2001/12/11 by Doo Ho Choi, Choi, Doo Ho, Ki Hyoung Ko +1 · 1 citation
Mathematics · #57M12 #57M25 #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:57M12 #msc:57M25
paper · pdf · doi:10.48550/arxiv.math/0112102
26 pages, 28 figures, a minor revision
arxiv created 2002/01/03 · arxiv updated 2009/11/30
A 1-bridge torus knot in a 3-manifold of genus ≤ 1 is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are similar to the Schubert's normal form and the Conway's normal form for 2-bridge knots. For a given Schubert's normal form we give algorithms to determine the number of components and to compute the fundamental group of the complement when the normal form determines a knot. We also give a description of the double branched cover of an ambient 3-manifold branched along a 1-bridge torus knot by using its Conway's normal form and obtain an explicit formula for the first homology of the double cover.