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

Morimoto, Kanji; Schultens, Jennifer

doi:10.1090/S0002-9939-99-05160-6

Tunnel numbers of small knots do not go down under connected sum
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by Kanji Morimoto and Jennifer Schultens PDF

Proc. Amer. Math. Soc. 128 (2000), 269-278 Request permission

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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