the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

waldhausen s classification theorem for finitely uniformizable 3 orbifolds we define a map between two orbifolds with respect to this map we generalize 3 manifold theory to 3 orbifolds as the main goal we generalize the waldhausen s classification theorem of haken 3 manifolds to finitely uniformizable 3 orbifolds for applications of the developed theory we introduce an invariant for links and tangles by using the orbifold fundamental group with the invariant we classify a class of links and show the untangling theorem

low temperature behavior of the multicomponent widom rowlison model on finite square lattices we consider the multicomponent widom rowlison with metropolis dynamics which describes the evolution of a particle system where m different types of particles interact subject to certain hard core constraints focusing on the scenario where the spatial structure is modeled by finite square lattices we study the asymptotic behavior of this interacting particle system in the low temperature regime analyzing the tunneling times between its m maximum occupancy configurations and the mixing time of the corresponding markov chain in particular we develop a novel combinatorial method that exploiting geometrical properties of the widom rowlinson configurations on finite square lattices leads to the identification of the timescale at which transitions between maximum occupancy configurations occur and shows how this depends on the chosen boundary conditions and the square lattice dimensions

non isospectral scattering problems and truncation for hierarchies burgers and dispersive water waves abstract non isospectral scattering problems have been proven useful for several reasons amongst them the information that they provide about painleve truncation for entire hierarchies of integrable partial differential equations pdes we show in this paper how our approach provides in a very straightforward manner truncation results for two hierarchies in 1 1 dimensions namely burgers hierarchy and the dispersive water wave hierarchy burgers equation is well known as a model for turbulence and the dispersive water wave equations as a system governing shallow water waves we also see how these results are easily extended to related hierarchies in 2 1 dimensions our results for the 2 1 dimensional burgers hierarchy and for the 1 1 and 2 1 dimensional dispersive water wave hierarchies are all new

stable maps and branched shadows of 3 manifolds in the early 1990s turaev introduced the notion of shadows as a combinatorial presentation of both 4 and 3 manifolds later costantino thurston revealed a strong relation between the stein factorizations of stable maps of 3 manifolds into the real plane and the shadows of the manifolds in fact a shadow can be seen locally as the stein factorization of a stable map in this paper we define the notion of stable map complexity for a compact orientable 3 manifold bounded by possibly empty tori counting with some weights the minimal number of singular fibers of codimension 2 of stable maps into the real plane and prove that this number equals the minimal number of vertices of its branched shadows in consequence we give a complete characterization of hyperbolic links in the 3 sphere whose exteriors have stable map complexity 1 in terms of dehn surgeries and also give an observation concerning the coincidence of the stable map complexity and shadow complexity using estimations of hyperbolic volumes

cusp geometry of fibered 3 manifolds let f be a surface and suppose that varphi colon f to f is a pseudo anosov homeomorphism fixing a puncture p of f the mapping torus m m varphi is hyperbolic and contains a maximal cusp c about the puncture p we show that the area and height of the cusp torus partial c is equal to the stable translation distance of varphi acting on the arc complex scr a f p up to an explicitly bounded multiplicative error our proof relies on elementary facts about the hyperbolic geometry of pleated surfaces in particular the proof of this theorem does not use any deep results from teichm uller theory kleinian group theory or the coarse geometry of scr a f p a similar result holds for quasi fuchsian manifolds n cong f times bbb r in that setting we find a combinatorial estimate for the area and height of the cusp annulus in the convex core of n up to explicitly bounded multiplicative and additive error as an application we show that covers of punctured surfaces induce quasi isometric embeddings of arc complexes

the number of reidemeister moves for splitting a link we give an explicit upper bound for the number of reidemeister moves for deforming a link diagram of a split link to be disconnected

the weiss tabor carnevale painleve test and burgers hierarchy the weiss tabor carnevale wtc painleve test and its recent perturbative extension provide necessary conditions for a partial differential equation to have the painleve property it follows that burgers hierarchy must pass the wtc painleve test the aim here is to prove this explicitly in addition the backlund transformation for burgers equation obtained by wtc via truncation is extended to the entire hierarchy the recursion operator is found to be related to a simple first order system

reflection subgroups of coxeter groups we use the geometry of the davis complex of a coxeter group to investigate finite index reflection subgroups ot coxeter groups the mam result is the following if g is an infinite indecomposable coxeter group and h g is a finite index reflection subgroup then the rank of h is not less than the rank of g this generalizes earlier results of the authors 2004 we also describe the relationship between the nerves of the group and the subgroup in the case of equal rank

reflection subgroups of coxeter systems let w r be a coxetcr system and t ij w wrw denote the corresponding set of reflections for u e w define n w tv tlf tw l w h w ere i denotes the length function of w r the principal result 3 3 of this paper is that if w is a reflection subgroup of w i e w w n t then e ti n z n w 1 is a set of coxeter generators for w in particular w is a coxeter group in 6 v deodhar gives a geometric proof of this last fact using properties of the root system of w it is easily seen that the coxeter gcncrators for w defined by deodhar coincide with those described above the idea of the proof here is that n may be regarded as a cocycle with n r r rer and coxeter systems are characterized among groups generated by involutions r by the existence of such a cocycle one then checks that w h n w n w w e w gives a suitable cocycle for w the same argument applies to a wider class of reflection systems g x characterized by the cxistcncc of a certain cocycle and so we give the proof for these the coxeter systems are precisely the reflection systems g x in which x consists of involutions the arrangement of this paper is as follows in section 2 we describe the main facts about reflection systems closely paralleling 1 chap iv sect 11 in section 3 we prove that reflection subgroups of a reflection system are themselves reflection systems in a canonical way 3 3 and obtain some additional facts about their canonical generators 3 4 3 11 in section 4 we specialize to the case of a coxeter system and give a geometric criterion 4 4 for a set of reflections to be the set of canonical generators of a reflection subgroup in terms of the inner products of the

a generalized bridge number for links in 3 manifolds in sch schubert introduced a new invariant of knots in the 3 sphere called the bridge number and showed that when reduced by 1 it is an additive invariant under the connected sum operation of knots several generalizations of schubert s theorem can be imagined indeed from schubert s perspective the behaviour of bridge number under addition is only a corollary of a more general theorem about bridge number of satellite knots boileau and lickorish have independently observed b l that the following generalization of schubert s theorem would be useful bridge number as defined by schubert begins with a projection of the knot or link onto a sphere in s 3 project instead onto a higher genus unknotted surface in s 3 one advantage of this approach is that it allows bridge number to be defined for a link in a 3 manifold other than 5 3 by projecting the link onto a heegaard surface for the 3 manifold here we explore how such an invariant behaves under connected sum using a line of argument similar to jaco s proof of haken s theorem that heegaard genus is additive ja a crucial additional ingredient needed is lemma4 6 some of the arguments were motivated by no in which as an aside a different generalization of schubert s theorem is attempted they want to show that the standard bridge number behaves well under knot decomposition even when the knot is decomposed by spheres intersecting the knot more than twice e g under tangle sums l besides recovering a new proof of schubert s theorem more in the spirit of jaco s proof of the haken lemma we are also able to prove an analogous theorem for bridge number defined with respect to genus one heegaard splittings of 5 3 or the lens space and also with respect to irreducible heegaard splittings of closed nonhaken 3 manifolds

coxeter groups and meridional rank of links we prove the meridional rank conjecture for twisted links and arborescent links associated to bipartite trees with even weights these links are substantial generalizations of pretzels and two bridge links respectively lower bounds on meridional rank are obtained via coxeter quotients of the groups of link complements matching upper bounds on bridge number are found using the wirtinger numbers of link diagrams a combinatorial tool developed by the authors

on hyperelliptic involutions of hyperbolic 3 manifolds by classical results a compact riemann surface or equivalently an orientable closed hyperbolic 2 manifold has at most one hyperelliptic involution i e the quotient by the involution is the 2 sphere a hyperelliptic involution is central in the automorphism or isometry group of such a hyperelliptic surface its fixed points are exactly the weierstrass points of the surface see e g fk in the present paper we consider hyperelliptic involutions of hyperbolic 3manifolds by definition these are isometric involutions whose quotient is the 3 sphere the situation is more complicated in dimension three in particular a hyperbolic 3 manifold can have more than one hyperelliptic involution our main result states that there is a universal bound on the number of non conjugate hyperelliptic involutions of closed hyperbolic 3 manifolds this bound is quite small and we compute it for various situations our methods also determine how different hyperelliptic involutions are related there remains a small gap between these upper bounds and the number of different involutionswhichwe can actually construct see r1 rz we remark that the question on the number of different hyperelliptic involutions of hyperbolic 3 manifolds has explicitly been asked in mv we obtain also results for hyperelliptic isometries of orders greater than two in particular for prime numbers p 2 we give a complete solution to the problem on the number of different hyperelliptic isometries of order p i e the upper bound we obtain is sharp let isom m denote the orientation preserving isometry group of a hyperbolic 3 manifoldm

phase transitions in the multicomponent widom rowlinson model and in hard cubes on the bcc lattice we use monte carlo techniques and analytical methods to study the phase diagram of the m component widom rowlinson model on the bcc lattice there are m species all with same fugacity z and a nearest neighbor hard core exclusion between unlike particles simulations show that for m 3 there is a crystal phase for z lying between zc m and zd m while for z zd m there are m demixed phases each consisting mostly of one species for m 2 there is a direct second order transition from the gas phase to the demixed phase while for m 3 the transition at zd m appears to be first order putting it in the potts model universality class for m large pirogov sinai theory gives zd m m 2 2 3m2 in the crystal phase the particles preferentially occupy one of the sublattices independent of species i e spatial symmetry but not particle symmetry is broken for m this transition approaches that of the one component hard cube gas with fugacity y zm we find by direct simulations of such a system a transition at yc 0 71 which is consistent with the simulation zc m for large m this transition appears to be always of the ising type

the singular manifold method revisited for many completely integrable partial differential equations pdes the singular manifold method of weiss allows the recovery of the lax pair and darboux transformation dt and so also the backlund transformation from a truncated painleve expansion recently the so called two singular manifold method has been proposed in order to handle pdes such as the modified korteweg de vries mkdv equation here we present a more natural extension of the weiss singular manifold method which makes use of only one singular manifold but is capable of dealing with such pdes in this approach we allow the possibility that the dt might in fact correspond to an infinite painleve expansion for a certain choice of the arbitrary coefficients this then leads us to a new and more consistent definition of singular manifold equation sme this can give smes different from those usually presented the summation of infinite painleve expansions is effected by seeking a truncation in a new riccati variable z the use o

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

dehn filling volume and the jones polynomial given a hyperbolic 3 manifold with torus boundary we bound the change in volume under a dehn filling where all slopes have length at least 2 this result is applied to give explicit diagrammatic bounds on the volumes of many knots and links as well as their dehn fillings and branched covers finally we use this result to bound the volumes of knots in terms of the coefficients of their jones polynomials

geometrization of 3 dimensional orbifolds this paper is devoted to the proof of the orbifold theorem if o is a compact connected orientable irreducible and topologically atoroidal 3 orbifold with nonempty ramification locus then o is geometric i e has a metric of constant curvature or is seifert fibred as a corollary any smooth orientationpreserving nonfree finite group action on s3 is conjugate to an orthogonal action

a geometric proof of the fintushel stern formula abstract the fintushel stern formula asserts that the casson invariant of a brieskorn homology sphere p q r equals 1 8 the signature of its milnor fiber we give a geometric proof of this formula as opposite to computational methods used in the original proof the formula is also refined to relate equivariant casson invariants to equivariant signatures

lower bounds on volumes of hyperbolic haken 3 manifolds we prove a volume inequality for 3 manifolds having c o metrics bent along a surface and satisfying certain curvature conditions the result makes use of perelman s work on the ricci flow and geometrization of closed 3 manifolds corollaries include a new proof of a conjecture of bonahon about volumes of convex cores of kleinian groups improved volume estimates for certain haken hyperbolic 3 manifolds and a lower bound on the minimal volume of orientable hyperbolic 3 manifolds an appendix compares estimates of volumes of hyperbolic 3 manifolds drilled along a closed embedded geodesic with experimental data

on floer homology and the berge conjecture on knots admitting lens space surgeries we complete the first step in a two part program proposed by baker grigsby and the author to prove that berge s construction of knots in the three sphere which admit lens space surgeries is complete the first step which we prove here is to show that a knot in a lens space with a threesphere surgery has simple in the sense of rank knot floer homology the second conjectured step involves showing that for a fixed lens space the only knots with simple floer homology belong to a simple finite family using results of baker we provide evidence for the conjectural part of the program by showing that it holds for a certain family of knots coupled with work of ni these knots provide the first infinite family of non trivial knots which are characterized by their knot floer homology as another application we provide a floer homology proof of a theorem of berge

generalized scaling reductions and painleve hierarchies we give an alternative derivation of two painleve hierarchies this is done by constructing generalized scaling reductions of the korteweg de vries and dispersive water wave hierarchies we also construct a generalized scaling reduction of burgers hierarchy

non standard components of the character variety for a family of montesinos knots we show that there are montesinos knots with n 1 geq 4 tangles whose character varieties contain arbitrarily many irreducible components of dimension d for any 1 le d le n 2 moreover these irreducible components can be chosen so that the trace of the meridian is non constant

finite group actions on 3 manifolds if g is a finite group acting smoothly on a closed surface f it is well known that g leaves invariant some riemannian metric of constant curvature on f thus any action of g on the 2 sphere s 2 is conjugate in diff s 2 to an orthogonal action if g acts on the torus sx s 1 there is a g invariant flat metric on s a s 1 and if g acts on a surface f with negative euler number then f admits a g invariant hyperbolic metric recently thurston th 1 th 2 th 3 has described the eight 3 dimensional geometries which provide geometric structures for closed 3 manifolds in the same way that the 2 sphere s 2 the euclidean plane e 2 and the hyperbolic plane h 2 provide geometric structures for surfaces see also the survey article by scott sc4 thurston also conjectured that if m is a closed 3 manifold with a geometric structure modelled on one of these eight geometries say x then any smooth action of a finite group g on m should leave invariant some metric on m inducing the geometry x we will say that g preserves the geometric structure on m in this case it should be noted that the restriction to smooth actions of g on m is essential for bing bi showed that there are involutions of s 3 whose fixed set is a wild 2 sphere however in dimension two it was proved by eilenberg ei that any action of a finite group on a surface is conjugate to a smooth action in this paper our main result asserts that thurston s conjecture holds for five of the eight geometries the result is the following

inverse variational problem and canonical structure of burgers equations it is demonstrated that burgers equations which are often believed to describe dissipative systems are non lagrangian following bateman s analysis of damped harmonic oscillator an action is defined to look for a lagrangian representation for equations in the burgers hierarchy the associated higher order lagrangian densities are found to be degenerate such that the hamiltonian structure could be studied by a repackaging of ostrogradski formalism and dirac s theory of constraints

nonmonotonic behavior in hard core and widom rowlinson models we give two examples of nonmonotonic behavior in symmetric systems exhibiting more than one critical point at which spontaneous symmetry breaking appears or disappears the two systems are the hard core model and the widom rowlinson model and both examples take place on a variation of the cayley tree bethe lattice devised by schonmann and tanaka we obtain similar though less constructive examples of non monotonicity via certain local modifications of any graph e g the square lattice which is known to have a critical point for either model en route we prove that the widom rowlinson model does behave monotonically on the ordinary cayley tree and we obtain an explicit description of its critical behavior there we conclude with some results about monotonicity of the phase transition phenomenon relative to graph structure

an alternate version of pirogov sinai theory a new approach to the pirogov sinai theory of phase transitions is developed not employing the contour models with a parameter the completeness of the phase diagram is proven

ordering and demixing transitions in multicomponent widom rowlinson models we use monte carlo techniques and analytical methods to study the phase diagram of multicomponent widom rowlinson models on a square lattice there are m species all with the same fugacity z and a nearest neighbor hard core exclusion between unlike particles simulations show that for m between two and six there is a direct transition from the gas phase at z mathit z mathit d m to a demixed phase consisting mostly of one species at zg mathit z mathit d m while for m ensuremath ge 7 there is an intermediate crystal phase for z lying between mathit z mathit c m and mathit z mathit d m in this phase which is driven by entropy particles independent of species preferentially occupy one of the sublattices i e spatial symmetry but not particle symmetry is broken the transition at mathit z mathit d m appears to be of the first order for m ensuremath ge 5 putting it in the potts model universality class for large m the transition between the crystalline and demixed phases at mathit z mathit d m can be proven to be of the first order with mathit z mathit d m ensuremath sim m 2 1 m while mathit z mathit c m is argued to behave as ensuremath lambda mathrm cr m with ensuremath lambda mathrm cr the critical value of the fugacity at which the one component hard core square lattice gas has a transition and to be always of the ising type explicit calculations for the bethe lattice with the coordination number q 4 give results similar to those for the square lattice except that the transition at mathit z mathit d m becomes first order at mg2 this happens for all q consistent with the model being in the potts universality class c 1995 the american physical society

the painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

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

heegaard genus and rank of augmented link complements we prove that the rank of the fundamental group of an augmented link complement equals its heegaard genus this is achieved by showing that the heegaard genus equals the number of link components a straightforward consequence is that these manifolds satisfy a conjecture of shalen

groups generated by reflections and aspherical manifolds not covered by euclidean space a coxeter system r v is a group r a coxeter group together with a set of generators v such that each element of v has order two and such that all relations in r are consequences of relations of the form vw m v w 1 where v w e v and m v w denotes the order of vw the m v w s which are positive integers or oo obviously determine the coxeter system up to isomorphism the list of finite coxeter groups is short and well known however general coxeter groups are much more flexible in fact the set v and the m v w s can be specified arbitrarily at least if v is finite subject only to the conditions m v v 1 and m v w m w v 2 if v w suppose that r v is a coxeter system that x is a hausdorff space and that xv v v is a locally finite family of closed subspaces indexed by v the xv are called the panels of x 2 there is a classical method for constructing a transformation group from these data for each x e x let v x denote the set of v in v such that x e xv for each subset s of v let rs be the subgroup generated by s and let xs be the face of x defined by

hyperbolic structures on 3 manifolds i deformation of acylindrical manifolds this is the first in a series of papers showing that haken manifolds have hyperbolic structures this first was published the second two have existed only in preprint form and later preprints were never completed this eprint is only an approximation to the published version which is the definitive form for part i and is provided for convenience only all references and quotations should be taken from the published version since the theorem numbering is different and not all corrections have been incorporated into the present version parts ii and iii will be made available as eprints shortly

exceptional and cosmetic surgeries on knots we show that the bridge distance of a knot determines a lower bound on the genera of essential surfaces and heegaard surfaces in the manifolds that result from non trivial dehn surgeries on the knot in particular knots with high bridge distance do not admit non trivial non hyperbolic surgeries or non trivial cosmetic surgeries we further show that if a knot has bridge distance at least 3 then its bridge number is bounded above by a function of seifert genus or indeed by the genus of almost any essential surface or heegaard surface in the surgered manifold

symmetry reductions of partial differential equations related to singular manifold expansions it is shown that all the information yielded by the reduction methods of bluman and cole 1974 and clarkson and kruskal 1989 can be obtained using the singular manifold expansion the burgers equation modified korteweg de vries equation caudrey dodd gibbon equation and the fitzhugh nagumo equation are used as illustrative examples several new exact solutions are presented

volume rigidity for finite volume manifolds we extend to finite volume manifolds some volume rigidity results of besson courtois gallot our main result is a volume theorem which shows that proper maps from finite volume manifolds with ricci curvature bounded from below to finite volume manifolds of pinched negative curvature decrease volume as a consequence we deduce a lower bound on the minimal volume of finite volume manifolds possessing a proper map to a finite volume manifold of pinched negative curvature and we show asymptotic isometry holds when the minimal volume is attained finally we prove that finite rank one locally symmetric manifolds minimize normalized entropy rigidity in their homotopy class

a unified approach to painleve expansions abstract the painleve test for partial differential equations developed by weiss tabor and carnevale wtc is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations a simple modification of the wtc procedure used for partial differential equations enables us to determine the lax pairs hirota equations and auto backlund transformations for ordinary differential equations including a new lax pair for an integrable case of the henon heiles system a detailed study of the kdv hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained the role of the singular branches is examined in detail and important new insights obtained in particular we find that each singular branch is simply a re expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce a parallel analysis is carried out for the akns hierarchy

universal bounds for hyperbolic dehn surgery this paper gives a quantitative version of thurston s hyperbolic dehn surgery theorem applications include the first universal bounds on the number of nonhyperbolic dehn fillings on a cusped hyperbolic 3 manifold and estimates on the changes in volume and core geodesic length during hyperbolic dehn filling the proofs involve the construction of a family of hyperbolic conemanifold structures using infinitesimal harmonic deformations and analysis of geometric limits

on the super additivity of tunnel number of knots abstract in the present paper we show a necessary and sufficient condition for knots k 1 k 2 in s 3 with some side condition to have the super additivity of tunnel number under connected sum i e t k 1 k 2 t k 1 t k 2 1

minimal index torsion free subgroups of kleinian groups a kleinian group is a discrete subgroup of psl 2 c the full group of orientation preserving isometries of 3 dimensional hyperbolic space in the language of t1 q h3 is a hyperbolic 3 orbifold that is a metric 3 orbifold in which all sectional curvatures are 1 and for which is the orbifold fundamental group see t1 for further details a fuchsian group is a discrete subgroup of psl 2 r and as such acts discontinuously on the hyperbolic plane we define to be of finite co volume resp co area if the volume resp area of the quotient orbifold q is finite by selberg s lemma if is a kleinian resp fuchsian group of finite covolume resp co area it contains a torsion free subgroup of finite index by definition any torsion free subgroup cannot contain any finite subgroups of so the index must be a multiple of the lowest common multiple of all orders of finite subgroups of see cfjr for a proof being of finite co volume implies there are only finitely many conjugacy classes of finite subgroups thus we make the following

the π orbifold group of a link on associe a un enlacement l dans s 3 un groupe o l appele le groupe de variete d orbite de l on etudie certaines proprietes du groupe o l reliees a la geometrie de l enlacement l

every knot has characterising slopes let k be a knot in the 3 sphere a slope p q is said to be characterising for k if whenever p q surgery on k is homeomorphic via an orientation preserving homeomorphism to p q surgery on another knot k in the 3 sphere then k and k are isotopic it was an old conjecture of gordon proved by kronheimer mrowka ozsvath and szabo that every slope is characterising for the unknot in this paper we show that every knot k has infinitely many characterising slopes confirming a conjecture of baker and motegi in fact p q is characterising for k provided p is at most q and q is sufficiently large

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

unknotting tunnels and seifert surfaces let k be a knot with an unknotting tunnel gamma and suppose that k is not a 2 bridge knot there is an invariant rho p q in mathbb q 2 mathbb z with p odd defined for the pair k gamma the invariant rho has interesting geometric properties it is often straightforward to calculate for example for k a torus knot and gamma an annulus spanning arc rho k gamma 1 although rho is defined abstractly it is naturally revealed when k cup gamma is put in thin position if rho neq 1 then there is a minimal genus seifert surface f for k such that the tunnel gamma can be slid and isotoped to lie on f one consequence is that if rho k gamma neq 1 then mathrm genus k 1 this confirms a conjecture of goda and teragaito for pairs k gamma with rho k gamma neq 1

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

an integral equation theory for the widom rowlinson mixture the widom rowlinson mixture is a two component fluid in which like species do not interact and unlike species interact via a hard core repulsion as the density is increased this fluid phase separates standard integral equation approaches such as the percus yevick or hypernetted chain or thermodynamically self consistent hybirds of these two make very inaccurate predictions for the location of this critical point in the three dimensional model in this article we suggest a family of new approximations for this model that rely on incorporating terms in the density expansion of the direct correlation function into the closure approximation we show that the simplest of these closures is significantly more accurate than previous theories for the structure and thermodynamics of the fluid

cusp shapes under cone deformation a horospherical torus about a cusp of a hyperbolic manifold inherits a euclidean similarity structure called a cusp shape we bound the change in cusp shape when the hyperbolic structure of the manifold is deformed via cone deformation preserving the cusp the bounds are in terms of the change in structure in a neighborhood of the singular locus alone we then apply this result to provide information on the cusp shape of many hyperbolic knots given only a diagram of the knot specifically we show there is a universal constant c such that if a knot admits a prime twist reduced diagram with at least c crossings per twist region then the length of the second shortest curve on the cusp torus is bounded the bound is linear in the number of twist regions of the diagram

word hyperbolic dehn surgery in the late 1970 s thurston dramatically changed the nature of 3 manifold theory with the introduction of his geometrisation conjecture and by proving it in the case of haken 3 manifolds 23 the conjecture for general closed orientable 3 manifolds remains perhaps the most important unsolved problem in the subject a weaker form of the conjecture 19 deals with the fundamental group of a closed orientable 3 manifold it proposes that either it contains z z as a subgroup or it is word hyperbolic in the sense of gromov 11 word hyperbolic groups are precisely those groups which satisfy a linear isoperimetric inequality they are of fundamental importance in geometric group theory and have very many useful properties

systematics of strongly self dominant higher order differential equations based on the painleve analysis of their singularities this paper presents a simple way of classifying higher order differential equations based on the requirements of the painleve property i e the presence of no movable critical points the fundamental building blocks for such equations may be generated by strongly self dominant differential equations of the type x nu xm u m n p p in which m and n are positive integers and p is a negative integer such differential equations having both a constant degree d and a constant value of the difference n m form a painleve chain however only three of the many possible painleve chains can have the painleve property among the three painleve chains that can have the painleve property one contains the burgers equation another contains the dominant terms of the first painleve transcendent the isospectral korteweg de vries equation and the isospectral boussinesq equation and the third contains the dominant terms of the second painleve transcendent and the isospectral modified cubic korteweg de vr

a perturbative painleve approach to nonlinear differential equations abstract we further improve the painleve test so that negative indices resonances can be treated we demand single valuedness not only for any pole like expansion as in the usual painleve test but also for every solution close to it represented as a perturbation series in a small parameter e order zero is the usual test order one already treated in a preliminary paper reduces to a linear fuchs analysis near a regular singularity and allows the introduction of all missing arbitrary coefficients higher orders lead to the analysis of a linear fuchsian type inhomogeneous equation we obtain an infinite sequence of necessary conditions for the absence of movable logarithmic branch points arising at every integer index whether positive or negative and at every order those arising at negative indices including 1 are new while some conditions may not arise before some high perturbation order we present several illustrative examples we discuss the understanding of negative indices and conclude that they are indistinguishable from positive indices just as in the fuchs theory in particular negative indices give rise to doubly infinite laurent series

a new derivation of painleve hierarchies abstract we give a new derivation of two painleve hierarchies this is done by extending the accelerating wave reductions of the korteweg de vries and dispersive water wave equations to their respective hierarchies we also consider the extension of this reduction of burgers equation to the burgers hierarchy