On the signature of positive braids and adjacency for torus knots

The objects of study in this thesis are knots. More precisely, positive braid knots, which include algebraic knots and torus knots. In the first part of this thesis, we compare two classical knot invariants - the genus g and the signature σ - for positive braid knots. Our main result on positive braid knots establishes a linear lower bound for the signature in terms of the genus. In the second part of the thesis, a positive braid approach is applied to the study of the local behavior of polynomial functions from the complex affine plane to the complex numbers. After endowing polynomial function germs with a suitable topology, the adjacency problem arises: for a fixed germ f, what classes of germs g can be found arbitrarily close to f? We introduce two purely topological notions of adjacency for knots and discuss connections to algebraic notions of adjacency and the adjacency problem.

Cutting Arcs For Torus Links And Trees

Among all torus links, we characterise those arising as links of simple plane curve singularities by the property that their fibre surfaces admit only a finite number of cutting arcs that preserve fibredness. The same property allows a characterisation of Coxeter-Dynkin trees (i.e., An , Dn , E6 , E7 and E8 ) among all positive tree-like Hopf plumbings.

Diagnosis of endometriosis with imaging: a review

Endometriosis corresponds to ectopic endometrial glands and stroma outside the uterine cavity. Clinical symptoms include dysmenorrhoea, dyspareunia, infertility, painful defecation or cyclic urinary symptoms. Pelvic ultrasound is the primary imaging modality to identify and differentiate locations to the ovary (endometriomas) and the bladder wall. Characteristic sonographic features of endometriomas are diffuse low-level internal echos, multilocularity and hyperchoic foci in the wall. Differential diagnoses include corpus luteum, teratoma, cystadenoma, fibroma, tubo-ovarian abscess and carcinoma. Repeated ultrasound is highly recommended for unilocular cysts with low-level internal echoes to differentiate functional corpus luteum from endometriomas. Posterior locations of endometriosis include utero-sacral ligaments, torus uterinus, vagina and recto-sigmoid. Sonographic and MRI features are discussed for each location. Although ultrasound is able to diagnose most locations, its limited sensitivity for posterior lesions does not allow management decision in all patients. MRI has shown high accuracies for both anterior and posterior endometriosis and enables complete lesion mapping before surgery. Posterior locations demonstrate abnormal T2-hypointense, nodules with occasional T1-hyperintense spots and are easier to identify when peristaltic inhibitors and intravenous contrast media are used. Anterior locations benefit from the possibility of MRI urography sequences within the same examination. Rare locations and possible transformation into malignancy are discussed.

Bipartite graphs and quasipositive surfaces

We present a simple combinatorial model for quasipositive surfaces and positive braids, based on embedded bipartite graphs. As a first application, we extend the well-known duality on standard diagrams of torus links to twisted torus links. We then introduce a combinatorial notion of adjacency for bipartite graph links and discuss its potential relation with the adjacency problem for plane curve singularities.

From doubled Chern–Simons–Maxwell lattice gauge theory to extensions of the toric code

We regularize compact and non-compact Abelian Chern–Simons–Maxwell theories on a spatial lattice using the Hamiltonian formulation. We consider a doubled theory with gauge fields living on a lattice and its dual lattice. The Hilbert space of the theory is a product of local Hilbert spaces, each associated with a link and the corresponding dual link. The two electric field operators associated with the link-pair do not commute. In the non-compact case with gauge group R, each local Hilbert space is analogous to the one of a charged “particle” moving in the link-pair group space R2 in a constant “magnetic” background field. In the compact case, the link-pair group space is a torus U(1)2 threaded by k units of quantized “magnetic” flux, with k being the level of the Chern–Simons theory. The holonomies of the torus U(1)2 give rise to two self-adjoint extension parameters, which form two non-dynamical background lattice gauge fields that explicitly break the manifest gauge symmetry from U(1) to Z(k). The local Hilbert space of a link-pair then decomposes into representations of a magnetic translation group. In the pure Chern–Simons limit of a large “photon” mass, this results in a Z(k)-symmetric variant of Kitaev’s toric code, self-adjointly extended by the two non-dynamical background lattice gauge fields. Electric charges on the original lattice and on the dual lattice obey mutually anyonic statistics with the statistics angle . Non-Abelian U(k) Berry gauge fields that arise from the self-adjoint extension parameters may be interesting in the context of quantum information processing.

The M-theory origin of global properties of gauge theories

We show that global properties of gauge groups can be understood as geometric properties in M-theory. Different wrappings of a system of N M5-branes on a torus reduce to four-dimensional theories with AN−1 gauge algebra and different unitary groups. The classical properties of the wrappings determine the global properties of the gauge theories without the need to impose any quantum conditions. We count the inequivalent wrappings as they fall into orbits of the modular group of the torus, which correspond to the S-duality orbits of the gauge theories.

Low-energy energetic neutral atom imaging of Io plasma and neutral tori

Io's plasma and neutral tori play significant roles in the Jovian magnetosphere. We present feasibility studies of measuring low-energy energetic neutral atoms (LENAs) generated from the Io tori. We calculate the LENA flux between 10 eV and 3 keV. The energy range includes the corotational plasma flow energy. The expected differential flux at Ganymede distance is typically 10(3)-10(5) cm(-2) s(-1) sr(-1) eV(-1) near the energy of the corotation. It is above the detection level of the planned LENA sensor that is to be flown to the Jupiter system with integration times of 0.01-1 s. The flux has strong asymmetry with respective to the Io phase. The observations will exhibit periodicities, which can be attributed to the Jovian magnetosphere rotation and the rotation of Io around Jupiter. The energy spectra will exhibit dispersion signatures, because of the non-negligible flight time of the LENAs from Io to the satellite. In 2030, the Jupiter exploration mission JUICE will conduct a LENA measurement with a LENA instrument, the Jovian Neutrals Analyzer (JNA). From the LENA observations collected by JNA, we will be able to derive characteristic quantities, such as the density, velocity, velocity distribution function, and composition of plasma-torus particles. We also discuss the possible physics to be explored by JNA in addition to the constraints for operating the sensor and analyzing the obtained dataset. (C) 2015 Elsevier Ltd. All rights reserved.