Algebraic & Geometric Topology 8 (2008) 953–969

DOI: 10.2140/agt.2008.8.953

Abstract

Given integers g ≥ 2, n ≥ 1 we prove that there exist a collection of knots, denoted by K_g,n, fulfilling the following two conditions:
(1) For any integer 2 ≤ h ≤ g, there exist infinitely many knots K in K_g,n with g(E(K)) = h.
(2) For any m ≤ n, and for any collection of knots K₁,…,K_m in K_g,n, the Heegaard genus is additive: g(E(#_i=1^m K_i)) = ∑_i=1^m g(E(K_i)).
This implies the existence of counterexamples to Morimoto's Conjecture [Math. Ann. 317 (2000) 489–508].

Keywords

Heegaard splitting, tunnel number, knot, composite knot

Mathematical Subject Classification

Primary: 57M25

Secondary: 57M27

References

Publication

Received: 1 May 2007
Revised: 24 April 2008
Accepted: 28 April 2008
Published: 5 July 2008

Authors