A Toolkit for the Construction and Understanding of 3-Manifolds

• Main Author: Lambert, Lee R.
• Format: Others
• Published: BYU ScholarsArchive 2010
• Subjects: Twist construction; bitwist construction; 3-manifolds; Dehn surgery; Heegaard splitting; Heegaard diagram; Mathematics
• Online Access: https://scholarsarchive.byu.edu/etd/2188; https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=3187&context=etd

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.