Abstract
The combination of the identity (3.2) and the skein-Jones substitution (2.5) was already used in Section 4.2 to translate the determination of the 3-braid link genus from P to V. A similar line of thought will now enable us to extend the other main result in [St2], the description of alternating links of braid index 3. This result was motivated by the work of Murasugi [Mu], and Birman’s problem in [Mo2] how to relate braid representations and diagrammatic properties of links. We will see how via (2.5) and the famous Kauffman–Murasugi–Thistlethwaite theorem [Ka2, Mu3, Th2] the Jones polynomial enters in a new way into the braid representation picture. The argument will lead to the braid index 3 result surprisingly easily, and then also to the classification for braid index 4 (which seems out of scope with the methods in [St2] alone). We also obtain a good description of the general (braid index) case.
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Stoimenow, A. (2017). Studying Alternating Links by Braid Index. In: Properties of Closed 3-Braids and Braid Representations of Links . SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-68149-8_6
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DOI: https://doi.org/10.1007/978-3-319-68149-8_6
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