Since our world is experienced locally in three-dimensional space, students of mathematics struggle to visualize and understand objects which do not fit into three-dimensional space. 3-manifolds are locally three-dimensional, but do not fit into 3-dimensional space and can be very complicated. Twist and bitwist are simple constructions that provide an easy path to both creating and understanding closed, orientable 3-manifolds. By starting with simple face pairings on a 3-ball, a myriad of 3-manifolds can be easily constructed. In fact, all closed, connected, orientable 3-manifolds can be developed in this manner. We call this work a tool kit to emphasize the ease with which 3-manifolds can be developed and understood applying the tools of twist and bitwist construction. We also show how two other methods for developing 3-manifolds–Dehn surgery and Heegaard splitting–are related to the twist and bitwist construction, and how one can transfer from one method to the others. One interesting result is that a simple bitwist construction on a 3-ball produces a group of manifolds called generalized Sieradski manifolds which are shown to be a cyclic branched cover of S^3 over the 2-braid, with the number twists determined by the hemisphere subdivisions. A slight change from bitwist to twist causes the knot to become a generalized figure-eight knot.
| Identifer | oai:union.ndltd.org:BGMYU2/oai:scholarsarchive.byu.edu:etd-3187 |
| Date | 13 July 2010 |
| Creators | Lambert, Lee R. |
| Publisher | BYU ScholarsArchive |
| Source Sets | Brigham Young University |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Theses and Dissertations |
| Rights | http://lib.byu.edu/about/copyright/ |
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