The Jones polynomial and boundary slopes of alternating knots

Cindy Curtis, The College of New Jersey

Abstract: The boundary slopes of essential surfaces in the complement of a knot have been a valuable tool in the study of the knot. These slopes play a key role in the definitions of the A-polynomial of the knot and of the Culler-Gordon-Luecke-Shalen norms of the knot. We show that for alternating knots the Jones polynomial detects the maximal and minimal integral boundary slopes among all essential surfaces in the knot complement. For alternating Montesinos knots, all boundary slopes of essential surfaces are integral, so the Jones polynomial detects the maximal and minimal boundary slopes.