Volume 8, issue 2 (2008)

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Knot exteriors with additive Heegaard genus and Morimoto's Conjecture

Tsuyoshi Kobayashi and Yo’av Rieck

Algebraic & Geometric Topology 8 (2008) 953–969
Abstract

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:

g(E(#i=1mK i)) = i=1mg(E(Ki)).

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 2000
Primary: 57M25
Secondary: 57M27
References
Publication
Received: 1 May 2007
Revised: 24 April 2008
Accepted: 28 April 2008
Published: 5 July 2008
Authors
Tsuyoshi Kobayashi
Department of Mathematics
Nara Women’s University
Kitauoya-Nishimachi
Nara, 630-8506
Japan
Yo’av Rieck
Department of Mathematical Sciences
University of Arkansas
Fayetteville, AR 72701