Mieczyslaw K. Dabkowski

Non-left-orderable 3-manifold groups

We show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched coverings of S3 branched along links. The figure eight knot provides simple nontrivial examples. The groups arising in these examples are known as Fibonacci groups which we show not to be left-orderable. Many other examples of non-orderable groups are obtained by taking 3-fold branched covers of S3 branched along various hyperbolic 2-bridge knots. The manifold obtained in such a way from the 52 knot is of special interest as it is conjectured to be the hyperbolic 3-manifold with the smallest volume.

Unexpected connections between Burnside groups and knot theory

In classical knot theory and the theory of quantum invariants substantial effort was directed toward the search for unknotting moves on links. We solve, in this article, several classical problems concerning unknotting moves. Our approach uses a concept, Burnside groups of links, that establishes an unexpected relationship between knot theory and group theory. Our method has the potential to be used in computational biology in the analysis of DNA via tangle embedding theory, as developed by D. W. Sumners [Sumners, D. W., ed. (1992) New Scientific Applications of Geometry and Topology (Am Math. Soc., Washington, DC) and Ernst, C. & Sumners, D. W. (1999) Math. Proc. Cambridge Philos. Soc. 126, 23-36].

Burnside obstructions to the Montesinos-Nakanishi 3-move conjecture

Yasutaka Nakanishi asked in 1981 whether a 3–move is an unknotting operation. In Kirby’s problem list, this question is called The Montesinos–Nakanishi 3– move conjecture. We define the nth Burnside group of a link and use the 3rd Burnside group to answer Nakanishi’s question; ie, we show that some links cannot be reduced to trivial links by 3–moves.

COMPACTNESS OF THE SPACE OF LEFT ORDERS

A left order on a magma (e.g. semigroup) is a total order of its elements that is left invariant under the magma operation. A natural topology can be introduced on the set of all left orders of an arbitrary magma. We prove that this topological space is compact. Interesting examples of nonassociative magmas, whose spaces of right orders we analyze, come from knot theory and are called quandles. Our main result establishes an interesting connection between topological properties of the space of left orders on a group, and the classical algebraic result by Conrad [4] and Łoś [13] concerning the existence of left orders.

Active Integrated Filters for RF-Photonic Channelizers

A theoretical study of RF-photonic channelizers using four architectures formed by active integrated filters with tunable gains is presented. The integrated filters are enabled by two- and four-port nano-photonic couplers (NPCs). Lossless and three individual manufacturing cases with high transmission, high reflection, and symmetric couplers are assumed in the work. NPCs behavior is dependent upon the phenomenon of frustrated total internal reflection. Experimentally, photonic channelizers are fabricated in one single semiconductor chip on multi-quantum well epitaxial InP wafers using conventional microelectronics processing techniques. A state space modeling approach is used to derive the transfer functions and analyze the stability of these filters. The ability of adapting using the gains is demonstrated. Our simulation results indicate that the characteristic bandpass and notch filter responses of each structure are the basis of channelizer architectures, and optical gain may be used to adjust filter parameters to obtain a desired frequency magnitude response, especially in the range of 1–5 GHz for the chip with a coupler separation of ∼9 mm. Preliminarily, the measurement of spectral response shows enhancement of quality factor by using higher optical gains. The present compact active filters on an InP-based integrated photonic circuit hold the potential for a variety of channelizer applications. Compared to a pure RF channelizer, photonic channelizers may perform both channelization and down-conversion in an optical domain.

Spaces of orders and their Turing degree spectra

Abstract We investigate computability theoretic and topological properties of spaces of orders on computable orderable groups. A left order on a group G is a linear order of the domain of G , which is left-invariant under the group operation. Right orders and bi-orders are defined similarly. In particular, we study groups for which the spaces of left orders are homeomorphic to the Cantor set, and their Turing degree spectra contain certain upper cones of degrees. Our approach unifies and extends Sikora’s (2004) [28] investigation of orders on groups in topology and Solomon’s (2002) [31] investigation of these orders in computable algebra. Furthermore, we establish that a computable free group F n of rank n > 1 has a bi-order in every Turing degree.

NEW INVARIANT OF 4-MOVES

Study of equivalence classes of links up to n-moves plays an important role in the theory of invariants based on the skein relation and, in particular, skein modules. In this paper, we consider Nakanishis 4-move conjecture [12]. The modification of the conjecture to 2-component link (homotopically trivial links) is a question proposed by Kawauchi [10]. We define a new invariant of links which is preserved by 4-moves and analyze its potential strength. In particular, we show that our invariant allows us to obtain results of [8, 9, 13] concerning 4-moves.

Turing degrees of nonabelian groups

For a countable structure A, the (Turing) degree spectrum of A is the set of all Turing degrees of its isomorphic copies. If the degree spectrum of A has the least degree d, then we say that d is the (Turing) degree of the isomorphism type of A. So far, degrees of the isomorphism types have been studied for abelian and metabelian groups. Here, we focus on highly nonabelian groups. We show that there are various centerless groups whose isomorphism types have arbitrary Turing degrees. We also show that there are various centerless groups whose isomorphism types do not have Turing degrees.

Experiment and Theory of an Active Optical Filter

The role of gain in an optical filter is advanced by good agreement between theory and experiment presented herein. The particular integrated photonic filter is composed of four semiconductor optical amplifiers and one four-port coupler located at the intersection of the amplifiers. The four-port coupler is realized using frustrated total internal reflection off a very thin slab of alumina embedded in the substrate. The delta function response of the filter is measured using an ultra-fast laser and cross-correlator, and the measured transfer functions agree well with a z-transform-based description of the device.

GROUPS WITH ORDERINGS OF ARBITRARY ALGORITHMIC COMPLEXITY

We give general sufficient conditions that a computable group admit bi-orderings of arbitrary computability-theoretic complexity in a strong sense. We apply this result to show that a large class of computable, finitely presented, residually nilpotent groups admit bi-orderings in every truth-table degree, a refinement of the Turing degrees. This class includes a wide variety of important groups such as finitely generated free groups, surface groups and certain nilpotent groups.