Algebraic & Geometric Topology

Knot exteriors with additive Heegaard genus and Morimoto's Conjecture

Tsuyoshi Kobayashi and Yo’av Rieck

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Given integers g2, n1 we prove that there exist a collection of knots, denoted by Kg,n, fulfilling the following two conditions:

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

(2) For any mn, and for any collection of knots K1,,KmKg,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].

Article information

Algebr. Geom. Topol., Volume 8, Number 2 (2008), 953-969.

Received: 1 May 2007
Revised: 24 April 2008
Accepted: 28 April 2008
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M27: Invariants of knots and 3-manifolds

Heegaard splitting tunnel number knot composite knot


Kobayashi, Tsuyoshi; Rieck, Yo’av. Knot exteriors with additive Heegaard genus and Morimoto's Conjecture. Algebr. Geom. Topol. 8 (2008), no. 2, 953--969. doi:10.2140/agt.2008.8.953.

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