Derivation of Schubert normal forms of 2-bridge knots from (1,1)-diagrams

A genus one 1-bridge knot (simply called a (1, 1)-knot) is a knot that can be decomposed into two trivial arcs embed in two solid tori in a genus one Heegaard splitting of a lens space. A (1,1)-knot can be described by a (1,1)-diagram D(a, b, c, r) determined by four integers a, b, c, and r. It is known that every 2-bride knot is a (1, 1)-knot and has a (1, 1)-diagram of the form D(a, 0, 1, r). In this talk, we give the dual diagram of D(a, 0, 1, r) explicitly and present how to derive a Schubert normal form of a 2-bridge knot from the dual diagram. This gives an alternative proof of the Grasselli and Mulazzani’s result asserting that D(a, 0, 1, r) is a (1, 1)-diagram of 2-bridge knot with a Schubert normal form b(2a+1, 2r).