Date of Degree
PhD (Doctor of Philosophy)
Dehn surgery and the notion of reducible manifolds are both important tools in the study of 3-manifolds. The Cabling Conjecture of Francisco González-Acuña and Hamish Short describes the purported circumstances under which Dehn surgery can produce a reducible manifold. This thesis extends the work of James Allen Hoffman, who proved the Cabling Conjecture for knots of bridge number up to four. Hoffman built upon the combinatorial machinery used by Cameron Gordon and John Luecke in their solution to the knot complement problem. The combinatorial approach starts with the graphs of intersection of a thin level sphere of the knot and the reducing sphere in the surgered manifold. Gordon and Luecke's proof then proceeds by induction on certain cycles. Hoffman provides more insight into the structure of the base case of the induction (i.e. in an innermost cycle or a graph containing no such cycles). Hoffman uses this structure in a case-by-case proof of the Cabling Conjecture for knots of bridge number up to four. We find trees with specific properties in the graph of intersection, and use them to provethe existence of structure which provides lower bounds on the number of the aforementioned innermost cycles. Our results combined with a recent lower bound on the number of vertices inside the innermost cycles succinctly prove the conjecture for bridge number up to five and suggests an approach to the conjecture for knots of higher bridge number.
The Earth's curvature is invisible to us, with our feet planted firmly on the ground. Yet the Earth is round. By the very roundness of the Earth, by our successes at leaving its surface, and by our study of the vastness through which it travels, we are compelled to be curious about the nature of 3-dimensional space.
Before we recognized the Earth as round, we first had to recognize that round things existed, and consider their properties. Such steps are also crucial in considering 3-dimensional spaces. A useful method for determining properties of a 3-dimensional space is to ascertain how precisely that space differs from normal 3-dimensional space, and to ask what types of differences lead to which kinds of properties.
The Cabling Conjecture is a suggested answer to a question of this form. In this work, the conjecture is proven (and the answer is therefore verified) for certain cases, and a framework is outlined which may be useful in proving the conjecture in general.
publicabstract, Geometric Topology
Copyright 2016 Colin Grove