A new representation of Links: Butterflies

With the idea of an eventual classification of 3-bridge links,\ we define a very nice class of 3-balls (called butterflies) with faces identified by pairs, such that the identification space is \(S^{3},\) and the image of a prefered set of edges is a link. Several examples are given. We prove that e...

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Published in:arXiv.org 2012-03
Main Authors: Hilden, H M, Montesinos, J M, Tejada, D M, Toro, M M
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Language:English
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Knot theory
Representations
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Montesinos, J M
Tejada, D M
Toro, M M
description With the idea of an eventual classification of 3-bridge links,\ we define a very nice class of 3-balls (called butterflies) with faces identified by pairs, such that the identification space is \(S^{3},\) and the image of a prefered set of edges is a link. Several examples are given. We prove that every link can be represented in this way (butterfly representation). We define the butterfly number of a link, and we show that the butterfly number and the bridge number of a link coincide. This is done by defining a move on the butterfly diagram. We give an example of two different butterflies with minimal butterfly number representing the knot \(8_{20}.\) This raises the problem of finding a set of moves on a butterfly diagram connecting diagrams representing the same link. This is left as an open problem.
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subjects Knot theory
Representations
title A new representation of Links: Butterflies
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