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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References
J.S. Birman, W.W. Menasco, Studying knots via braids VI: a non-finiteness theorem. Pac. J. Math. 156, 265–285 (1992)
P.R. Cromwell, Homogeneous links. J. Lond. Math. Soc. (series 2) 39, 535–552 (1989)
T. Kanenobu, Examples on polynomial invariants of knots and links II. Osaka J. Math. 26(3), 465–482 (1989)
L.H. Kauffman, State models and the Jones polynomial. Topology 26, 395–407 (1987)
H.R. Morton, (ed.), Problems, in Braids, Santa Cruz, 1986, ed. by J.S. Birman, A.L. Libgober. Contemporary Mathematics, vol. 78 (Cambridge University Press, Cambridge, 1986), pp. 557–574
K. Murasugi, J. Przytycki, The skein polynomial of a planar star product of two links. Math. Proc. Camb. Philos. Soc. 106(2), 273–276 (1989)
K. Murasugi, J. Przytycki, An index of a graph with applications to knot theory. Mem. Am. Math. Soc. 106(508) (1993)
K. Murasugi, On the braid index of alternating links. Trans. Am. Math. Soc. 326(1), 237–260 (1991)
K. Murasugi, Jones polynomial and classical conjectures in knot theory. Topology 26, 187–194 (1987)
Y. Ohyama, On the minimal crossing number and the braid index of links. Canad. J. Math. 45(1), 117–131 (1993)
A. Stoimenow, The skein polynomial of closed 3-braids. J. Reine Angew. Math. 564, 167–180 (2003)
A. Stoimenow, Diagram Genus, Generators and Applications. Monographs and Research Notes in Mathematics (T&F/CRC Press, Boca Raton, 2016). ISBN 9781498733809
A. Stoimenow, A. Vdovina, Counting alternating knots by genus. Math. Ann. 333, 1–27 (2005)
M.B. Thistlethwaite, A spanning tree expansion for the Jones polynomial. Topology 26, 297–309 (1987)
P. Vogel, Representation of links by braids: a new algorithm. Comment. Math. Helv. 65, 104–113 (1990)
S. Yamada, The minimal number of Seifert circles equals the braid index. Invent. Math. 88, 347–356 (1987)
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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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