Almost all 2-bridge knots are curly

Abstract

A spiral knot or link diagram (introduced in [C. Adams, R. Hudson, Rachel, R. Morrison, W. George, L. Starkston, S. Taylor and O. Turanova, The spiral index of knots, Math. Proc. Cambridge Philos. Soc. 149(2) (2010) 297–315]) is an oriented knot or link diagram where, when traversing through the planar diagram, the curvature does not change sign. An oriented knot or link type is called curly if it admits a spiral diagram with fewer maxima than the braid index of that knot or link type. We prove that all 2-bridge knots and links with a braid index greater than three are curly. We also show that many alternating oriented Montesinos links are curly.