The Existence of a Meridional Curve in Closed Incompressible Surfaces in Fully Alternating Link Complements

Wei Lin

Chinese Annals of Mathematics, Series B ›› 2024, Vol. 45 ›› Issue (1) : 73 -80.

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Chinese Annals of Mathematics, Series B ›› 2024, Vol. 45 ›› Issue (1) : 73 -80. DOI: 10.1007/s11401-024-0004-x
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The Existence of a Meridional Curve in Closed Incompressible Surfaces in Fully Alternating Link Complements

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Abstract

Menasco showed that a closed incompressible surface in the complement of a non-split prime alternating link in S 3 contains a circle isotopic in the link complement to a meridian of the links. Based on this result, he was able to argue the hyperbolicity of non-split prime alternating links in S 3. Adams et al. showed that if FS × I L is an essential torus, then F contains a circle which is isotopic in S × I \ L to a meridian of L. The author generalizes his result as follows: Let S be a closed orientable surface, L be a fully alternating link in S × I. If FS × I \ L is a closed essential surface, then F contains a circle which is isotopic in S × I \ L to a meridian of L.

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Fully alternating / Incompressible surfaces / Meridionally incompressible

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Wei Lin. The Existence of a Meridional Curve in Closed Incompressible Surfaces in Fully Alternating Link Complements. Chinese Annals of Mathematics, Series B, 2024, 45(1): 73-80 DOI:10.1007/s11401-024-0004-x

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