Is it possible to get a braid representation for a general Montesinos link with small number of strands? I know by Alexander's theorem it is possible to braid any link but is it possible to find a braid of index three for a Montesinos link?
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No, the braid index of a Montesinos link can be as large as one wishes.
One way to see this is as follows.
There is a lower bound for the braid index in terms of the Homfly polynomial due to Morton and Franks-Williams. In terms of the Jones polynomial it says $$b\ge\frac{d}{2}+1,$$ where $b$ is the braid index and $d$ the degree of the Jones polynomial (i.e., the difference between largest and smallest exponent).
On the other hand, Stoimenow has computed the Jones polynomial of Montesinos links in https://projecteuclid.org/download/pdf_1/euclid.jmsj/1191591855 and from his computations one sees that the degree $d$ can be as large as one wishes.
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4$\begingroup$ another way of seeing this is the following: A lower bound for the braid index is the bridge number (this is obvious) and the bridge number of Montesinos knots was computed by Boileau and Zieschang. $\endgroup$ Commented Feb 13, 2017 at 17:44
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$\begingroup$ How do you derive the bound from the Morton-Franks-Williams bound? It seems not correct to me. The torus knot T(5,2) has braid index 2 but Jones polynomial: t^2+ t^4-t^5+ t^6-t^7 $\endgroup$ Commented Apr 16, 2023 at 16:12
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$\begingroup$ This is explicitly stated as a corollary on the 3rd page of Franks-Williams 1987 paper in Trans.AMS. Yet your example seems to be right - confusingly. The paper is at ams.org/journals/tran/1987-303-01/S0002-9947-1987-0896009-2/… $\endgroup$– ThiKuCommented Apr 17, 2023 at 18:47
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$\begingroup$ Okay, I see, the confusion just comes from the fact, that what Franks-Williams call the Jones polynomial is not what everybody else calls by this name. So this answer seems to be wrong. Still the argument in Richard Weidmanns comment should be correct. $\endgroup$– ThiKuCommented Apr 17, 2023 at 18:50