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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Tunnel numbers of small knots
do not go down under connected sum


Authors: Kanji Morimoto and Jennifer Schultens
Journal: Proc. Amer. Math. Soc. 128 (2000), 269-278
MSC (1991): Primary 57M25, 57N10
Published electronically: September 9, 1999
MathSciNet review: 1641065
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Abstract: Let $K_1$ and $K_2$ be two knots in $S^3$ and $t(K_1)$, $t(K_2)$ the tunnel numbers of them. In this paper, we show that if both $K_1$ and $K_2$ are small, then $t(K_1 \# K_2) \ge t(K_1) + t(K_2)$. Moreover we show that $t(K_1 \# K_2 \# \cdots \# K_n)\linebreak \ge t(K_1) + t(K_2) + \cdots + t(K_n)$ for any small knots $K_1, K_2, \cdots , K_n$.


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Additional Information

Kanji Morimoto
Affiliation: Department of Mathematics, Takushoku University Tatemachi, Hachioji, Tokyo 193, Japan
Email: morimoto@la.takushoku-u.ac.jp

Jennifer Schultens
Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
Email: jcs@mathcs.emory.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05160-6
PII: S 0002-9939(99)05160-6
Keywords: Knots, connected sum, tunnel number
Received by editor(s): March 1, 1998
Published electronically: September 9, 1999
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 1999 American Mathematical Society