a knot invariant via representation spaces a beautiful construction of a casson on the representation spaces corresponding to a heegard splitting of an oriented homology 3 sphere m gives rise to an integer invariant m of m this invariant generalizes the rohlin invariant and gives stricking corollaries in low dimension al topology defined as an intersection number of appropriate subspaces this invariant m can be roughly thought of as the number of conjugacy classes of irreducible representations of n m into su2 counted with signs a detailed discussion of this invariant can be found in an expose by s akbulut and j mccarthy 1 further works on casson s invariant include the generalizations by k walker as well as s boyer and a nicas to rational homology 3 spheres 14 3 and by s cappell r lee and e miller to representatio ns into sun 4 the works of c taubes 13 and a flore 6 interpret casson s invariant as the euler number of the instanton homology or flore homology of m in this paper analogous to casson s original construction we will define an intersection number of the representation spaces corresponding to a braid representative of a knot k in s this intersection number turns out to be an integer knot invariant see theorem 1 8 the representations of the knot group s k used in our construction seem to be mysterious they are representations of s3 k into su2 such that all meridians of k are represented by trace zero matrices call such a representation of the knot group a trace free representation then roughly speaking our knot invariant h k is the number of conjugacy classes of irreducible trace free representations of s3 k counted with signs our knot invariant h k can be computed via knot diagrams it turns out that our algorithm of computing h k by using the skein model is the same as the algorithm of computing sgn k given by j h conway 5

quadratic forms and the graph isomorphism problem abstract we relate the graph isomorphism problem to the classical problem of equivalence of integer quadratic forms we give some necessary number theoretic conditions that the isomorphic graphs have to satisfy

theory of acoustic eigenmodes in parabolic cylindrical enclosures we present an exact method for calculating eigenmodes and eigenfrequencies for an acoustic enclosure ae defined by two confocal parabolic cylinders and two plane surfaces using parabolic cylinder coordinates rigid wall boundary conditions are assumed however the model can easily be extended to include the more general case with mixed boundary conditions a discussion of eigenmode symmetry properties and mode frequency dependence on ae shape is given the latter problem is addressed by considering a series of symmetrical and asymmetrical aes having the same volume

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

natural mode analysis of hollow and annular elliptical cylindrical cavities analytical solutions for natural frequencies and mode shapes of elliptical cylindrical acoustic cavities were obtained by solving the homogeneous wave equation in elliptical cylindrical co ordinates both hollow cylinder and annular cylinder cases were considered the characteristic equation was obtained in terms of mathieu functions natural frequencies and mode shapes were calculated for cavities of the size used as a typical hermetic compressor shell or a small automotive muffler the results were compared with those of circular cylindrical cavities which have the same cross sectional area and volume the influence of the eccentricity on the general characteristics of cylindrically shaped acoustic devices is discussed

some np complete problems similar to graph isomorphism the graph isomorphism problem has so far resisted attempts at determining its complexity status it has not been shown to be np complete nor in p in this paper several altered or generalized versions of the isomorphism problem are presented and shown to be np complete one of these is the problem of determining whether a given graph has a fixed point free automorphism some speculation is made on the possible implications of these results on deciding the complexity status of isomorphism various classes and hierarchies of problems in np are discussed

a counterexample to a conjecture of friedland abstract this paper is concerned with a conjecture of friedland s friedland rational orthogonal similarity of rational symmetric matrices linear algebra appl 192 1993 109 114 a method for constructing counterexamples to the above conjecture is provided

on the geometry of the slice of trace free sl_2 mathbb c characters of a knot group let k be a knot in an integral homology 3 sphere with exterior e k and let b 2 denote the two fold branched cover of branched along k we construct a map from the slice of trace free rm sl 2 mathbb c characters of 1 e k to the rm sl 2 mathbb c character variety of 1 b 2 when this map is surjective it describes the slice as the two fold branched cover over the rm sl 2 mathbb c character variety of b 2 with branched locus given by the abelian characters whose preimage is precisely the set of metabelian characters we show that each metabelian character can be represented as the character of a binary dihedral representation of 1 e k the map is shown to be surjective for all 2 bridge knots and all pretzel knots of type p q r an extension of this framework to n fold branched covers is also described

on culler shalen seminorms and dehn filling if f is a finitely generated discrete group and g a complex algebraic lie group the g character variety of r is an affine algebraic variety whose points correspond to characters of representations of r with values in g marc culler and peter shalen developed the theory of sl2 c character varieties of finitely generated groups and applied their results to study the topology of 3 dimensional manifolds in the papers 6 7 8 consider the exterior m of a hyperbolic knot lying in a closed connected orientable 3 manifold the mostow rigidity theorem implies that the holonomy representation p iri m isom h3 psl2 c is unique up to conjugation and taking complex conjugates the orientability of m can be used to show 1 lifts to a representation p e hom ri m sl2 c whose character determines an essentially unique point of xp of x iri m the sl2 c character variety of irl m culler and shalen 8 proved that the component x0 of x x1ri m which contains x is a curve one of their major contributions was to show how x0 determines a norm on h1 wm r which encodes many topological properties of m in particular it provides information on the dehn fillings of m their construction may be applied to arbitrary curves in the sl2 c character variety of a connected compact orientable irreducible 3manifold whose boundary is a torus though in this generality one can only guarantee that it will define a seminorm the first half of this paper is devoted to the development of the general theory of culler shalen seminorms defined for curves of psl2 c characters by working over psl2 c we obtain a theory that is more generally applicable than its sl2 c counterpart while being only mildly more difficult to set up in the second half of this paper we apply the theory of culler shalen seminorms to study the dehn filling operation in particular we examine the relationship between fillings which yield manifolds having a positive dimen

plane curves associated to character varieties of 3 manifolds consider a compact 3 manifold m with boundary consisting of a single torus the papers cs1 cs2 cgls discuss the variety of characters of sl2 c representations of zl m and some of the ways in which the topological structure of m is reflected in the algebraic geometry of the character variety we will describe in this paper a certain affine algebraic curve dm which is naturally associated to the character variety of m a basis b 9 1 for the peripheral subgroup of m determines an embedding p8 of dm into c x c with coordinates l and m the closure in c 2 of ps dm is a plane algebraic curve and therefore is defined by a polynomial am n0 m that after certain normalizations is uniquely determined up to multiplication by constants the polynomial am b is effectively computable and is an invariant of the manifold m together with the choice of basis b the results in this paper describe how geometric properties of the character variety and hence topological properties of m are reflected by the polynomial am b in the case that m is the complement of a knot k in a homology 3 sphere we may take the basis b to consist of the longitude and meridian of k with the usual orientation conventions this basis is well defined modulo the involution which inverts both the longitude and meridian it will follow from the construction of dm that if the basis b is obtained from b by inverting both generators then the regular maps pb and pb have the same image thus ar am s am b is an invariant of the knot the polynomial am b displays in a striking way information about the incompressible surfaces in m this involves the newton polygon of am b

integral quadratic forms and graphs abstract the structure of an undirected graph is completely determined by a symmetric matrix its adjacency matrix with respect to an ordering of its vertices and that matrix can be used to define an integral quadratic form the main purpose of this paper is to raise this question what can quadratic forms tell us about graphs as an initial answer the theory of quadratic forms will be applied to the graph isomorphism problem the essential definitions and facts from the theory of quadratic forms will be sketched without proof

the principal rank characteristic sequence over various fields abstract given an n n matrix its principal rank characteristic sequence is a sequence of length n 1 of 0s and 1s where for k 0 1 n a 1 in the kth position indicates the existence of a principal submatrix of rank k and a 0 indicates the absence of such a submatrix the principal rank characteristic sequences for symmetric matrices over various fields are investigated with all such attainable sequences determined for all n over any field with characteristic 2 a complete list of attainable sequences for real symmetric matrices of order 7 is reported

orthogonal similarity and pairs of quadratic forms abstract the trace takes bilinear forms over a separable field extension to certain bilinear forms over the base field this paper shows how to compute the inverse of that process the construction implies that two known ways of classifying rational symmetric matrices under orthogonal similarity are actually the same

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

spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems in numerous science and engineering applications a partial differential equation has to be solved on some fairly regular domain that allows the use of the method of separation of variables in several orthogonal coordinate systems separation of variables applied to the helmholtz laplace or schrodinger equation leads to a multiparameter eigenvalue problem mep important cases include mathieu s system lame s system and a system of spheroidal wave functions although multiparameter approaches are exploited occasionally to solve such equations numerically meps remain less well known and the variety of available numerical methods is not wide the classical approach of discretizing the equations using standard finite differences leads to algebraic meps with large matrices which are difficult to solve efficiently the aim of this paper is to change this perspective we show that by combining spectral collocation methods and new efficient numerical methods for algebraic meps it is possible to solve such problems both very efficiently and accurately we improve on several previous results available in the literature and also present a matlab toolbox for solving a wide range of problems

rational orthogonal similarity of rational symmetric matrices abstract in this paper the hasse minkowski theorem is used over a number field to give necessary and sufficient conditions for the orthogonal similarity of two symmetric matrices over the field of rational numbers