A quadratic bound on the number of boundary slopes of essential surfaces with bounded genus

Li, Tao; Qiu, Ruifeng; Wang, Shicheng

doi:10.1007/s10711-009-9443-5

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2020

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“…In particular, if ∂M is a torus, there is a finiteness result about boundary slopes of essential surfaces in M as follows. For a general case, there are some results on it, see [6,7,13].…”

Section: Preliminariesmentioning

confidence: 99%

On tunnel numbers of a cable knot and its companion

Wang

Zou

2020

Preprint

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Let K be a nontrivial knot in S 3 and t(K) its tunnel number. For any (p ≥ 2, q)-slope in the torus boundary of a closed regular neighborhood of K in S 3 , denoted by K , it is a nontrivial cable knot in S 3 . Though t(K ) ≤ t(K) + 1, Example 1.1 in Section 1 shows that in some case, t(K ) ≤ t(K). So it is interesting to know when t(K ) = t(K) + 1.After using some combinatorial techniques, we prove that (1) for any nontrivial cable knot K and its companion K, t(K ) ≥ t(K); (2) if either K admits a high distance Heegaard splitting or p/q is far away from a fixed subset in the Farey graph, then t(K ) = t(K) + 1. Using the second conclusion, we construct a satellite knot and its companion so that the difference between their tunnel numbers is arbitrary large. 2010 Mathematics Subject Classification. 57M27.

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Section: Preliminariesmentioning

confidence: 99%