Some sufficient conditions for tunnel numbers of connected sum of two knots not to go down |
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| Authors: | Guo Qiu Yang Feng Chun Lei |
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| Institution: | [1]School of Astronautics, Harbin Institute of Technology, Harbin 150001, P. R. China [2]Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China [3]School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, P. R. China |
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| Abstract: | In this paper, we show the following result: Let K
i
be a knot in a closed orientable 3-manifold M
i
such that (M
i
,K
i
) is not homeomorphic to (S
2 ×S
1, x
0 ×S
1), i = 1, 2. Suppose that the Euler Characteristic of any meridional essential surface in each knot complement E(K
i
) is less than the difference of one and twice of the tunnel number of K
i
. Then the tunnel number of their connected sum will not go down. If in addition that the distance of any minimal Heegaard
splitting of each knot complement is strictly more than 2, then the tunnel number of their connected sum is super additive. |
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| Keywords: | Tunnel number Heegaard splitting Heegaard distance meridional surface |
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