Unknotting Numbers are not Realized in Minimal Projections for a Class of Rational Knots

Abstract

In previous results, Bleiler and Nakanishi produced an example of a knot where the unknotting number was not realized in a minimal projection of the knot. Bernhard generalied this example to an infi{}nite class of examples with Conway notation $\left(2j+1,1,2j\right)$ with j $\geq$ 2. In this paper we examine the entire class of knots given in Conway notation by (2j + 1, 2k + 1, 2j) where j $\geq$ 1 and k $\geq$ 0 and we determine that a large class of knots of this form have the unknotting number not realized in a minimal projection. We also produce an infi{}nite class of two component links with unknotting number gap arbitrarily large.