knot concordance whitney towers and l 2 signatures we construct many examples of nonslice knots in 3 space that cannot be distinguished from slice knots by previously known invariants using whitney towers in place of embedded disks we define a geometric filtration of the 3 dimensional topological knot concordance group the bottom part of the filtration exhibits all classical concordance invariants including the cassongordon invariants as a first step we construct an infinite sequence of new obstructions that vanish on slice knots these take values in the l theory of skew fields associated to certain universal groups finally we use the dimension theory of von neumann algebras to define an l 2 signature and use this to detect the first unknown step in our obstruction theory

holomorphic disks and knot invariants abstract we define a floer homology invariant for knots in an oriented three manifold closely related to the heegaard floer homologies for three manifolds defined in an earlier paper we set up basic properties of these invariants including an euler characteristic calculation and a description of the behavior under connected sums then we establish a relationship with hf for surgeries along the knot applications include calculation of hf of three manifolds obtained by surgeries on some special knots in s 3 and also calculation of hf for certain simple three manifolds which fiber over the circle

an instanton invariant for 3 manifolds to an oriented closed 3 dimensional manifoldm withh1 m 0 we assign a 8 graded homology groupi m whose euler characteristic is twice casson s invariant the definition uses a construction on the space of instantons onm

sasaki einstein manifolds and volume minimisation we study a variational problem whose critical point determines the reeb vector field for a sasaki einstein manifold this extends our previous work on sasakian geometry by lifting the condition that the manifolds are toric we show that the einstein hilbert action restricted to a space of sasakian metrics on a link l in a calabi yau cone x is the volume functional which in fact is a function on the space of reeb vector fields we relate this function both to the duistermaat heckman formula and also to a limit of a certain equivariant index on x that counts holomorphic functions both formulae may be evaluated by localisation this leads to a general formula for the volume function in terms of topological fixed point data as a result we prove that the volume of a sasaki einstein manifold relative to that of the round sphere is always an algebraic number in complex dimension n 3 these results provide via ads cft the geometric counterpart of a maximisation in four dimensional superconformal field theories we also show that our variational problem dynamically sets to zero the futaki invariant of the transverse space the latter being an obstruction to the existence of a kahler einstein metric

the cr geometry of weighted extremal kahler and sasaki metrics we establish an equivalence between conformally einstein maxwell kahler 4 manifolds recently studied in apostolov et al j reine angew math 721 109 147 2016 apostolov and maschler j eur math soc 21 1319 1360 2019 futaki and ono j math soc jpn 70 1493 1521 2018 koca et al ann glob anal geom 50 29 46 2016 lebrun einstein maxwell equations extremal kahler metrics and seiberg witten theory in the many facets of geometry a tribute to nigel hitchin oxford university press oxford pp 17 33 2009 lebrun j geom phys 91 171 2015 and lebrun commun math phys 344 621 653 2016 and extremal kahler 4 manifolds in the sense of calabi extremal kahler metrics seminar on differential geometry annals of mathematics studies vol 102 pp 259 290 princeton university press princeton 1982 with nowhere vanishing scalar curvature the corresponding pairs of kahler metrics arise as transversal kahler structures of sasaki metrics compatible with the same cr structure and having commuting sasaki reeb vector fields this correspondence extends to higher dimensions using the notion of a weighted extremal kahler metric apostolov et al in levi kahler reduction of cr structures product of spheres and toric geometry arxiv 1708 05253 apostolov et al in weighted extremal kahler metrics and the einstein maxwell geometry of projective bundles arxiv 1808 02813 lahdili in j geom anal 29 542 568 2019 lahdili in int math res not arxiv 1710 00235 lahdili in proc lond math soc 119 1065 1114 2019 illuminating and uniting several explicit constructions in kahler and sasaki geometry it also leads to new existence and non existence results for extremal sasaki metrics suggesting a link between the notions of relative weighted k stability of a polarized variety introduced in apostolov et al weighted extremal kahler metrics and the einstein maxwell geometry of projective bundles arxiv 1808 02813 and lahdili proc lond math soc 119 1065 1114 2019 and relative k stability of the kahler cone corresponding to a sasaki polarization studied in boyer and van coevering math res lett 25 1 19 2018 and collins and szekelyhidi j differ geom 109 81 109 2018

on the quantum filtration of the khovanov rozansky cohomology abstract we prove the quantum filtration on the khovanov rozansky link cohomology h p with a general degree n 1 monic potential polynomial p x is invariant under reidemeister moves and construct a spectral sequence converging to h p that is invariant under reidemeister moves whose e 1 term is isomorphic to the khovanov rozansky sl n cohomology h n then we define a generalization of the rasmussen invariant and study some of its properties we also discuss relations between upper bounds of the self linking number of transversal links in standard contact s 3

density expansion for particle particle correlations in time dependent physical clusters we present a theory for the particle particle correlations in physical clusters for which bonding between particles is determined by a connectivity distance and a permanency time a generalized mayer density expansion for the cluster pair correlation function is found as well as an ornstein zernike like relation we can rely on this formalism to study clustering in realistic models by applying techniques of liquid state theory

the moduli space of fano manifolds with kahler ricci solitons abstract we construct a canonical hausdorff complex analytic moduli space of fano manifolds with kahler ricci solitons this enlarges the moduli space of fano manifolds with kahler einstein metrics we discover a moment map picture for kahler ricci solitons and give complex analytic charts on the topological space consisting of kahler ricci solitons by studying differential geometric aspects of this moment map some stacky words and arguments on gromov hausdorff convergence help to glue them together in the holomorphic manner

conformally einstein maxwell kahler metrics and structure of the automorphism group let m g be a compact kahler manifold and f a positive smooth function such that its hamiltonian vector field k j mathrm grad g f for the kahler form omega g is a holomorphic killing vector field we say that the pair g f is conformally einstein maxwell kahler metric if the conformal metric tilde g f 2 g has constant scalar curvature in this paper we prove a reductiveness result of the reduced lie algebra of holomorphic vector fields for conformally einstein maxwell kahler manifolds extending the lichnerowicz matsushima theorem for constant scalar curvature kahler manifolds more generally we consider extensions of calabi functional and extremal kahler metrics and prove an extension of calabi s theorem on the structure of the lie algebra of holomorphic vector fields for extremal kahler manifolds the proof uses a hessian formula for the calabi functional under the set up of donaldson fujiki picture

a slice genus lower bound from sl n khovanov rozansky homology we show that a generic perturbation of the doubly graded khovanov rozansky knot homology gives rise to a lower bound on the slice genus of a knot we prove a theorem about obtainable presentations of surfaces embedded in 4 space which we use to simplify significantly our algebraic computations

instantons concordance and whitehead doubling we use moduli spaces of instantons and chern simons invariants of flat connections to prove that the whitehead doubles of 2 2 n 1 torus knots are independent in the smooth knot concordance group that is they freely generate a subgroup of infinite rank call two knots in the 3 sphere concordant if they arise as the boundary of a smooth and properly embedded cylinder in the 3 sphere times an interval concordance is clearly an equivalence relation modulo this relation the set of knots forms an abelian group c with the role of addition played by connected sum and inverse given by considering the mirror image with reversed orientation this concordance group is a much studied object well motivated by its role as a gateway into the mysterious world of 4 dimensional topology indeed even in this relative situation of studying 3 dimensional manifolds in relation to the 4 dimensional manifolds they bound one can observe the distinction between the smooth and topological categories in dimension four moreover the group structure afforded by passing to concordance paves a clearer path through the often intractable field of knot theory our results focus on a particular satellite operation positive untwisted whitehead doubling and its effect on the concordance group see figure 1 and section 3 1 for a definition our motivation comes from the following conjecture to state it recall that a knot is slice if it is concordant to the unknot or equivalently if it bounds a smooth and properly embedded disk in the 4 ball

on conformally kahler einstein manifolds we prove that any compact complex surface with c1 0 admits an einstein metric which is conformally related to a kahler metric the key new ingredient is the existence of such a metric on the blow up cp2 2cp2 of the complex projective plane at two distinct points

absolutely graded floer homologies and intersection forms for four manifolds with boundary abstract in ozsvath and szabo holomorphic triangles and invariants for smooth four manifolds math sg 0110169 2001 we introduced absolute gradings on the three manifold invariants developed in ozsvath and szabo holomorphic disks and topological invariants for closed three manifolds math sg 0101206 ann of math 2001 to appear coupled with the surgery long exact sequences we obtain a number of three and four dimensional applications of this absolute grading including strengthenings of the complexity bounds derived in ozsvath and szabo holomorphic disks and three manifold invariants properties and applications math sg 0105202 ann of math 2001 to appear restrictions on knots whose surgeries give rise to lens spaces and calculations of hf for a variety of three manifolds moreover we show how the structure of hf constrains the exoticness of definite intersection forms for smooth four manifolds which bound a given three manifold in addition to these new applications the techniques also provide alternate proofs of donaldson s diagonalizability theorem and the thom conjecture for cp 2

khovanov homology and the slice genus we use lee s work on the khovanov homology to define a knot invariant s we show that s k is a concordance invariant and that it provides a lower bound for the smooth slice genus of k as a corollary we give a purely combinatorial proof of the milnor conjecture

braids transversal links and the khovanov rozansky theory we establish some inequalities for the khovanov rozansky cohomologies of braids these give new upper bounds of the self linking numbers of transversal links in standard contact s 3 which are sharper than the well known bound given by the homfly polynomial we also introduce a sequence of transversal link invariants and discuss some of their properties

instanton homology of seifert fibred homology three spheres for an oriented integral homology 3 sphere 2 a casson has introduced an integer invariant a 2 that is defined by using the space 2 of conjugacy classes of irreducible representations of 2 into su 2 see 1 this invariant a 2 can be computed from a surgery or heegard description of 2 and satisfies a 2 ju 2 mod 2 where i 2 is the kervaire milnor rochlin invariant of 2 this powerful new invariant was used to settle an outstanding problem in 3 manifold topology namely showing that if 2 is a homotopy 3 sphere then ju 2 o c taubes 17 utilizing gauge theoretic considerations has reinterpreted casson s invariant in terms of flat connections refining this approach a floer 13 has recently defined another invariant of 2 its instanton homology which takes the form of an abelian group 2 with a natural z8 grading that is an enhancement of a 2 in that

representations of knot groups in su 2 this paper is a study of the structure of the space r k of representations of classical knot groups into su 2 let r k equal the set of conjugacy classes of irreducible representations in gi we interpret the relations in a presentation of the knot group in terms of the geometry of su 2 using this technique we calculate r k for k equal to the torus knots twist knots and the whitehead link we also determine a formula for the number of binary dihedral representations of an arbitrary knot group we prove using techniques introduced by culler and shalen that if the dimension of r k is greater than 1 then the complement in sj of a tubular neighborhood of k contains closed nonboundary parallel incompressible surfaces we also show how for certain nonprime and doubled knots r k has dimension greater than one in gii we calculate the zariski tangent space tp r k for an arbitrary knot k at a reducible representation p using a technique due to weil we prove that for all but a finite number of the reducible representations dimtp r k 3 these nonexceptional representations possess neighborhoods in r k containing only reducible representations at the exceptional representations which correspond to real roots of the alexander polynomial dim tp r k 3 2k for a positive integer k in those examples analyzed in this paper these exceptional representations can be expressed as limits of arcs of irreducible representations we also give an interpretation of these extra tangent vectors as representations in the group of euclidean isometries of the plane 0 introduction this paper is a study of the space r k of representations of the fundamental group of the complement of a knot k into the lie group su 2 the major results are as follows 1 a calculation of the topological type of r k for k a torus knot a twist knot or the whitehead link 2 a determination of the zariski tangent space to r k at a reducible representation for an arbitrary knot k and 3 a proposition and some examples relating the dimension of r k to incompressible surfaces in the complement of k while the idea of representing 3 manifold groups in su 2 is a fairly recent one the representation of these groups in other groups has a substantial history we will not attempt to chronicle this history in detail but will comment here on received by the editors january 18 1989 and in revised form july 25 1989 1980 mathematics subject classification 1985 revision primary 57m05 57m25

gauge theory on asymptotically periodic 4 manifolds il existe une famille non denombrable de classes de diffeomorphismes des 4 varietes orientees homeomorphes a r 4