Symmetric ribbon disks

We study the ribbon discs that arise from a symmetric union presentation of a ribbon knot. A natural notion of symmetric ribbon number is introduced and compared with the classical ribbon number. We show that the gap between these numbers can be arbitrarily large by constructing an infinite family of ribbon knots with ribbon number 2 and arbitrarily large symmetric ribbon number. The proof is based on a particularly simple description of symmetric unions in terms of certain band diagrams which leads to an upper bound for the Heegaard genus of their branched double covers.Comment: 9 pages, 10 figures. Few typos corrected. Final version published in JKT

Automated Discovery of Internet Censorship by Web Crawling

Censorship of the Internet is widespread around the world. As access to the web becomes increasingly ubiquitous, filtering of this resource becomes more pervasive. Transparency about specific content that citizens are denied access to is atypical. To counter this, numerous techniques for maintaining URL filter lists have been proposed by various individuals and organisations that aim to empirical data on censorship for benefit of the public and wider censorship research community. We present a new approach for discovering filtered domains in different countries. This method is fully automated and requires no human interaction. The system uses web crawling techniques to traverse between filtered sites and implements a robust method for determining if a domain is filtered. We demonstrate the effectiveness of the approach by running experiments to search for filtered content in four different censorship regimes. Our results show that we perform better than the current state of the art and have built domain filter lists an order of magnitude larger than the most widely available public lists as of Jan 2018. Further, we build a dataset mapping the interlinking nature of blocked content between domains and exhibit the tightly networked nature of censored web resources

Knot concordance and homology sphere groups

We study two homomorphisms to the rational homology sphere group. If $\psi$ denotes the inclusion homomorphism from the integral homology sphere group, then using work of Lisca we show that the image of $\psi$ intersects trivially with the subgroup of the rational homology sphere group generated by lens spaces. As corollaries this gives a new proof that the cokernel of $\psi$ is infinitely generated, and implies that a connected sum $K$ of 2-bridge knots is concordant to a knot with determinant 1 if and only if $K$ is smoothly slice. Furthermore, if $\beta$ denotes the homomorphism from the knot concordance group defined by taking double branched covers of knots, we prove that the kernel of $\beta$ contains a $\mathbb{Z}^{\infty}$ summand by analyzing the Tristram-Levine signatures of a family of knots whose double branched covers all bound rational homology balls.Comment: 14 pages, 6 figures. Version 3 contains minor changes. This is essentially the version accepted for publication by International Mathematics Research Notices (IMRN

Rational approximation to the fractional Laplacian operator in reaction-diffusion problems

This paper provides a new numerical strategy to solve fractional in space reaction-diffusion equations on bounded domains under homogeneous Dirichlet boundary conditions. Using the matrix transform method the fractional Laplacian operator is replaced by a matrix which, in general, is dense. The approach here presented is based on the approximation of this matrix by the product of two suitable banded matrices. This leads to a semi-linear initial value problem in which the matrices involved are sparse. Numerical results are presented to verify the effectiveness of the proposed solution strategy

Rational cobordisms and integral homology

We consider the question of when a rational homology 3-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology embeds in any other element in the same class. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite rank cokernels. Further consequences include a divisibility condition between the determinants of a connected sum of 2-bridge knots and any other knot in the same concordance class. Lastly, we use knot Floer homology combined with our main result to obstruct Dehn surgeries on knots from being rationally cobordant to lens spaces.Comment: 19 pages, final version to appear in Compositio Mathematic

Embedding 3-manifolds in spin 4-manifolds

An invariant of orientable 3-manifolds is defined by taking the minimum $n$ such that a given 3-manifold embeds in the connected sum of $n$ copies of $S^2 \times S^2$, and we call this $n$ the embedding number of the 3-manifold. We give some general properties of this invariant, and make calculations for families of lens spaces and Brieskorn spheres. We show how to construct rational and integral homology spheres whose embedding numbers grow arbitrarily large, and which can be calculated exactly if we assume the 11/8-Conjecture. In a different direction we show that any simply connected 4-manifold can be split along a rational homology sphere into a positive definite piece and a negative definite piece.Comment: 27 pages, 14 figures. This is the final version. We made several corrections and small improvements, some suggested by the referee. This paper has been accepted for publication by the Journal of Topolog