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

a combinatorial description of knotted surfaces and their isotopies abstract we discuss the diagrammatic theory of knot isotopies in dimension 4 we project a knotted surface to a three dimensional space and arrange the surface to have generic singularities upon further projection to a plane we examine the singularities in this plane as an isotopy is performed and give a finite set of local moves to the singular set that can be used to connect any two isotopic knottings we show how the notion of projections of isotopies can be used to give a combinatoric description of knotted surfaces that is sufficient for categorical applications in this description knotted surfaces are presented as sequences of words in symbols and there is a complete list of moves among such sequences that relate the symbolic representations of isotopic knotted surfaces

a link invariant from the symplectic geometry of nilpotent slices we define an invariant of oriented links in s 3 using the symplectic geometry of certain spaces which arise naturally in lie theory more specifically we present a knot as the closure of a braid which in turn we view as a loop in configuration space fix an affine subspacesm of the lie algebra sl2m c which is a transverse slice to the adjoint action at a nilpotent matrix with two equal jordan blocks the adjoint quotient map restricted to sm gives rise to a symplectic fibre bundle over configuration space an inductive argument constructs a distinguished lagrangian submani

knot floer homology detects fibred knots ozsvath and szabo conjectured that knot floer homology detects fibred knots in s 3 we will prove this conjecture for null homologous knots in arbitrary closed 3 manifolds namely if k is a knot in a closed 3 manifold y y k is irreducible and hat hfk y k is monic then k is fibred the proof relies on previous works due to gabai ozsvath szabo ghiggini and the author a corollary is that if a knot in s 3 admits a lens space surgery then the knot is fibred

concordance homomorphisms from knot floer homology abstract we modify the construction of knot floer homology to produce a one parameter family of homologies thfk for knots in s 3 these invariants can be used to give homomorphisms from the smooth concordance group c to z giving bounds on the four ball genus and the concordance genus of knots we give some applications of these homomorphisms

combinatorial cobordism maps in hat heegaard floer theory in a previous article sarkar and wang 15 gave a combinatorial description of the hat version of heegaard floer homology for three manifolds given a cobordism between two connected three manifolds there is an induced map between their heegaard floer homologies assume that the first homology group of each boundary component surjects onto the first homology group of the cobordism modulo torsion under this assumption we present a procedure for finding the rank of the induced heegaard floer map combinatorially in the hat version

knots links and 4 manifolds in this paper we investigate the relationship between isotopy classes of knots and links in s and the di eomorphism types of homeomorphic smooth 4 manifolds as a corollary of this initial investigation we begin to uncover the surprisingly rich structure of di eomorphism types of manifolds homeomorphic to the k3 surface in order to state our theorems we need to view the seiberg witten invariant of a smooth 4 manifold as a multivariable laurent polynomial to do this recall that the seiberg witten invariant of a smooth closed oriented 4 manifold x with b2 x 1 is an integer valued function which is de ned on the set of spinc structures over x cf w km ko1 t1 in case h1 x z has no 2 torsion which will be the situation in this paper there is a natural identi cation of the spinc structures of x with the characteristic elements of h2 x z in this case we view the seiberg witten invariant as

cuspidal curves and heegaard floer homology we give bounds on the gap functions of the singularities of a cuspidal plane curve of arbitrary genus generalising recent work of borodzik and livingston we apply these inequalities to unicuspida

monopoles and lens space surgeries monopole floer homology is used to prove that real projective three space cannot be obtained from dehn surgery on a nontrivial knot in the three sphere to obtain this result we use a surgery long exact sequence for monopole floer homology together with a nonvanishing theorem which shows that monopole floer homology detects the unknot in addition we apply these techniques to give information about knots which admit lens space surgeries and to exhibit families of three manifolds which do not admit taut foliations

equivalence of heegaard floer homology and embedded contact homology via open book decompositions we sketch the proof of the equivalence between the hat versions of heegaard floer homology and embedded contact homology ech the key point is to express these two floer homology theories in terms of an open book decomposition of the ambient manifold

linear groups with orders having certain large prime divisors in this paper we obtain a classification of those subgroups of the finite general linear group gld q with orders divisible by a primitive prime divisor of qe 1 for some in the course of the analysis we obtain new results on modular representations of finite almost simple groups in particular in the last section we obtain substantial extensions of the results of landazuri and seitz on small cross characteristic representations of some of the finite classical groups 1991 mathematics subject classification primary 20g40 secondary 20c20 20c33 20c34 20e99

an odd khovanov homotopy type abstract for each link l s 3 and every quantum grading j we construct a stable homotopy type x o j l whose cohomology recovers ozsvath rasmussen szabo s odd khovanov homology h i x o j l kh o i j l following a construction of lawson lipshitz sarkar of the even khovanov stable homotopy type furthermore the odd khovanov homotopy type carries a z 2 action whose fixed point set is a desuspension of the even khovanov homotopy type we also construct a potentially new even khovanov homotopy type with a z 2 action with fixed point set a desuspension of x o j l

heegaard floer homology of matsumoto s manifolds abstract we consider a homology sphere m n k 1 k 2 presented by two knots k 1 k 2 with linking number 1 and framing 0 n we call the manifold matsumoto s manifold we show that m n t 2 3 k 2 never bounds any contractible 4 manifold if n 2 k 2 holds we also give a formula of ozsvath szabo s invariant as the total sum of the euler numbers of the reduced filtration we compute the invariants of the twisted whitehead doubles of torus knots and correction terms of the branched covers of the whitehead doubles by using owens and strle s obstruction we show that the 12 twisted whitehead double of the 2 7 torus knot and the 20 twisted whitehead double of the 3 7 torus knot are not slice but the double branched covers bound rational homology 4 balls these are new examples having a gap between what a knot is slice and what a double branched cover bounds a rational homology 4 ball

existence results for translation nets ii abstract let d be a translation bruck net of order s and degree r with translation group g and let s q 1 q n be the prime power factorization of s we prove that r min q i 1 i 1 n moreover equality can be realized provided that g is the direct sum of elementary abelian groups similar results are obtained for translation s r nets and more generally for groups with large systems of large subgroups as a corollary we considerably strengthen a known result on symmetric translation nets by showing that any translation s s net has parameters of the form s p i p j for some prime p and also necessarily has elementary abelian translation group

heegaard floer correction terms and rational genus bounds given an element in the first homology of a rational homology 3 sphere y one can consider the minimal rational genus of all knots in this homology class this defines a function on h1 y z which was introduced by turaev as an analogue of thurston norm we will give a lower bound for this function using the correction terms in heegaard floer homology as a corollary we show that floer simple knots in l spaces are genus minimizers in their homology classes hence answer questions of turaev and rasmussen about genus minimizers in lens spaces

on the functoriality of khovanov floer theories abstract we introduce the notion of a khovanov floer theory we prove that every page after e 1 of the spectral sequence accompanying a khovanov floer theory is a link invariant and that an oriented link cobordism induces a map on each page which is an invariant of the cobordism up to smooth isotopy rel boundary we then prove that the spectral sequences relating khovanov homology to heegaard floer homology and singular instanton knot homology are induced by khovanov floer theories and are therefore functorial in the manner described above as had been conjectured for some time

exponential growth of colored homfly pt homology abstract we define reduced colored gl n link homologies and use deformation spectral sequences to characterize their dependence on color and rank we then define reduced colored homfly pt homologies and prove that they arise as large n limits of gl n homologies together these results allow proofs of many aspects of the physically conjectured structure of the family of type a link homologies in particular we verify a conjecture of gorsky gukov and stosic about the growth of colored homfly pt homology

link homology theories from symplectic geometry for each positive integer n khovanov and rozansky constructed an invariant of links in the form of a doubly graded cohomology theory whose euler characteristic is the link polynomial we use lagrangian floer cohomology on some suitable affine varieties to build a similar series of link invariants and we conjecture them to be equal to those of khovanov and rozansky after a collapse of the bigrading our work is a generalization of that of seidel and smith who treated the case n 2

holomorphic disks and three manifold invariants properties and applications in 27 we introduced floer homology theories hf y s hf y s hf y t hf y s and hf red y s associated to closed oriented three manifolds y equipped with a spiny structures s spin c y in the present paper we give calculations and study the properties of these invariants the calculations suggest a conjectured relationship with seiberg witten theory the properties include a relationship between the euler characteristics of hf and turaev s torsion a relationship with the minimal genus problem thurston norm and surgery exact sequences we also include some applications of these techniques to three manifold topology

interlacing eigenvalues and graphs abstract we give several old and some new applications of eigenvalue interlacing to matrices associated to graphs bounds are obtained for characteristic numbers of graphs such as the size of a maximal co clique the chromatic number the diameter and the bandwidth in terms of the eigenvalues of the standard adjacency matrix or the laplacian matrix we also deal with inequalities and regularity results concerning the structure of graphs and block designs

quantum gl1 1 and tangle floer homology abstract we identify the grothendieck group of the tangle floer dg algebra with a tensor product of certain u q gl 1 1 representations under this identification up to a scalar factor the map on the grothendieck group induced by the tangle floer dg bimodule associated to a tangle agrees with the reshetikhin turaev homomorphism for that tangle we also introduce dg bimodules which act on the grothendieck group as the generators e and f of u q gl 1 1

cobordisms of sutured manifolds and the functoriality of link floer homology abstract it has been a central open problem in heegaard floer theory whether cobordisms of links induce homomorphisms on the associated link floer homology groups we provide an affirmative answer by introducing a natural notion of cobordism between sutured manifolds and showing that such a cobordism induces a map on sutured floer homology this map is a common generalization of the hat version of the closed 3 manifold cobordism map in heegaard floer theory and the contact gluing map defined by honda kazez and matic we show that sutured floer homology together with the above cobordism maps forms a type of tqft in the sense of atiyah applied to the sutured manifold cobordism complementary to a decorated link cobordism our theory gives rise to the desired map on link floer homology hence link floer homology is a categorification of the multi variable alexander polynomial we outline an alternative definition of the contact gluing map using only the contact element and handle maps finally we show that a weinstein sutured manifold cobordism preserves the contact element

a combinatorial description of knot floer homology given a grid presentation of a knot or link k in the three sphere we describe a heegaard diagram for the knot complement in which the heegaard surface is a torus and all elementary domains are squares using this diagram we obtain a purely combinatorial description of the knot floer homology of k

some applications of gabai s internal hierarchy any haken 3 manifold possibly with boundary consisting of tori can be transformed into a surface s 1 by a series of splitting and regluing along incompressible surfaces this fact was proved by gabai as an application of his sutured manifold theory the first half of this paper provides a few technical details in the proof in the second half of this paper some applications of gabai s theorem to heegaard floer homology are given we refine the known results about the thurson norm and fibrations we also give some classification results for floer simple knots in manifolds with positive b 1

higher dimensional algebra iv 2 tangles just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions compact surfaces smoothly embedded in r4 can be described as certain 2 morphisms in the 2 category of 2 tangles in 4 dimensions using the work of carter rieger and saito we prove that this 2 category is the free semistrict braided monoidal 2 category with duals on one unframed self dual object by this universal property any unframed self dual object in a braided monoidal 2 category with duals determines an invariant of 2 tangles in 4 dimensions

tight contact structures on some small seifert fibered 3 manifolds we classify tight contact structures on the small seifert fibered 3 manifolds m 1 r1 r2 r3 with ri 0 1 q and r1 r2 the result is obtained by combining convex surface theory with computations of contact ozsvath szabo invariants we also show that some of the tight contact structures on the manifolds considered are nonfillable justifying the use of heegaard floer theory

a generalized cameron kantor theorem abstract in this paper we prove that the non trivial unipotent characters of psl n q are reducible over any proper subgroup with the only exception of the unipotent character of psl 4 2 of degree 14 this result can be viewed as a generalization of a theorem of cameron and kantor

on subdesigns of symmetric designs the proof of haemers and shrikhande uses an eigenvalue technique due to haemers 6 it is not clear whether or not 1 1 c holds provided that is a design with the parameters given above we will prove this in sect 2 by counting arguments which turn out to be much more delicate than one would expect at first this also leads to another inequality for the parameters of and which in general will not coincide with 1 1 b there are also a few further reasons to study this situation again to begin with eigenvalue techniques are less elementary than counting also a result such as theorem 1 1 should have a geometric interpretation in fact the second

combinatorial heegaard floer homology and nice heegaard diagrams abstract we consider a stabilized version of hf of a 3 manifold y i e the u 0 variant of heegaard floer homology for closed 3 manifolds we give a combinatorial algorithm for constructing this invariant starting from a heegaard decomposition for y and give a topological proof of its invariance properties

an invariant of tangle cobordisms we construct a new invariant of tangle cobordisms the invariant of a tangle is a complex of bimodules over certain rings well defined up to chain homotopy equivalence the invariant of a tangle cobordism is a homomorphism between complexes of bimodules assigned to boundaries of the cobordism

knot homology via derived categories of coherent sheaves i the mathfrak sl 2 case using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum mathfrak sl m 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

heegaard floer homology and fibred 3 manifolds given a closed 3 manifold y we show that the heegaard floer homology determines whether y fibres over the circle with a fibre of negative euler characteristic this is an analogue of an earlier result about knots proved by ghiggini and the author

atlas of finite groups maximal subgroups and ordinary characters for simple groups this atlas covers groups from the families of the classification of finite simple groups recently updated incorporating corrections

heegaard floer homology and concordance bounds on the thurston norm we prove that twisted correction terms in heegaard floer homology provide lower bounds on the thurston norm of certain cohomology classes determined by the strong concordance class of a 2 component link l in s 3 we then specialise this procedure to knots in s 2 times s 1 and obtain a lower bound on their geometric winding number furthermore we produce an obstruction for a knot in s 3 to have untwisting number 1 we then provide an infinite family of null homologous knots with increasing geometric winding number on which the bound is sharp

an ozsvath szabo floer homology invariant of knots in a contact manifold abstract using the knot floer homology filtration we define invariants associated to a knot in a three manifold possessing non vanishing floer co homology classes in the case of the ozsvath szabo contact invariant we obtain an invariant of knots in a contact three manifold this invariant provides an upper bound for the thurston bennequin plus rotation number of any legendrian realization of the knot we use it to demonstrate the first systematic construction of prime knots in contact manifolds other than s 3 with negative maximal thurston bennequin invariant perhaps more interesting our invariant provides a criterion for an open book to induce a tight contact structure a corollary is that if a manifold possesses contact structures with distinct non vanishing ozsvath szabo invariants then any fibered knot can realize the classical eliashberg bennequin bound in at most one of these contact structures

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

sutured floer homology fibrations and taut depth one foliations for an oriented irreducible 3 manifold m with non empty toroidal boundary we describe how sutured floer homology sfh can be used to determine all fibred classes in h 1 m furthermore we show that the sfh of a balanced sutured manifold m gamma detects which classes in h 1 m admit a taut depth one foliation such that the only compact leaves are the components of r gamma the latter had been proved earlier by the first author under the extra assumption that h 2 m 0 the main technical result is that we can obtain an extremal spin c structure s i e one that is in a corner of the support of sfh via a nice and taut sutured manifold decomposition even when h 2 m not equal 0 assuming the corresponding group sfh m gamma s has non trivial euler characteristic

holomorphic disks and topological invariants for closed three manifolds the aim of this article is to introduce certain topological invariants for closed oriented three manifolds y equipped with a spiny structure given a heegaard splitting of y u 0o u u 1 these theories are variants of the lagrangian floer homology for the g fold symmetric product of relative to certain totally real subspaces associated to u 0 and u 1

a categorification of the jones polynomial author s khovanov mikhail abstract we construct a bigraded cohomology theory of links whose euler characteristic is the jones polynomial

decompositions of difference sets abstract we characterize those symmetric designs with a singer group g which admit a quasi regular g invariant partition into strongly induced symmetric subdesigns in terms of the corresponding difference sets the set associated with the larger design can be decomposed into a difference set describing the small designs and a suitable relative difference set this generalizes the decomposition of the classical design with the complements of hyperplanes in pg m 1 q as blocks into sub designs arising from pg d 1 q whenever d divides m parametrically these geometrical examples provide the only known examples of the situation we are studying but there are many nonisomorphic examples with the same parameters namely the complements of the classical gmw designs and some generalizations we also discuss the possibilities for obtaining new difference sets in this way and point out a connection to the recent constructions of ionin for symmetric designs

on the transverse invariant for bindings of open books let t y be a transverse knot which is the binding of some open book t for the ambient contact manifold y in this paper we show that the transverse invariant t t fk y k defined in lossz09 is nonvanishing for such transverse knots this is true regardless of whether or not is tight we also prove a vanishing theorem for the invariants l and t as a corollary we show that if t is an open book with connected binding then the complement of t has no giroux torsion

on the heegaard floer homology of a surface times a circle we make a detailed study of the heegaard floer homology of the product of a closed surface sg of genus g with s1 we determine completely in the case which for g3 was previously unknown we show that in this case hf8 is closely related to the cohomology of the total space of a certain circle bundle over the jacobian torus of sg and furthermore that contains nontrivial 2 torsion whenever g3 and this is the first example known to the authors of torsion in coefficient heegaard floer homology our methods also give new information on the action of h1 sg s1 on when is nonzero

kauffman states bordered algebras and a bigraded knot invariant abstract we define and study a bigraded knot invariant whose euler characteristic is the alexander polynomial closely connected to knot floer homology the invariant is the homology of a chain complex whose generators correspond to kauffman states for a knot diagram the definition uses decompositions of knot diagrams to a collection of points on the line we associate a differential graded algebra to a partial knot diagram we associate modules over the algebra the knot invariant is obtained from these modules by an appropriate tensor product

link homologies and the refined topological vertex we establish a direct map between refined topological vertex and sl n homological invariants of the of hopf link which include khovanov rozansky homol ogy as a special case this relation provides an exact answer for homological invariants of the hopf link whose components are colored by arbitrary representations of sl n at present the mathematical formulation of such homological invariants is available only for the fundamental representation the khovanov rozansky theory and the rela tion with the refined topological vertex should be useful for categorizing quantum group invariants associated with other representations r1 r2 our result is a first direct veri fication of a series of conjectures which identifies link homologies with the hilbert space of bps states in the presence of branes where the physical interpretation of gradings is in terms of charges of the branes ending on lagrangian branes

a spanning tree cohomology theory for links abstract in their recent preprint baldwin ozsvath and szabo defined a twisted version with coefficients in a novikov ring of a spectral sequence previously defined by ozsvath and szabo from khovanov homology to heegaard floer homology of the branched double cover along a link in their preprint they give a combinatorial interpretation of the e 3 term of their spectral sequence the main purpose of the present paper is to prove directly that this e 3 term is a link invariant we also give some concrete examples of computation of the invariant

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

knot floer homology detects genus one fibred knots ozsv ath and szab o conjectured that knot floer homology detects fibred knots we propose a strategy to approach this conjecture based on contact topology and gabai s theory of sutured manifold decomposition we implement this strategy for genus one knots obtaining as a corollary that if rational surgery on a knot k gives the poincar e homology sphere sigma 2 3 5 then k is the left handed trefoil knot

khovanov homology open books and tight contact structures abstract we define the reduced khovanov homology of an open book s and identify a distinguished contact element in this group which may be used to establish the tightness or non fillability of contact structures compatible with s our construction generalizes the relationship between the reduced khovanov homology of a link and the heegaard floer homology of its branched double cover as an application we give combinatorial proofs of tightness for several contact structures which are not stein fillable lastly we investigate a comultiplication structure on the reduced khovanov homology of an open book which parallels the comultiplication on heegaard floer homology defined in baldwin 2008 4

khovanov seidel quiver algebras and ozsvath szabo s bordered theory abstract in the 2016 preprint kauffman states bordered algebras and a bigraded knot invariant ozsvath and szabo introduced a set of algebraic constructions in the spirit of bordered heegaard floer homology their constructions can be used to compute knot floer homology algebraically for knots in the 3 sphere in this paper we investigate a relationship between ozsvath szabo s bordered theory and the algebras and bimodules constructed by khovanov and seidel in quivers floer cohomology and braid group actions 2002 specifically we show that the khovanov seidel quiver algebras are isomorphic to quotients of idempotent truncations of some of ozsvath szabo s algebras furthermore we show that the dg bimodule associated to a braid generator by khovanov seidel with the right action restricted using the quotient map is homotopy equivalent to ozsvath szabo s da bimodule with the left action induced using the quotient map