Bitwist manifolds and two-bridge knots

James W. Cannon, William J. Floyd, LeeR Lambert, Walter R. Parry and Jessica S. Purcell

Vol. 284 (2016), No. 1, 1–39

DOI: 10.2140/pjm.2016.284.1

Abstract

We give uniform, explicit, and simple face-pairing descriptions of all the branched cyclic covers of the $3$-sphere, branched over two-bridge knots. Our method is to use the bitwisted face-pairing constructions of Cannon, Floyd, and Parry; these examples show that the bitwist construction is often efficient and natural. Finally, we give applications to computations of fundamental groups and homology of these branched cyclic covers.

Though LeeR Lambert spent his life as an actuary and a musician and was a loving \null father of nine girls and one boy, he had always wanted to earn an advanced degree as a mathematician. With the encouragement of his wife, he earned his Ph.D. in mathematics at the age of 68. Many of the results of this paper appeared in his Ph.D. dissertation at Brigham Young University. At the age of 71, LeeR died of bone cancer. We miss you, LeeR. \rightskip=14pt \leftskip=14pt

Keywords

bitwist manifolds, two-bridge knots, branched cyclic covers

Mathematical Subject Classification 2010

Primary: 57M12, 57M25

Milestones

Received: 26 May 2015

Revised: 28 January 2016

Accepted: 18 February 2016

Published: 10 July 2016