Browsing by Subject "Low-dimensional topology"
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Item The combinatorics of reducible Dehn surgeries(2015-05) Zufelt, Nicholas Troy; Gordon, Cameron, 1945-; Gompf, Robert; Luecke, John; Reid, Alan; Thompson, AbigailWe study reducible Dehn surgeries on nontrivial knots in S³. The conjectured classification of such surgeries is known as the Cabling Conjecture, and partial progress toward the conjecture often comes in the form of a statement that an arbitrary reducible surgery resembles a cabled reducible surgery. One such resemblance is the Two Summands Conjecture: Dehn surgery on a knot in S³ can only produce a manifold with at most two irreducible connected summands. In the event that a reducible surgery on a knot K in S³ of slope r produces a manifold with more than two such summands, we show that |r| ≤ b, where b denotes the bridge number of K. As a consequence, we rule out this possibility for knots with b ≤ 5 and for positive braid closures. We also study reducible Dehn surgeries without the assumption that the reducible manifold contains more than two connected summands. Specifically, if P is an essential planar surface in the exterior of a hyperbolic knot which completes to a reducing sphere in this surgery, then it is shown that the number of boundary components of P is at least ten.Item On toroidal knots, chirality, and molecules(2014-09-29) Garza, RamiroItem Primitive/primitive and primitive/Seifert knots(2011-05) Guntel, Brandy Jean; Gordon, Cameron, 1945-; Allcock, Daniel; Luecke, John; Morrison, Phil; Reid, AlanBerge introduced knots that are primitive/primitive with respect to the standard genus 2 Heegaard surface, F, for the 3-sphere; surgery on such knots at the surface slope yields a lens space. Later Dean described a similar class of knots that are primitive/Seifert with respect to F; surgery on these knots at the surface slope yields a Seifert fibered space. The examples Dean worked with are among the twisted torus knots. In Chapter 3, we show that a given knot can have distinct primitive/Seifert representatives with the same surface slope. In Chapter 4, we show that a knot can also have a primitive/primitive and a primitive/Seifert representative that share the same surface slope. In Section 5.2, we show that these two results are part of the same phenomenon, the proof of which arises from the proof that a specific class of twisted torus knots are fibered, demonstrated in Section 5.1.Item Properties of commensurability classes of hyperbolic knot complements(2011-05) Hoffman, Neil Reardon; Reid, Alan W.; Adams, Colin C.; Gordon, Cameron M.; Knopf, Daniel; Luecke, John E.This thesis investigates the topology and geometry of hyperbolic knot complements that are commensurable with other knot complements. In chapter 3, we provide an infinite family examples of hyperbolic knot complements commensurable with exactly two other knot complements. In chapter 4, we exhibit an obstruction to knot complements admitting exceptional surgeries in conjunction with hidden symmetries. Finally, in chapter 5, we discuss the role of surfaces embedded in 3-orbifolds as it relates to hidden symmetries.Item Some Constructions Involving Surgery on Surfaces in 4-manifolds(2015-12) Larson, Kyle Lee; Gompf, Robert E., 1957-; Akbulut, Selman; Gordon, Cameron; Luecke, John; Perutz, TimothyThis dissertation concerns embedded surfaces in smooth 4-manifolds and especially surgery on those surfaces. These cut and paste operations are a powerful tool in the study of smooth 4-manifolds, and we study these operations in several new contexts. We give applications to several different areas of low-dimensional topology, including embedding 3-manifolds into 4-manifolds, broken Lefschetz fibrations, slice knots, and the relationship between knots and 2-knots.