Knot exteriors with additive Heegaard genus and Morimoto’s Conjecture

Kobayashi, Tsuyoshi; Rieck, Yo’av

doi:10.2140/agt.2008.8.953

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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

DOI: 10.2140/agt.2008.8.953

Abstract

Given integers $g \geq 2$ , $n \geq 1$ we prove that there exist a collection of knots, denoted by $K_{g, n}$ , fulfilling the following two conditions:

(1) For any integer $2 \leq h \leq g$ , there exist infinitely many knots $K \in K_{g, n}$ with $g (E (K)) = h$ .

(2) For any $m \leq n$ , and for any collection of knots $K_{1}, \dots, K_{m} \in K_{g, n}$ , the Heegaard genus is additive:

g (E (#_{i = 1}^{m} K_{i})) = \sum_{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 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

Volume 8, issue 2 (2008)