Knots of genus one or on the number of alternating knots of given genus (Q2718983)

A Gauss diagram of a knot diagram is an oriented circle with arrows connecting points on the circle mapped to a crossing in the knot diagram. The arrows are oriented so the arrows point from the preimage of an undercrossing to the preimage of the overcrossing. In this article Gauss diagrams belonging to knots with particular properties -- such as positive (or alternating) knots of a fixed genus -- are analyzed. Given an upper bound on the number of crossings, bounds are established on the number of twist reduced Gauss diagrams of such knots. Among several corollaries of the results on Gauss diagrams are the following statements: As a function of crossing number there are only polynomially many alternating (or positive) knots of a given genus (or given unknotting number). Any non-hyperbolic genus one knot except the trefoil does not have a minimal canonical Seifert surface.