algebraic convergence of schottky groups a discrete faithful representation of the free group on g generators f g into isom h 3 is said to be a schottky group if h 3 d is homeomorphic to a handlebody h g where d is the domain of discontinuity for s action on the sphere at infinity for h 3 schottky space s g the space of all schottky groups is parameterized by the quotient of the teichmuller space f s g of the closed surface of genus g by mod 0 h g where mod 0 h g is the group of isotopy classes of homeomorphisms of s g which extend to homeomorphisms of h g which are homotopic to the identity masur exhibited a domain o h g of discontinuity for mod 0 h g s action on pl s g the space of projective measured laminations on s g so b h g o h g mod 0 h g may be appended to s g as a boundary

on the minimum ropelength of knots and links the ropelength of a knot is the quotient of its length by its thickness the radius of the largest embedded normal tube around the knot we prove existence and regularity for ropelength minimizers in any knot or link type these are c 1 1 curves but need not be smoother we improve the lower bound for the ropelength of a nontrivial knot and establish new ropelength bounds for small knots and links including some which are sharp

local connectivity kleinian groups and geodesics on the blowup of the torus let n h 3 be a hyperbolic 3 manifold with free fundamental group 1 n h a bi such that a b is parabolic we show that the limit setof n is always locally connected more precisely letbe a compact surface of genus 1 with a single boundary component equipped with the fuchsian action of 1 on the circle s 1 1 we show that for any homotopy equivalence f n there is a natural continuous map

hard variants of stable marriage the stable marriage problem and its many variants have been widely studied in the literature gusfield and irving the stable marriage problem structure and algorithms mit press cambridge ma 1989 roth and sotomayor two sided matching a study in game theoretic modeling and analysis econometric society monographs vol 18 cambridge university press cambridge 1990 knuth stable marriage and its relation to other combinatorial problems crm proceedings and lecture notes vol 10 american mathematical society providence ri 1997 partly because of the inherent appeal of the problem partly because of the elegance of the associated structures and algorithms and partly because of important practical applications such as the national resident matching program roth j political economy 92 6 1984 991 and similar large scale matching schemes here we present the first comprehensive study of variants of the problem in which the preference lists of the participants are not necessarily complete and not necessarily totally ordered we show that under surprisingly restrictive assumptions a number of these variants are hard and hard to approximate the key observation is that in contrast to the case where preference lists are complete or strictly ordered or both a given problem instance may admit stable matchings of different sizes in this setting examples of problems that are hard are finding a stable matching of maximum or minimum size determining whether a given pair is stableeven if the indifference takes the form of ties on one side only the ties are at the tails of lists there is at most one tie per list and each tie is of length 2 and finding or approximating both an egalitarian and a minimum regret stable matching however we give a 2 approximation algorithm for the problems of finding a stable matching of maximum or minimum size we also discuss the significant implications of our results for practical matching schemes

quasi fuchsian seifert surfaces let k subset s 3 be a non fibered knot with hyperbolic complement given a seifert surface of minimal genus for k we prove that it corresponds to a quasi fuchsian group by showing it has no accidental parabolics in particular this shows that any lift of the seifert surface to the universal cover is a quasi disk and its limit set is a quasi circle in the sphere at infinity

geodesic currents and length compactness for automorphisms of free groups we prove a compactness theorem for automorphisms of free groups namely we show that the set of automorphisms keeping the length of the uniform current bounded is compact up to conjugation this implies that the spectrum of the length of the images of the uniform current is discrete proving a conjecture of i kapovich

the extended bethe ansatz for infinite s 1 2 quantum spin chains with non nearest neighbor interaction it is shown that the description of the states of infinites 1 2 interacting spin systems with the hamiltonian open image in new window can be performed by studying the hyperbolic calogero sutherland eigenvalue problem the construction of multimagnon wave functions in eachn magnon sector is based on solutions of the set of linear algebraic equations which determine also the structure of zonal spherical functions on symmetric spacesxn sl n h sp n of negative curvature the usual bethe ansatz for the xxx heisenberg model corresponds to asymptotic forms of these wave functions at small values ofa 1 or large distances between spins turned over the ferromagnetic ground state

approximability results for stable marriage problems with ties we consider instances of the classical stable marriage problem in which persons may include ties in their preference lists we show that in such a setting strong lower bounds hold for the approximability of each of the problems of finding an egalitarian minimum regret and sex equal stable matching we also consider stable marriage instances in which persons may express unacceptable partners in addition to ties in this setting we prove that there are constants such that each of the problems of approximating a maximum and minimum cardinality stable matching within factors of respectively is np hard under strong restrictions we also give an approximation algorithm for both problems that has a performance guarantee expressible in terms of the number of lists with ties this significantly improves on the best known previous performance guarantee for the case that the ties are sparse our results have applications to large scale centralized matching schemes

on the connection between the one dimensional s 1 2 heisenberg chain and haldane shastry model extra integrals of motion and the lax representation are found for interacting spin systems with the hamiltonianh j 2 j k 1 j k n p j k j k where one of the periods of the weierstrassp function is equal ton the heisenberg and haldane shastry chains appear as limiting cases of these systems at some values of the second period the simplest eigenvectors and eigenvalues ofh corresponding to the scattering of two spin waves are presented explicitly for these finite dimensional systems and for their infinite dimensional version

marked length spectral rigidity for flat metrics in this paper we prove that the space of at metrics nonpos itively curved euclidean cone metrics on a closed oriented surface is marked length spectrally rigid in other words two at metrics assigning the same lengths to all closed curves dier by an isometry isotopic to the identity

on the unknotting number of minimal diagrams answering negatively a question of bleiler we give examples of knots where the difference between minimal and maximal unknotting number of minimal crossing number diagrams grows beyond any extent

non realizability and ending laminations proof of the density conjecture we give a complete proof of the bers sullivan thurston density conjecture in the light of the ending lamination theorem it suffices to prove that any collection of possible ending invariants is realized by some algebraic limit of geometrically finite hyperbolic manifolds the ending invariants are either riemann surfaces or filling laminations supporting masur domain measured laminations and satisfying some mild additional conditions with any set of ending invariants we associate a sequence of geometrically finite hyperbolic manifolds and prove that this sequence has a convergent subsequence we derive the necessary compactness theorem combining the rips machine with non existence results for certain small actions on real trees of free products of surface groups and free groups we prove then that the obtained algebraic limit has the desired conformal boundaries and the property that none of the filling laminations is realized by a pleated surface in order to be able to apply the ending lamination theorem we have to prove that this algebraic limit has the desired topological type and that these non realized laminations are ending laminations that this is the case is the main novel technical result of this paper loosely speaking we show that a filling masur domain lamination which is not realized is an ending lamination

laminations mesurees de plissage des varietes hyperboliques de dimension 3 for a hyperbolic metric on a 3 dimensional manifold the boundary of its convex core is a surface which is almost everywhere totally geodesic but which is bent along a family of disjoint geodesics the locus and intensity of this bending is described by a measured geodesic lamination which is a topological object we consider two problems the topological characterization of those measured geodesic laminations which can occur as bending measured laminations of hyperbolic metrics and the uniqueness problem which asks whether a hyperbolic metric is uniquely determined by its bending measured lamination

the volume of hyperbolic alternating link complements if a hyperbolic link has a prime alternating diagram d then we show that the link complement s volume can be estimated directly from d we define a very elementary invariant of the diagram d its twist number t d and show that the volume lies between v 3 t d 2 2 and v 3 10t d 10 where v 3 is the volume of a regular hyperbolic ideal 3 simplex as a consequence the set of all hyperbolic alternating and augmented alternating link complements is a closed subset of the space of all complete finite volume hyperbolic 3 manifolds in the geometric topology

on invariants of graphs with applications to knot theory to each weighted graph two invariants a polynomial p x y z and the signature are defined the various partial degress of p x y z and are expressed in terms of maximal spanning graphs of furthermore one unexpected property of tutte s dichromate is proved these results are applied to knots or links in s 3

foliations with good geometry the goal of this article is to show that there is a large class of closed hyperbolic 3 manifolds admitting codimension one foliations with good large scale geometric properties we obtain results in two directions first there is an internal result a possibly singular foliation in a manifold is quasi isometric if when lifted to the universal cover distance along leaves is efficienit up to a bounded multiplicative distortion in measuring distance in the universal cover this means that leaves reflect very well the geometry in the large of the universal cover and are geometrically tight this is the best geometric behavior we previously proved that nonsingular codimension one foliations in closed hyperbolic 3 manifolds can never be quasi isometric in this article we produce a large class of singular quasi isometric codimension one foliations in closed hyperbolic 3 manifolds the foliations are stable and unstable foliations of pseudo anosov flows our second result is an external result relating nonsingular foliations in hyperbolic 3 manifolds with their limit sets in the universal cover that is showing that leaves in the universal cover have good asymptotic behavior let g be a reebless finite depth foliation in a closed hyperbolic 3 manifold then g is not quasi isometric but suppose that g is transverse to a quasigeodesic pseudo anosov flow with quasi isometric stable and unstable foliations which are given by the internal result we then show that the lifts of leaves of g to the universal cover extend continuously to the sphere at infinity and we also produce infinitely many examples satisfying the hypothesis the main tools used to prove these results are first a link between geometric properties of stable unstable foliations of pseudo anosov flows and the topology of these foliations in the tuniversal cover and second a topological theory of the joint structure of the pseudo anosov foliations in the universal cover reebless codimension one foliations are extremely useful for understanding the topology of 3 manifolds for instance they imply that the manifold is irreducible ro its universal cover is homeomorphic to r3 pa leaves are 7rl injective no and transversals are never null homotopic no hence they reflect topological properties of the manifold as for which manifolds have reebless foliations gabai gal ga2 ga3 proved that any compact oriented irreducible 3 manifold with nonzero first betti number has many reebless finite depth foliations roughly a

hausdorff dimension and dendritic limit sets let be a singly degenerate closed surface group acting properly discontinuously on hyperbolic 3 space h3 such that h3 has positive injectivity radius it is known that the limit set is a dendrite of hausdorff dimension 2 we show that the cut point set of the limit set has hausdorff dimension strictly less than 2

multiple bumping of components of deformation spaces of hyperbolic 3 manifolds let m be a hyperbolizable 3 manifold with nonempty incompressible boundary of negative euler characteristic suppose that b 1 b k is a collection of components of the interior of the space of complete marked hyperbolic 3 manifolds homotopy equivalent to m such that for any i j inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i we prove that there is a geometrically finite hyperbolic structure on int m which is in the closure of each b i moreover we show that this structure can be constructed so as to admit quasiconformal deformations which also lie in the closure of every b i

iteration of mapping classes and limits of hyperbolic 3 manifolds let mod s be an element of the mapping class group of a surface s we classify algebraic and geometric limits of sequences q i x y i 1 of quasi fuchsian hyperbolic 3 manifolds ranging in a bers slice when has infinite order with finite order restrictions there is an essential subsurface d s so that the geometric limits have homeomorphism type s d 0 typically has pseudo anosov restrictions and d has components with negative euler characteristic these components correspond to new asymptotically periodic simply degenerate ends of the geometric limit we show there is an s 1 depending on and bounded in terms of s so that q si x y i 1 converges algebraically and geometrically and we give explicit quasi isometric models for the limits

on continuous extensions of grafting maps the definition of the grafting operation for quasifuchsian groups is extended by bromberg preprint to all b groups in this paper we show that the extended grafting maps behave as continuous maps for every sequence which converges standardly to a boundary group although the maps are not continuous in general as a consequence of this result we extend goldman s grafting theorem for quasifuchsian groups to all boundary b groups

projective structures with degenerate holonomy and the bers density conjecture we prove the bers density conjecture for singly degenerate kleinian surface groups without parabolics

ubiquity of geometric finiteness in boundaries of deformation spaces of hyperbolic 3 manifolds we show that geometrically finite kleinian groups are dense in the boundary of the quasiconformal deformation space of any geometrically finite kleinian group

bennequin s inequality and the positivity of the signature we use an algorithm for special diagrams to prove a bennequin type inequality tor the signature of an arbitrary link diagram related to its murasugi sum decomposition we apply this inequality to show that the signature of a non trivial positive 3 braid knot is greater than its genus and that the signature of a positive braid link is minorated by an increasing function of its negated euler characteristic the latter property is conjectured to extend to positive links

hyperbolic knots spanning accidental seifert surfaces of arbitrarily high genus a method for constructing hyperbolic knots each of which bounds accidental incompressible seifert surfaces of arbitrarily high genus is given

divergence free fields energy and asymptotic crossing number in more than one context the equations of classical physics lead to a closed 2 form co on 3 dimensional space which is a conserved quantity in the sense that it pulls back under time evolution in the physics literature co is written as a divergence free vector field w appendix a describes two settings in which a divergence free w is conserved or frozen one is the magnetic field in ideal magneto hydrodynamics and the other is curl velocity in incompressible inviscid fluid flow if ht denotes the transport diffeomorphisms of a moving fluid in which the field is imbedded with ho id wt denotes the field in the fluid at time t and x0 is the space variable at time t 0 then

connectivity of the space of ending laminations we prove that for any closed surface of genus at least four and any punctured surface of genus at least two the space of ending laminations is connected a theorem of e klarreich 28 theorem 1 3 implies that this space is homeomorphic to the gromov boundary of the complex of curves it follows that the boundary of the complex of curves is connected in these cases answering the conjecture of p storm other applications include the rigidity of the complex of curves and connectivity of spaces of degenerate kleinian groups

free vs locally free kleinian groups we prove that kleinian groups whose limit sets are cantor sets of hausdorff dimension 0 examples of non free purely hyperbolic kleinian groups whose limit set is a cantor set of hausdorff dimension 1 epsilon

hitchin characters and geodesic laminations for a closed surface s the hitchin component hitn s is a preferred component of the character variety consisting of group homomorphisms from the fundamental group 1 s to the lie group psln r we construct a parametrization of the hitchin component that is well adapted to a geodesic laminationon the surface this is a natural extension of thurston s parametrization of the teichmuller space t s by shear coordinates associated to corresponding to the case n 2 however significantly new ideas are needed in this higher dimensional case the article concludes with a few applications

statistical regularities of self intersection counts for geodesics on negatively curved surfaces let s be a compact surface with constant negative curvature 1 from among all closed geodesics on of length t choose one at random and letnt be the number of its self intersections we prove that for a certain constant 0 the random variable nt t t has a limit distribution as t we conjecture that for surfaces of variable negative curvature the order of magnitude of typical variations is t rather than t we also prove analogous results for generic geodesics that is geodesics whose initial tangent vectors are chosen randomly according to normalized liouville measure date november 10 2011 1991 mathematics subject classification primary 57m05 secondary 53c22 37d40

the weil petersson metric and volumes of 3 dimensional hyperbolic convex cores we present a coarse interpretation of the weil petersson distance dwp x y between two finite area hyperbolic riemann surfaces x and y using a graph of pants decompositions introduced by hatcher and thurston the combinatorics of the pants graph reveal a connection between riemann surfaces and hyperbolic 3 manifolds conjectured by thurston the volume of the convex core of the quasi fuchsian manifold q x y with x and y in its conformal boundary is comparable to the weil petersson distance dwp x y in applications we relate the weil petersson distance to the hausdorff dimen sion of the limit set and the lowest eigenvalue of the laplacian for q x y and give a new finiteness criterion for geometric limits mathematics department university of chicago 5734 s university ave chicago il 60637 e mail address brock math uchicago edu

crosscap numbers and the jones polynomial abstract we give sharp two sided linear bounds of the crosscap number non orientable genus of alternating links in terms of their jones polynomial our estimates are often exact and we use them to calculate the crosscap numbers for several infinite families of alternating links and for several alternating knots with up to twelve crossings we also discuss generalizations of our results for classes of non alternating links

an efficient algorithm for the optimal stable marriage in an instance of size n of the stable marriage problem each of n men and n women ranks the members of the opposite sex in order of preference a stable matching is a complete matching of men and women such that no man and woman who are not partners both prefer each other to their actual partners under the matching it is well known 2 that at least one stable matching exists for every stable marriage instance however the classical gale shapley algorithm produces a marriage that greatly favors the men at the expense of the women or vice versa the problem arises of finding a stable matching that is optimal under some more equitable or egalitarian criterion of optimality this problem was posed by knuth 6 and has remained unsolved for some time here the objective of maximizing the average or equivalently the total satisfaction of all people is used this objective is achieved when a person s satisfaction is measured by the position of his her partner in his her preference list by exploiting the structure of the set of all stable matchings and using graph theoretic methods an o n 4 algorithm for this problem is derived

a 4 dimensional kleinian group we give an example of a 4 dimensional kleinian group which is finitely generated but not finitely presented and is a subgroup of a cocompact kleinian group

convergence of freely decomposable kleinian groups we consider a compact orientable hyperbolic 3 manifold with a compressible boundary suppose that we are given a sequence of geometrically finite hyperbolic metrics whose conformal boundary structures at infinity diverge to a projective lamination we prove that if this limit projective lamination is doubly incompressible then the sequence has compact closure in the deformation space as a consequence we generalise thurston s double limit theorem and solve his conjecture on convergence of function groups affirmatively

engulfing and subgroup separability for hyperbolic groups if a group is subgroup separable otherwise known as locally extended residually finite or lerf one can pass from immersions to embeddings in some finite covering space we show that a certain engulfing property gives subgroup separability for a large and useful class of subgroups of hyperbolic 3 manifold groups

on the density of geometrically finite kleinian groups the density conjecture of bers sullivan and thurston predicts that each complete hyperbolic 3 manifold m with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3 manifolds we prove that the conjecture obtains for each complete hyperbolic 3manifold with no cusps and incompressible ends

the classification of kleinian surface groups i models and bounds we give the first part of a proof of thurston s ending lamination conjecture in this part we show how to construct from the end invariants of a kleinian surface group a lipschitz model for the thick part of the corresponding hyperbolic manifold this enables us to describe the topological structure of the thick part and to give a priori geometric bounds

dehn surgery on knots in d dehn considered the following method for constructing 3 manifolds remove a solid torus neighborhood n k of some knot x in the 3 sphere s and sew it back differently in particular he showed that taking x to be the trefoil one could obtain infinitely many non simply connected homology spheres o in this way let mx s n k then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus omk that bounds a meridional disk in the re attached solid torus we denote the resulting closed oriented 3 manifold by mk and say that it is obtained by r dehn surgery on x more generally one can consider the manifolds ml obtained by r dehn surgery on a fc component link l xi u u kk in s where r r it turns out that every closed oriented 3 manifold can be constructed in this way wal lie thus a good understanding of dehn surgery might lead to progress on general questions about the structure of 3 manifolds starting with the case of knots it is natural to extend the context a little and consider the manifolds m r obtained by attaching a solid torus v to an arbitrary compact oriented irreducible every 2 sphere bounds a 3 ball 3 manifold m with dm an incompressible torus where r is the isotopy class slope on dm of the boundary of a meridional disk of v we say that m r is the result of r dehn filling on m an observed feature of this construction is that

hausdorff dimension and conformal dynamics i strong convergence of kleinian groups this paper investigates the behavior of the hausdorff dimensions of the limit sets n and of a sequence of kleinian groups n where m h is geometrically finite we show if n strongly then a mn h 3 n is geometrically finite for all n 0 b n in the hausdorff topology and c h dim n h dim if h dim 1 on the other hand we give examples showing the dimension can vary discontinuously under strong limits when h dim 1 continuity can be recovered by requiring that accidental parabolics converge radially similar results hold for higher dimensional manifolds applications are given to quasifuchsian groups and their limits

the classification of punctured torus groups thurston s ending lamination conjecture proposes that a flnitely generated kleinian group is uniquely determined up to isometry by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends we present a proof of this conjecture for punctured torus groups these are free two generator kleinian groups with parabolic commutator which should be thought of as representations of the fundamental group of a punctured torus as a consequence we verify the conjectural topological description of the deformation space of punctured torus groups including bers conjecture that the quasi fuchsian groups are dense in this space and prove a rigidity theorem two punctured torus groups are quasi conformally conjugate if and only if they are topologically conjugate

the bianchi groups are separable on geometrically finite subgroups let d be a square free positive integer and od the ring of integers in q p id the main result of this paper is that the groups psl 2 od are

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

efficient algorithms for generalized stable marriage and roommates problems we consider a generalization of the stable roommates problem sr in which preference lists may be partially ordered and forbidden pairs may be present denoted by srpf this includes as a special case a corresponding generalization of the classical stable marriage problem sm denoted by smpf by extending previous work of feder we give a two step reduction from srpf to 2 sat this has many consequences including fast algorithms for a range of problems associated with finding optimal stable matchings and listing all solutions given variants of sr and sm for example given an smpf instance i we show that there exists an o m succinct certificate for the unsolvability of i an o m algorithm for finding all the super stable pairs in i an o m kn algorithm for listing all the super stable matchings in i an o m 1 5 algorithm for finding an egalitarian super stable matching in i and an o m algorithm for finding a minimum regret super stable matching in i where n is the number of men m is the total length of the preference lists and k is the number of super stable matchings in i analogous results apply in the case of srpf

bundles and finite foliations by a hyperbolic manifold we shall always mean a complete orientable hyperbolic manifold of nite volume we recall that if is a kleinian group then it is said to be geometrically nite if there is a nite sided convex fundamental domain for the action of on hyperbolic space otherwise is geometrically in nite if happens to be a surface group then we say it is quasi fuchsian if the limit set for the group action is a jordan curve c and preserves the components of s n c the starting point for this work is the following theorem which is a combination of theorems due to marden thurston and bonahon

cores of hyperbolic 3 manifolds and limits of kleinian groups troels jorgensen conjectured that the algebraic and geometric limits of an algebraically convergent sequence of isomorphic kleinian groups agree if there are no new parabolics in the algebraic limit we prove that this conjecture holds in most cases in particular we show that it holds when the domain of discontinuity of the algebraic limit of such a sequence is non empty we further show with the same assumptions that the limit sets of the groups in the sequence converge to the limit set of the algebraic limit as a corollary we verify the conjecture for finitely generated kleinian groups which are not non trivial free products of surface groups and infinite cyclic groups these results are extensions of similar results for purely loxodromic groups thurston previously established these results in the case when the kleinian groups are freely indecomposable using different techniques from ours ohshika has proven versions of these results for purely loxodromic function groups

length spectra and degeneration of flat metrics in this paper we consider flat metrics semi translation structures on surfaces of finite type there are two main results the first is a complete description of when a set of simple closed curves is spectrally rigid that is when the length vector determines a metric among the class of flat metrics secondly we give an embedding into the space of geodesic currents and use this to obtain a compactification for the space of flat metrics the geometric interpretation is that flat metrics degenerate to mixed structures on the surface part flat metric and part measured foliation

exact multimagnon states in one dimensional ferromagnetic spin chains with a short range interaction the authors propose the procedure of constructing the exact wavefunctions of m magnons on an infinite one dimensional lattice in the one parameter model with an exponentially decreasing interaction between spins for m or 4 the problem is solved explicitly the bethe wavefunctions in the heisenberg xxx chain appear in the limit of nearest neighbour interaction

tait s conjectures and odd crossing number amphicheiral knots we give a brief historical overview of the tait conjectures made 120 years ago in the course of his pioneering work in tabulating the simplest knots and solved a century later using the jones polynomial we announce the solution again based on a substantial study of the jones polynomial of one possibly his last remaining problem of tait with the construction of amphicheiral knots of almost all odd crossing numbers an application to the non triviality problem for the jones polynomial is also outlined 1 the first knot tables knot theory originated in the late 19th century at that time w thomson lord kelvin p g tait and j maxwell propagated the vortex atom theory in an attempt to explain the structure of the universe they believed that a supersubstance ether makes up all of matter and atoms are knotted tubes of ether knotting is hereby understood as tying a piece of rope and then identifying both its ends so that the tying cannot be any more undone thus in the realm of constructing a periodic table of elements tait began the catalogization of the simplest knots he depicted knots as we still do today by diagrams consisting of a smooth plane curve with transverse self intersections or crossings at each crossing one of the two strands passes over the other the term simplest refers to the number of crossings of the diagram we can define the crossing number of a knot as the minimal crossing number of all its diagrams and say that tait sought the list of knots with given small crossing number the list meant to present each knot by exactly one diagram this entails that knots from different diagrams should be inequivalent in the sense that one cannot turn a closed piece of rope knotted one way into one knotted the other way without cutting the rope the simplest knots are shown in figure 1 the leftmost one of crossing number 0 is the trivial knot or unknot it has some special importance much like the unit element in a group tait completed the list up to 7 crossings little kirkman later conway co and others took over and continued his work in the modern computer age tables have reached the knots of 17 crossings with millions of entries even though tait s vortex atom theory has long been dismissed an account on knot tabulation is received by the editors may 30 2007 2000 mathematics subject classification primary 57m25 secondary 01a55 01a60

three fast algorithms for four problems in stable marriage the stable marriage problem is a well known problem of matching n men to n women to achieve a certain type of stability the o n 2 time gale shapley gs algorithm for finding two particular but extreme stable marriages out of a possibly exponential number of stable marriages is also well known in this paper we consider four problems concerned with finding information about the set of all stable marriages and with finding stable marriages other than those obtained by the gale shapley algorithm in particular we give an o n 2 time algorithm which for any problem instance of n men and n women finds every man woman pair that is contained in at least one stable marriage we show that the same algorithm finds all the rotations for the problem instance in o n 2 time we give an o n 2 n s time and o n 2 space bounded algorithm which is time and space optimal to enumerate all stable marriages where s is the set of them and we give an o n 2 time algorithm to find the mini