Knot exteriors with additive Heegaard genus and Morimoto’s Conjecture

Kobayashi, Tsuyoshi¹ ; Rieck, Yo’av²

¹Department of Mathematics, Nara Women’s University, Kitauoya-Nishimachi, Nara, 630-8506, Japan
²Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701

Algebraic and Geometric Topology, Tome 8 (2008) no. 2, pp. 953-969

Résumé

Given integers g ≥ 2, n ≥ 1 we prove that there exist a collection of knots, denoted by Kg,n, fulfilling the following two conditions:

(1) For any integer 2 ≤ h ≤ g, there exist infinitely many knots K ∈Kg,n with g(E(K)) = h.

(2) For any m ≤ n, and for any collection of knots K1,…,Km ∈Kg,n, the Heegaard genus is additive:

This implies the existence of counterexamples to Morimoto’s Conjecture [Math. Ann. 317 (2000) 489–508].

Détail

DOI : 10.2140/agt.2008.8.953

Keywords: Heegaard splitting, tunnel number, knot, composite knot

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