Alternating surgeries

This thesis is concerned with the question of when the double branched cover of an alternating knot can arise by Dehn surgery on a knot in S^3. We approach this problem using a surgery obstruction, first developed by Greene, which combines Donaldson's Diagonalization Theorem with the $d$-invariants of Ozsvath and Szabo's Heegaard Floer homology. This obstruction shows that if the double branched cover of an alternating knot or link L arises by surgery on S^3, then for any alternating diagram the lattice associated to the Goeritz matrix takes the form of a changemaker lattice. By analyzing the structure of changemaker lattices, we show that the double branched cover of L arises by non-integer surgery on S^3 if and only if L has an alternating diagram which can be obtained by rational tangle replacement on an almost-alternating diagram of the unknot. When one considers half-integer surgery the resulting tangle replacement is simply a crossing change. This allows us to show that an alternating knot has unknotting number one if and only if it has an unknotting crossing in every alternating diagram. These techniques also produce several other interesting results: they have applications to characterizing slopes of torus knots; they produce a new proof for a theorem of Tsukamoto on the structure of almost-alternating diagrams of the unknot; and they provide several bounds on surgeries producing the double branched covers of alternating knots which are direct generalizations of results previously known for lens space surgeries. Here, a rational number p/q is said to be characterizing slope for K in S^3 if the oriented homeomorphism type of the manifold obtained by p/q-surgery on K determines K uniquely. The thesis begins with an exposition of the changemaker surgery obstruction, giving an amalgamation of results due to Gibbons, Greene and the author. It then gives background material on alternating knots and changemaker lattices. The latter part of the thesis is then taken up with the applications of this theory.

Investigating the use of LiDAR scanning as a method for the measurement of timber distortion features

Measuring the extent to which a piece of structural timber has distorted at a macroscopic scale is fundamental to assessing its viability as a structural component. From the sawmill to the construction site, as structural timber dries, distortion can render it unsuitable for its intended purposes. This rejection of unusable timber is a considerable source of waste to the timber industry and the wider construction sector. As such, ensuring accurate measurement of distortion is a key step in addressing ineffciencies within timber processing. Currently, the FRITS frame method is the established approach used to gain an understanding of timber surface profile. The method, while reliable, is dependent upon relatively few measurements taken across a limited area of the overall surface, with a great deal of interpolation required. Further, the process is unavoidably slow and cumbersome, the immobile scanning equipment limiting where and when measurements can be taken and constricting the process as a whole. This thesis seeks to introduce LiDAR scanning as a new, alternative approach to distortion feature measurement. In its infancy as a measurement technique within timber research, the practicalities of using LiDAR scanning as a measurement method are herein demonstrated, exploiting many of the advantages the technology has over current approaches. LiDAR scanning creates a much more comprehensive image of a timber surface, generating input data multiple magnitudes larger than that of the FRITS frame. Set-up and scanning time for LiDAR is also much quicker and more flexible than existing methods. With LiDAR scanning the measurement process is freed from many of the constraints of the FRITS frame and can be done in almost any environment. For this thesis, surface scans were carried out on seven Sitka spruce samples of dimensions 48.5x102x3000mm using both the FRITS frame and LiDAR scanner. The samples used presented marked levels of distortion and were relatively free from knots. A computational measurement model was created to extract feature measurements from the raw LiDAR data, enabling an assessment of each piece of timber to be carried out in accordance with existing standards. Assessment of distortion features focused primarily on the measurement of twist due to its strong prevalence in spruce and the considerable concern it generates within the construction industry. Additional measurements of surface inclination and bow were also made with each method to further establish LiDAR's credentials as a viable alternative. Overall, feature measurements as generated by the new LiDAR method compared well with those of the established FRITS method. From these investigations recommendations were made to address inadequacies within existing measurement standards, namely their reliance on generalised and interpretative descriptions of distortion. The potential for further uses of LiDAR scanning within timber researches was also discussed.