Knot adjacency and fibering

It is known that the Alexander polynomial detects fibered knots and 3-manifolds that fiber over the circle. In this note, we show that when the Alexander polynomial becomes inconclusive, the notion of knot adjacency can be used to obtain obstructions to the fibering of knots and of 3-manifolds. As an application, given a fibered knot , we construct infinitely many non-fibered knots that share the same Alexander module with . Our construction also provides, for every , examples of irreducible 3-manifolds that cannot be distinguished by the Cochran-Melvin finite type invariants of order .

     

Milnor link invariants and quantum 3-manifold invariants

Let be the 3-manifold invariant of Le, Murakami and Ohtsuki. We show that , where denotes terms of degree , if M is a homology 3-sphere obtained from by surgery on an n-component Brunnian link whose Milnor -invariants of length vanish.?We prove a realization theorem which is a partial converse to the above theorem.?Using the Milnor filtration on links, we define a new bifiltration on the vector space with basis the set of oriented diffeomorphism classes of homology 3-spheres. This includes the Milnor level 2 filtration defined by Ohtsuki. We show that the Milnor level 2 and level 3 filtrations coincide after reindexing. Received: October 23, 1998.     

An Algorithm of Witten‘s Invariants of SOme 3—manifolds

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Construction of exotic smooth structures

In this article, we construct infinitely many simply connected, nonsymplectic and pairwise nondiffeomorphic 4-manifolds starting from the elliptic surfaces E(n) and applying the sequence of knot surgery, ordinary blowups and rational blowdown. We also compute the Seiberg-Witten invariants of these manifolds.     

Classification of tight contact structures on small Seifert 3-manifolds with

We classify positive, tight contact structures on closed Seifert fibered 3-manifolds with base , three singular fibers and .

     

The theorem of Kronheimer-Mrowka

A proof of the conjecture of Thom (algebraic curves in the complex projective plane minimize genus within their homology class) due to Kronheimer-Mrowka is presented. The proof uses the mod-2 Seiberg-Witten invariants.     

Seiberg-Witten invariants — An expository account

The Seiberg-Witten monopole equations, and a new invariant for 4-manifolds which results from these equations, are introduced in this paper.     

A Survey on Mixed Spin P-Fields

The mixed spin P-fields (MSP for short) theory sets up a geometric platform to relate Gromov-Witten invariants of the quintic three-fold and Fan-Jarvis-Ruan-Witten invariants of the quintic polynomial in five variables.It starts with Wittens vision and the P-fields treatment of GW invariants and FJRW invariants.Then it briefly discusses the master space technique and its application to the set-up of the MSP moduli.Some key results in MSP theory are explained and some examples are provided.     

Spin Spaces, Lipschitz Groups, and Spinor Bundles

It is shown that every bundle M of complex spinormodules over the Clifford bundle Cl(g) of a Riemannian space(M, g) with local model (V, h)is associated with an lpin(Lipschitz) structure on M, this being a reduction of theO(h)-bundle of all orthonormal frames on M to the Lipschitzgroup Lpin(h) of all automorphisms of a suitably defined spinspace. An explicit construction is given of the total space of theLpin(h)-bundle defining such a structure. If the dimension mof M is even, then the Lipschitz group coincides with the complexClifford group and the lpin structure can be reduced to a pin _c structure. If m = 2n – 1, then a spinor module on M is of the Cartan type: its fibres are 2 _n -dimensional anddecomposable at every point of M, but the homomorphism of bundlesof algebras Cl(g) End globally decomposes if, andonly if, M is orientable. Examples of such bundles are given. Thetopological condition for the existence of an lpin structure on anodd-dimensional Riemannian manifold is derived and illustrated by theexample of a manifold admitting such a structure, but no pin _c structure.     

Noncomplex smooth 4-manifolds with genus-2 Lefschetz fibrations

We construct noncomplex smooth 4-manifolds which admit genus-2 Lefschetz fibrations over . The fibrations are necessarily hyperelliptic, and the resulting 4-manifolds are not even homotopy equivalent to complex surfaces. Furthermore, these examples show that fiber sums of holomorphic Lefschetz fibrations do not necessarily admit complex structures.

     

Duality Maps of Finite Abelian Groups and Their Applications to Spin Models

Duality maps of finite abelian groups are classified. As a corollary, spin models on finite abelian groups which arise from the solutions of the modular invariance equations are determined as tensor products of indecomposable spin models. We also classify finite abelian groups whose Bose-Mesner algebra can be generated by a spin model.     

Rigidity of Geometrically Finite Hyperbolic Cone-Manifolds

In a recent paper Hodgson and Kerckhoff prove a local rigidity theorem for finite volume, three-dimensional hyperbolic cone-manifolds. In this paper we extend this result to geometrically finite cone-manifolds. Our methods also give a new proof of a local version of the classical rigidity theorem for geometrically finite hyperbolic 3-manifolds.     

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 efficient 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 be a Reebless, finite depth foliation in a closed hyperbolic 3-manifold. Then is not quasi-isometric, but suppose that 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 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 universal cover, and second a topological theory of the joint structure of the pseudo-Anosov foliation in the universal cover.

     

Quasifree States in Some One-Dimensional Quantum Spin Models

We use numerical methods to investigate the SU _q(N) Perk–Schultz spin chain at the special quantum parameter value q=–e ^i/N . We discover simple laws applicable to a considerable part of the Hamiltonian spectrum, which in particular contains the energy of the ground state and the nearest excitations. The phenomenological formulas obtained resemble formulas for the spectrum of the free-fermion model. We formulate several hypotheses, some of which can be justified by constructing exact solutions of the system of Bethe-ansatz equations for finite-length chains. We obtain two sets of solutions of these equations. The first corresponds to the special value of the quantum parameter q and, in particular, describes the model ground state, which is antiferromagnetic. The second set of solutions describes a part of the spectrum belonging to the sectors where the numbers n _i of particles of different types (i=0,1,...,N–1) do not exceed unity for all the types except one. For this set, we obtain a simple spectrum at arbitrary values of q. It is hypothesized that this spectrum and the solutions of the Bethe-ansatz equations found in a closed form are intimately related to the existence of a special eigenstate for the transfer matrix of the auxiliary inhomogeneous SU _q(N–1) vertex model that is involved in constructing the system of Bethe-ansatz equations of a matrioshka structure. Indirect arguments based on combinatorial properties of the wave function of the relevant state are given to support this hypothesis.     

Some Remarks on Almost and Stable Almost Complex Manifolds

In the first part we give necessary and sufficient conditions for the existence of a stable almost complex structure on a 10-manifold M with H₁(M;?) = 0 and no 2-torsion in H₁(M;?) for i = 2,3. Using the Classification Theorem of Donaldson we give a reformulation of the conditions for a 4-manifold to be almost complex in terms of Betti numbers and the dimension of the ±-eigenspaces of the intersection form. In the second part we give general conditions for an almost complex manifold to admit infinitely many almost complex structures and apply these to symplectic manifolds, to homogeneous spaces and to complete intersections.     

Volumes of hyperbolic Löbell 3-manifolds

In 1931 F. Löbell constructed the first example of a closed orientable three-dimensional hyperbolic manifold. In the present paper we study properties of closed hyperbolic 3-manifolds generalizing Löbell's classical example. Explicit formulas for the volumes of these manifolds in terms of the Lobachevski function are obtained.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 17–23, July, 1998.This research was partially supported by GARC-KOSEF (Global Analysis Research Center of National Seoul University) and by the Russian Foundation for Basic Research under grant No. 95-01-01410.     

Twisted Extensions of Spin Models

A spin model is one of the statistical mechanical models which were introduced by V.F.R. Jones to construct invariants of links. In this paper, we give a new construction of spin models of size 4n from a given spin model of size n. The process is similar to taking tensor product with a spin model of size four, but we add some sign exchange. This construction also gives symmetric four-weight spin models of the type introduced by E. Bannai and E. Bannai.     

High Spin Particles with Spin-Mass Coupling

The classical and quantum model of high spin particles is analyzed within the manifestly covariant framework. The model is obtained by supplementing the standard Lagrange function for relativistic point particle by additional terms governing the dynamics of internal degrees of freedom. They are described by (3, 1) Clifford algebra (Majorana) spinors. The covariant quantization leads to the spectrum of the particles with the masses depending on their spins. The particles (and anti-particles) appear to be orphaned as their potential anti-particle partners are of different mass.