vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv I Orthogonal Graph Drawing 1 1

The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in three-dimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues. Because the domain of application is mathematics, topological drawing is also concerned with the correct representation and display of these objects on a computer. By correctness we mean that the essential topological features of objects are maintained during interaction. We have chosen to limit the scope of topological drawing to knot theory, a domain that consists essentially of one class of object (embedded circles in three-dimensional space) yet is rich enough to contain a wide variety of difficult problems of research interest. In knot theory, two embedded circles (knots) are considered equivalent if one may be smoothly deformed into the other without any cuts or self-intersections. This notion of equivalence may be thought of as the heart of knot theory. We present methods for the computer construction and interactive manipulation of a

We announce results about flat (linkless) embeddings of graphs in 3space. A piecewise-linear embedding of a graph in 3-space is called flat if every circuit of the graph bounds a disk disjoint from the rest of the graph. We have shown: (i) An embedding is flat if and only if the fundamental group of the complement in 3-space of the embedding of every subgraph is free. (ii) If two flat embeddings of the same graph are not ambient isotopic, then they differ on a subdivision of K5 or K3,3. (iii) Any flat embedding of a graph can be transformed to any other flat embedding of the same graph by “3-switches”, an analog of 2-switches from the theory of planar embeddings. In particular, any two flat embeddings of a 4-connected graph are either ambient isotopic, or one is ambient isotopic to a mirror image of the other. (iv) A graph has a flat embedding if and only if it has no minor isomorphic to one of seven specified graphs. These are the graphs that can be obtained from K6 by means of Y∆- and ∆Y-exchanges.

Abstract. A clear understanding of topology of higher-dimensional objects is important in many branches of both pure and applied mathematics. In this survey we attempt to present some results of higher-dimensional topology in a way which makes clear the visual and algebraic constructions appear naturally in the study of geometric problems. Before giving a general construction, we illustrate the main ideas in simple but important particular cases, in which the essence is not veiled by technicalities. More specifically, we present several classical and modern results on the embedding and knotting of manifolds in Euclidean space. We state many concrete results (in particular, recent explicit classification of knotted tori). Their statements (but not proofs!) are simple and accessible to non-specialists. We outline a general approach to embeddings via the classical van Kampen-Shapiro-Wu-Haefliger-Weber ’deleted product ’ obstruction. This approach reduces the isotopy classification of embeddings to the homotopy classification of equivariant maps, and so implies the above concrete results. We describe the revival of interest in this beautiful branch of topology, by presenting new results in this area (of Freedman, Krushkal, Teichner, Segal, Spie˙z and the author): a generalization the Haefliger-Weber embedding theorem below the metastable dimension range and examples showing that other analogues of this theorem are false outside the metastable dimension range. 1.

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In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the “rough ” structure of graphs excluding a fixed minor. This result was used to prove Wagner’s Conjecture that finite graphs are well-quasi-ordered under the graph minor relation. Recently, a number of beautiful results that use this structural result have appeared. Some of these along with some other recent advances on graph minors are surveyed. Keywords: Graph minor theory, Tree-width, Tree-decomposition, Path-decomposition, Complete

All rights reserved. This dissertation may not be reproduced in whole or in part, by mimeograph or other means, without the permission of the author. Supervisor: M. R. Fellows A substantial part of the history of graph theory deals with the study and classi-cation of sets of graphs that share common properties. One predominant trend is to characterize graph families by sets of minimal forbidden graphs (within some partial ordering on the graphs). For example, the famous Kuratowski Theorem classi es the planar graph family by two forbidden graphs (in the topological partial order). Most, if not all, of the current approaches for nding these forbidden substructure characterizations use extensive and specialized case analysis. Thus, until now, for a xed graph family,thistype of mathematical theorem proving often required months or even years of human e ort. The main focus of this dissertation is to develop a practical theory for automating (with distributed computer programming) this clas-sic part of graph theory. We extend and (more importantly) implement avariation

We prove that every embedding of K 10 in R 3 contains a non-split link of three components.

Abstract. The vanishing of Van Kampen’s obstruction is known to be necessary and sufficient for embeddability of a simplicial n-complex into R 2n for n ̸ = 2, and it was recently shown to be incomplete for n = 2. We use algebraic-topological invariants of four-manifolds with boundary to introduce a sequence of higher embedding obstructions for a class of 2-complexes in R 4. 1.

Abstract. We introduce new sufficient conditions for intrinsic knotting and linking. A graph on n vertices with at least 4n −9 edges is intrinsically linked. A graph on n vertices with at least 5n − 14 edges is intrinsically knotted. We also classify graphs that are 0, 1, or 2 edges short of being complete partite graphs with respect to intrinsic linking and intrinsic knotting. In addition, we classify intrinsic knotting of graphs on 8 vertices. 1.