Given integers , we prove that there exist a collection of knots, denoted by , fulfilling the following two conditions:
(1) For any integer , there exist infinitely many knots with .
(2) For any , and for any collection of knots , the Heegaard genus is additive:
This implies the existence of counterexamples to Morimoto’s Conjecture [Math. Ann. 317 (2000) 489–508].
"Knot exteriors with additive Heegaard genus and Morimoto's Conjecture." Algebr. Geom. Topol. 8 (2) 953 - 969, 2008. https://doi.org/10.2140/agt.2008.8.953