CM-fields with relative class number one

We will show that the normal CM-fields with relative class number one are of degrees . Moreover, if we assume the Generalized Riemann Hypothesis, then the normal CM-fields with relative class number one are of degrees , and the CM-fields with class number one are of degrees . By many authors all normal CM-fields of degrees with class number one are known except for the possible fields of degree or . Consequently the class number one problem for normal CM-fields is solved under the Generalized Riemann Hypothesis except for these two cases.

     

Cusp size bounds from singular surfaces in hyperbolic 3-manifolds

Singular maps of surfaces into a hyperbolic 3-manifold are utilized to find upper bounds on meridian length, -curve length and maximal cusp volume for the manifold. This allows a proof of the fact that there exist hyperbolic knots with arbitrarily small cusp density and that every closed orientable 3-manifold contains a knot whose complement is hyperbolic with maximal cusp volume less than or equal to 9. We also find particular upper bounds on meridian length, -curve length and maximal cusp volume for hyperbolic knots in depending on crossing number. Particular improved bounds are obtained for alternating knots.

     

Admissible measures in one dimension

In this note we show that -admissible measures in one dimension (i.e. doubling measures admitting a -Poincaré inequality) are precisely the Muckenhoupt -weights.

     

Concordance crosscap numbers of knots and the Alexander polynomial

For a knot the concordance crosscap number, , is the minimum crosscap number among all knots concordant to . Building on work of G. Zhang, which studied the determinants of knots with , we apply the Alexander polynomial to construct new algebraic obstructions to . With the exception of low crossing number knots previously known to have , the obstruction applies to all but four prime knots of 11 or fewer crossings.

     

On approximations of rank one -perturbations

Approximations of rank one -perturbations of self-adjoint operators by operators with regular rank one perturbations are discussed. It is proven that in the case of arbitrary not semibounded operators such approximations in the norm resolvent sense can be constructed without any renormalization of the coupling constant. Approximations of semibounded operators are constructed using rank one non-symmetric regular perturbations.

     

Slice knots with distinct Ozsváth-Szabó and Rasmussen invariants

As proved by Hedden and Ording, there exist knots for which the Ozsváth-Szabó and Rasmussen smooth concordance invariants, and , differ. The Hedden-Ording examples have nontrivial Alexander polynomials and are not topologically slice. It is shown in this note that a simple manipulation of the Hedden-Ording examples yields a topologically slice Alexander polynomial one knot for which and differ. Manolescu and Owens have previously found a concordance invariant that is independent of both and on knots of polynomial one, and as a consequence have shown that the smooth concordance group of topologically slice knots contains a summand isomorphic to . It thus follows quickly from the observation in this note that this concordance group contains a summand isomorphic to .

     

Splicing and the Casson invariant

We establish a formula for the Casson invariant of spliced sums of homology spheres along knots. Along the way, we show that the Casson invariant vanishes for spliced sums along knots in .

     

Cohomogeneity one actions on noncompact symmetric spaces of rank one

We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces , . For the quaternionic hyperbolic spaces , , we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classification problem was essentially solved by Élie Cartan.

     

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 .

     

Exactness of one relator groups

A discrete group is -exact if the reduced crossed product with converts a short exact sequence of --algebras into a short exact sequence of -algebras. A one relator group is a discrete group admitting a presentation where is a countable set and is a single word over . In this short paper we prove that all one relator discrete groups are -exact. Using the Bass-Serre theory we also prove that a countable discrete group acting without inversion on a tree is -exact if the vertex stabilizers of the action are -exact.

     

Nearly monotone spline approximation in

It is shown that the rate of -approximation of a non-decreasing function in , , by ``nearly non-decreasing" splines can be estimated in terms of the third classical modulus of smoothness (for uniformly spaced knots) and third Ditzian-Totik modulus (for Chebyshev knots), and that estimates in terms of higher moduli are impossible. It is known that these estimates are no longer true for ``purely" monotone spline approximation, and properties of intervals where the monotonicity restriction can be relaxed in order to achieve better approximation rate are investigated.

     

Centered complexity one Hamiltonian torus actions

We consider symplectic manifolds with Hamiltonian torus actions which are ``almost but not quite completely integrable": the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are ``centered" and the moment map is proper. In particular, this classifies the preimages under the moment map of all sufficiently small open sets, which is an important step towards global classification. As an application, we construct a full packing of each of the Grassmannians and by two equal symplectic balls.

     

Construction of non-alternating knots

We investigate the behaviour of Rasmussen's invariant  under the sharp operation on knots and obtain a lower bound for the sharp unknotting number. This bound leads us to an interesting move that transforms arbitrary knots into non-alternating knots.

     

Polynomial splittings of metabelian von Neumann rho-invariants of knots

We show that if the connected sum of two knots with coprime Alexander polynomials has vanishing von Neumann -invariants associated with certain metabelian representations, then so do both knots. As an application, we give a new example of an infinite family of knots which are linearly independent in the knot concordance group.

     

A geometric characterization of Vassiliev invariants

It is a well-known paradigm to consider Vassiliev invariants as polynomials on the set of knots. We prove the following characterization: a rational knot invariant is a Vassiliev invariant of degree if and only if it is a polynomial of degree on every geometric sequence of knots. Here a sequence with is called geometric if the knots coincide outside a ball , inside of which they satisfy for all and some pure braid . As an application we show that the torsion in the braid group over the sphere induces torsion at the level of Vassiliev invariants: there exist knots in that can be distinguished by -invariants of finite type but not by rational invariants of finite type. In order to obtain such torsion invariants we construct over a universal Vassiliev invariant of degree for knots in .

     

Tenth degree number fields with quintic fields having one real place

In this paper, we enumerate all number fields of degree of discriminant smaller than in absolute value containing a quintic field having one real place. For each one of the (resp. found fields of signature (resp. the field discriminant, the quintic field discriminant, a polynomial defining the relative quadratic extension, the corresponding relative discriminant, the corresponding polynomial over , and the Galois group of the Galois closure are given.

In a supplementary section, we give the first coincidence of discriminant of (resp. nonisomorphic fields of signature (resp. .

     

The class number one problem for some non-abelian normal CM-fields of degree 48

We prove that there is precisely one normal CM-field of degree 48 with class number one which has a normal CM-subfield of degree 16: the narrow Hilbert class field of with .

     

A classification and examples of rank one chain domains

A chain order of a skew field is a subring of so that implies Such a ring has rank one if , the Jacobson radical of is its only nonzero completely prime ideal. We show that a rank one chain order of is either invariant, in which case corresponds to a real-valued valuation of or is nearly simple, in which case and are the only ideals of or is exceptional in which case contains a prime ideal that is not completely prime. We use the group of divisorial of with the subgroup of principal to characterize these cases. The exceptional case subdivides further into infinitely many cases depending on the index of in Using the covering group of and the result that the group ring is embeddable into a skew field for a skew field, examples of rank one chain orders are constructed for each possible exceptional case.

     

Realizing alternating groups as monodromy groups of genus one covers

We prove that if , a generic Riemann surface of genus 1 admits a meromorphic function (i.e., an analytic branched cover of ) of degree such that every branch point has multiplicity and the monodromy group is the alternating group . To prove this theorem, we construct a Hurwitz space and show that it maps (generically) onto the genus one moduli space.

     

Exponential Gelfond-Khovanskii formula in dimension one

Gelfond and Khovanskii found a formula for the sum of the values of a Laurent polynomial over the zeros of a system of Laurent polynomials in . We expect that a similar formula holds in the case of exponential sums with real frequencies. Here we prove such a formula in dimension one.