We construct knots in with Heegaard splittings of arbitrarily high distance, in any genus. As an application, for any positive integers and we find a tunnel number knot in the three-sphere which has no –decomposition.
"High distance knots." Algebr. Geom. Topol. 7 (3) 1471 - 1483, 2007. https://doi.org/10.2140/agt.2007.7.1471