The size of triangulations supporting a given link

Let T be a triangulation of S^3 containing a link L in its 1-skeleton. We give an explicit lower bound for the number of tetrahedra of T in terms of the bridge number of L. Our proof is based on the theory of almost normal surfaces.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper13.abs.htm

Schrijver graphs and projective quadrangulations

In a recent paper [J. Combin. Theory Ser. B}, 113 (2015), pp. 1-17], the authors have extended the concept of quadrangulation of a surface to higher dimension, and showed that every quadrangulation of the $n$-dimensional projective space $P^n$ is at least $(n+2)$-chromatic, unless it is bipartite. They conjectured that for any integers $k\geq 1$ and $n\geq 2k+1$, the Schrijver graph $SG(n,k)$ contains a spanning subgraph which is a quadrangulation of $P^{n-2k}$. The purpose of this paper is to prove the conjecture

Converting between quadrilateral and standard solution sets in normal surface theory

The enumeration of normal surfaces is a crucial but very slow operation in algorithmic 3-manifold topology. At the heart of this operation is a polytope vertex enumeration in a high-dimensional space (standard coordinates). Tollefson's Q-theory speeds up this operation by using a much smaller space (quadrilateral coordinates), at the cost of a reduced solution set that might not always be sufficient for our needs. In this paper we present algorithms for converting between solution sets in quadrilateral and standard coordinates. As a consequence we obtain a new algorithm for enumerating all standard vertex normal surfaces, yielding both the speed of quadrilateral coordinates and the wider applicability of standard coordinates. Experimentation with the software package Regina shows this new algorithm to be extremely fast in practice, improving speed for large cases by factors from thousands up to millions.Comment: 55 pages, 10 figures; v2: minor fixes only, plus a reformat for the journal styl

Averages of Fourier coefficients of Siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables

Let $-d$ be a a negative discriminant and let $T$ vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant $-d$. We prove an asymptotic formula for $d \to \infty$ for the average over $T$ of the number of representations of $T$ by an integral positive definite quaternary quadratic form and obtain results on averages of Fourier coefficients of linear combinations of Siegel theta series. We also find an asymptotic bound from below on the number of binary forms of fixed discriminant $-d$ which are represented by a given quaternary form. In particular, we can show that for growing $d$ a positive proportion of the binary quadratic forms of discriminant $-d$ is represented by the given quaternary quadratic form.Comment: v5: Some typos correcte

Quadrilateral-octagon coordinates for almost normal surfaces

Normal and almost normal surfaces are essential tools for algorithmic 3-manifold topology, but to use them requires exponentially slow enumeration algorithms in a high-dimensional vector space. The quadrilateral coordinates of Tollefson alleviate this problem considerably for normal surfaces, by reducing the dimension of this vector space from 7n to 3n (where n is the complexity of the underlying triangulation). Here we develop an analogous theory for octagonal almost normal surfaces, using quadrilateral and octagon coordinates to reduce this dimension from 10n to 6n. As an application, we show that quadrilateral-octagon coordinates can be used exclusively in the streamlined 3-sphere recognition algorithm of Jaco, Rubinstein and Thompson, reducing experimental running times by factors of thousands. We also introduce joint coordinates, a system with only 3n dimensions for octagonal almost normal surfaces that has appealing geometric properties.Comment: 34 pages, 20 figures; v2: Simplified the proof of Theorem 4.5 using cohomology, plus other minor changes; v3: Minor housekeepin

Maximal admissible faces and asymptotic bounds for the normal surface solution space

The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but non-linear and non-convex constraint. The main results of this paper are significant improvements upon the best known asymptotic bounds on the number of admissible vertices, using polytopes in both the standard normal surface coordinate system and the streamlined quadrilateral coordinate system. To achieve these results we examine the layout of admissible points within these polytopes. We show that these points correspond to well-behaved substructures of the face lattice, and we study properties of the corresponding "admissible faces". Key lemmata include upper bounds on the number of maximal admissible faces of each dimension, and a bijection between the maximal admissible faces in the two coordinate systems mentioned above.Comment: 31 pages, 10 figures, 2 tables; v2: minor revisions (to appear in Journal of Combinatorial Theory A

Treatment of retracted, postsurgical scars and reduction of locoregional edema using a combined three-dimensional approach of liposuction lipofilling, dissecting cannulas, and suspension sutures

Background : Post-surgical abdominal and inguinal scars are a frequent challenge in plastic surgery. There are limited non-invasive alternatives to address depressed and retracted scars. The associated retraction and fibrosis might cause lymphatic dysfunction with subsequent regional edema. The authors describe a combined surgical approach of liposuction, the use of dissecting cannulas, lipofilling, and Scarpa’s fascia suspension sutures in a prospective case series. Methods: The proposed procedure was performed in 22 consecutive patients between November 2012 and May 2015. Complications were assessed according to the Clavien-Dindo scale. Postoperative psychosocial, edema reduction, and patient satisfaction outcomes were gathered and analyzed based on blinded questionaries (Rosenberg Self-Esteem scale and a Cosmetic Procedures Screening Questionnaire (COPS)). Results: At 6 months, no major complications and 27.2% minor complications (Clavien-Dindo 1) were recorded. Four patients had superficial infections that settled with oral antibiotics and two patients developed a seroma. A significant improvement in self-esteem, aesthetic satisfaction, and social competence was found postoperatively in all patients. The novel technique reduced regional edema and scar-related self-consciousness. Patient satisfaction was rated very high, and all patients would recommend this surgery for abdominal or inguinal retracted scars. Conclusions: This study shows that the proposed technique is a safe minimally invasive alternative for the treatment of abdominal and inguinal retracted scars. The relatively high rate of minor complications is mainly due to the strict definition of the scale used. The results showed an improvement of local edema and high patient satisfaction. Level of evidence: Level IV, therapeuti

Relevance and Recent Developments of Chitosan in Peripheral Nerve Surgery

Developments in tissue engineering yield biomaterials with different supporting strategies to promote nerve regeneration. One promising material is the naturally occurring chitin derivate chitosan. Chitosan has become increasingly important in various tissue engineering approaches for peripheral nerve reconstruction, as it has demonstrated its potential to interact with regeneration associated cells and the neural microenvironment, leading to improved axonal regeneration and less neuroma formation. Moreover, the physiological properties of its polysaccharide structure provide safe biodegradation behavior in the absence of negative side effects or toxic metabolites. Beneficial interactions with Schwann cells (SC), inducing differentiation of mesenchymal stromal cells to SC-like cells or creating supportive conditions during axonal recovery are only a small part of the effects of chitosan. As a result, an extensive body of literature addresses a variety of experimental strategies for the different types of nerve lesions. The different concepts include chitosan nanofibers, hydrogels, hollow nerve tubes, nerve conduits with an inner chitosan layer as well as hybrid architectures containing collagen or polyglycolic acid nerve conduits. Furthermore, various cell seeding concepts have been introduced in the preclinical setting. First translational concepts with hollow tubes following nerve surgery already transferred the promising experimental approach into clinical practice. However, conclusive analyses of the available data and the proposed impact on the recovery process following nerve surgery are currently lacking. This review aims to give an overview on the physiologic properties of chitosan, to evaluate its effect on peripheral nerve regeneration and discuss the future translation into clinical practice

On the critical pair theory in abelian groups : Beyond Chowla's Theorem

We obtain critical pair theorems for subsets S and T of an abelian group such that |S+T| < |S|+|T|+1. We generalize some results of Chowla, Vosper, Kemperman and a more recent result due to Rodseth and one of the authors.Comment: Submitted to Combinatorica, 23 pages, revised versio

A survey of χ\chi-boundedness

If a graph has bounded clique number, and sufficiently large chromatic number, what can we say about its induced subgraphs? Andr\'as Gy\'arf\'as made a number of challenging conjectures about this in the early 1980's, which have remained open until recently; but in the last few years there has been substantial progress. This is a survey of where we are now