Computing rotation and self-linking numbers in contact surgery diagrams

We give an explicit formula to compute the rotation number of a nullhomologous Legendrian knot in contact (1/n)-surgery diagrams along Legendrian links and obtain a corresponding result for the self-linking number of transverse knots. Moreover, we extend the formula by Ding-Geiges-Stipsicz for computing the d3-invariant to (1/n)-surgeries.Comment: 19 pages, 6 figures; V2: Added the section on the d3-invariant and fixed a few misprints; V3: Minor corrections and clarifications; V4: Added a missing "n" in the formula for computing the Euler class of the contact structure in Theorem 5.

Microstructural and physical aspects of heat treated wood, part 1: softwood

Heat treatment of wood is an effective method to improve the dimensional stability and durability against biodegradation. Optimisation of a two-stage heat treatment process at relatively mild conditions (<200°C) and its effect on the anatomical structure of softwoods were investigated by means of a light and scanning electron microscopic analysis. Heat treatment did have an effect on the anatomical structure of wood, although this depends on the wood species considered and on the process method and conditions used. Softwood species with narrow annual rings and/or an abrupt transition from earlywood into latewood were sensitive to tangential cracks in the latewood section. Radial cracks occurred mainly in impermeable wood species such as Norway spruce, caused by large stresses in the wood structure during treatment. Sapwood of treated pine species revealed some damage to parenchyma cells in the rays and epithelial cells around resin canals, whereas this phenomenon has not been noticed in the heartwood section. Treated radiata pine resulted in a very open and permeable wood structure limiting the applications of this species. Broken cell walls perpendicular to the fibre direction resulting in transverse ruptures have been noticed in treated softwood species. This contributes to abrupt fractures of treated wood as observed in bending tests which can lead to considerably different failure behavior after impact or mechanical stress. In some treated softwood species maceration (small cracks between tracheids) was noticed after heat treatment. Heat treatment did not cause damage to the ray parenchyma pit membranes, bordered pits and large window pit membranes; the margo fibrils appeared without damage. Compared to the other softwood timbers tested European grown Douglas fir was the timber that stands heat treatment the best

Trisecting a 4-dimensional book into three chapters

We describe an algorithm that takes as input an open book decomposition of a closed oriented 4-manifold and outputs an explicit trisection diagram of that 4-manifold. Moreover, a slight variation of this algorithm also works for open books on manifolds with non-empty boundary and for 3-manifold bundles over the circle. We apply this algorithm to several simple open books, demonstrate that it is compatible with various topological constructions, and argue that it generalizes and unifies several previously known constructions.Comment: 29 pages, 15 figure

Complexity of equal 0-surgeries

We say that two knots are friends if they share the same 0-surgery. Two friends with different sliceness status would provide a counterexample to the 4-dimensional smooth Poincar\'e conjecture. Here we create a census of all friends with small crossing numbers c and tetrahedral complexities t, and compute their smooth 4-genera. In particular, we compute the minimum of c(K)+c(K') and of t(K)+t(K') among all friends K and K'. Along the way, we classify all 0-surgeries of knots of at most 15 crossings.Comment: 13 pages, 1 figure, 5 Table

Contact surgery graphs

We define a graph encoding the structure of contact surgery on contact 3-manifolds and analyze its basic properties and some of its interesting subgraphs.Comment: 11 pages, 3 figure