springer 1998 we show that a necessary condition for y to be a subgradient of an eigenvalue function f at x is that y commutes with x we also give a number of other necessary conditions for y based on the schur form and the jordan form of x in the case of the spectral abscissa we refine these conditions and we precisely identify the case where subdifferential regularity holds we conclude by introducing the notion of a semistable program maximize a linear function on the set of square matrices subject to linear equality constraints together with the constraint that the real parts of the eigenvalues of the solution matrix are non positive semistable programming is a nonconvex generalization of semidefinite programming using our analysis we derive a necessary condition for a local maximizer of a semistable program and we give a generalization of the complementarity condition familiar from semidefinite programming

infinitesimal fixed points in modules with good filtration let g be the lie algebra of the connected semisimple algebraic group g defined over an algebraically closed field k of characteristic p p 0 if m is a g module with good filtration d we consider the module m g of fixed points under g as a module for the image of g under the frobenius homomorphism g g this module is denoted m g 1 cf j and it has been conjectured that it also has a good filtration d we will give an example to show that this is too optimistic in the other direction we prove something stronger in rank 1 namely we show that if b is a borel subgroup in sl2 or psl2 and m is a b module with relative schubert filtration cf vdk then so is mb 1 some preliminary work was done at the university of virginia where i much enjoyed the hospitality of brian parshall and leonard scott

a non commutative formula for the colored jones function the colored jones function of a knot is a sequence of laurent polynomials that encodes the jones polynomial of a knot and its parallels it has been understood in terms of representations of quantum groups and witten gave an intrinsic quantum field theory interpretation of the colored jones function as the expectation value of wilson loops of a 3 dimensional gauge theory the chern simons theory we present the colored jones function as an evaluation of the inverse of a non commutative fermionic partition function this result is in the form familiar in quantum field theory namely the inverse of a generalized determinant our formula also reveals a direct relation between the alexander polynomial and the colored jones function of a knot and immediately implies the extensively studied melvin morton rozansky conjecture first proved by bar natan and the first author about 10 years ago our results complement recent work of huynh and le who also give a non commutative formulae for the colored jones function of a knot starting from a non commutative formula for the r matrix of the quantum group u q mathfrak sl 2 see huynh and le in math gt 0503296

a contribution of the trivial connection to the jones polynomial and witten s invariant of 3d manifolds ii we extend the results of our previous paper 1 from knots to links by using a formula for the jones polynomial of a link derived recently by n reshetikhin we establish a relation between the parameters of this formula and the multivariable alexander polynomial this relation is illustrated by an example of a torus link we check that our expression for the alexander polynomial satisfies some of its basic properties finally we derive a link surgery formula for the loop corrections to the trivial connection contribution to witten s invariant of rational homology spheres

extensions for finite chevalley groups iii rational and generic cohomology abstract let g be a connected reductive algebraic group and b be a borel subgroup defined over an algebraically closed field of characteristic p 0 in this paper the authors study the existence of generic g cohomology and its stability with rational g cohomology groups via the use of methods from the authors earlier work new results on the vanishing of g and b cohomology groups are presented furthermore vanishing ranges for the associated finite group cohomology of g f q are established which generalize earlier work of hiller in addition to stability ranges for generic cohomology which improve on seminal work of cline parshall scott and van der kallen

infinitesimal 1 parameter subgroups and cohomology this is the first of two papers in which we determine the spectrum of the cohomology algebra of infinitesimal group schemes over a field k of characteristic p 0 whereas sfb is concerned with detection of cohomology classes the present paper introduces the graded algebra k vr g of functions on the scheme of infinitesimal 1 parameter subgroups of height r on an affine group scheme g and demonstrates that this algebra is essentially a retract of h g k provided that g is an infinitesimal group scheme of height kr this work is a continuation of f s in which the existence of certain universal extension classes was established thereby enabling the proof of finite generation of h g k for any finite group scheme g over k the role of the scheme of infinitesimal 1 parameter subgroups of g was foreshadowed in f p where h g 1 k was shown to be isomorphic to the coordinate algebra of the scheme of p nilpotent elements of g lie g for g a smooth reductive group g 1 the first frobenius kernel of g and p char k sufficiently large indeed p nilpotent elements of g correspond precisely to infinitesimal 1 parameter subgroups of g 1 much of our effort in this present paper involves the analysis of the restriction of the universal extension classes to infinitesimal 1 parameter subgroups in 1 we construct the affine scheme vr g of homomorphisms from ga r to g which we call the scheme of infinitesimal 1 parameter subgroups of height r in g in the special case r 1 this is the scheme of p nilpotent elements of the p restricted lie algebra lie g for various classical groups g vr g is the scheme of r tuples of p nilpotent pairwise commuting elements of lie g more generally an embedding g c gln determines a closed embedding of vr g into the scheme of r tuples of p nilpotent pairwise commuting elements of gl1 lie gln the relationship between k vr g the coordinate algebra of vr g and h g k is introduced in theorem 1 14 a natural homomorphism of graded k algebras

the colored jones polynomials and the simplicial volume of a knot we show that the set of colored jones polynomials and the set of generalized alexander polynomials defined by akutsu deguchi and ohtsuki intersect non trivially moreover it is shown that the intersection is at least includes the set of kashaev s quantum dilogarithm invariants for links therefore kashaev s conjecture can be restated as follows the colored jones polynomials determine the hyperbolic volume for a hyperbolic knot modifying this we propose a stronger conjecture the colored jones polynomials determine the simplicial volume for any knot if our conjecture is true then we can prove that a knot is trivial if and only if all of its vassiliev invariants are trivial in 13 r m kashaev defined a family of complex valued link invariants indexed by integers n 2 using the quantum dilogarithm later he calculated the asymptotic behavior of his invariant and observed that for the three simplest hyperbolic knots it grows as exp vol k n 2 when n goes to the infinity where vol k is the hyperbolic volume of the complement of a knotk 14 this amazing result and his conjecture that the same also holds for any hyperbolic knot have been almost ignored by mathematicians since his definition of the invariant is too complicated though it uses only elementary tools the aim of this paper is to reveal his mysterious definition and to show that his invariant is nothing but a specialization of the colored jones polynomial the colored jones polynomial is defined for colored links each component is decorated with an irreducible representation of the lie algebra sl 2 c the original jones polynomial corresponds to the case that all the colors are identical to the 2 dimensional fundamental representation we show that kashaev s invariant with parameter n coincides with the colored jones polynomial in a certain normalization with every color the n dimensional representation evaluated at the primitive n th root of unity we have to normalize the colored jones polynomial so that the value for the trivial knot is one for otherwise it always vanishes on the other hand there are other colored polynomial invariants the generalized multivariable alexander polynomial defined by y akutsu t deguchi and t ohtsuki 1 they used the same lie algebra sl 2 c but a different hierarchy of representations their invariants are parameterized by c 1 parameters an integer date february 1 2008 1991 mathematics subject classification 57m25 57m50 17b37 81r50

knot polynomials and vassiliev s invariants a fundamental relationship is established between jones knot invariants and vassiliev s knot invariants since vassiliev s knot invariants have a firm grounding in classical topology one obtains as a result a first step in understanding the jones polynomial by topological methods

a unified quantum so 3 invariant for rational homology 3 spheres given a rational homology 3 sphere m with h1 m z b and a link l inside m colored by odd numbers we construct a unified invariant im l belonging to a modification of the habiro ring where b is inverted our unified invariant dominates the whole set of the so 3 witten reshetikhin turaev invariants of the pair m l i fb 1a ndl im coincides with habiro s invariant of integral homology 3 spheres for b 1 the unified in variant defined by the third author is determined by im important applica tions are the new ohtsuki series perturbative expansions of im dominating quantum so 3 invariants at roots of unity whose order is not a power of a prime these series are not known to be determined by the lmo invariant

abelian bf theories and knot invariants in the context of the batalin vilkovisky formalism a new observable for the abelian bf theory is proposed whose vacuum expectation value is related to the alexander conway polynomial the three dimensional case is analyzed explicitly and it is proved to be anomaly free moreover at the second order in perturbation theory a new formula for the second coefficient of the alexander conway polynomial is obtained an account on the higher dimensional generalizations is also given