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Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

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

Author(s): Kanji Morimoto; Jennifer Schultens
Journal: Proc. Amer. Math. Soc. 128 (2000), 269-278.
MSC (1991): Primary 57M25, 57N10
Posted: 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 $t(K_1 \# K_2 \# \cdots \# K_n) \ge t(K_1) + t(K_2) + \cdots + t(K_n)$ 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: 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
Posted: September 9, 1999
Communicated by: Ronald A. Fintushel
Copyright of article: Copyright 1999, American Mathematical Society




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