A diagrammatic approach for determining the braid index of alternating links

Abstract

This paper concerns the braid index of an alternating link. It is well known that the braid index of any link equals the minimum number of Seifert circles among all link diagrams representing it. For an alternating link represented by a reduced alternating diagram [Formula: see text], it is known that [Formula: see text], the number of Seifert circles in [Formula: see text], equals the braid index [Formula: see text] of [Formula: see text] if [Formula: see text] contains no lone crossings, where a crossing in [Formula: see text] is called a lone crossing if it is the only crossing between two Seifert circles in [Formula: see text]. If [Formula: see text] contains lone crossings, then one can reduce the number of Seifert circles in [Formula: see text] using link-type preserving moves such as the Murasugi–Przytycki operation. Let [Formula: see text] be the maximum number of Seifert circles in [Formula: see text] that can be reduced, then we have [Formula: see text]. On the other hand, if the [Formula: see text]-span of the HOMFLY polynomial of [Formula: see text] satisfies the equality [Formula: see text]-span[Formula: see text], then the Morton–Frank–Williams (MFW) inequality [Formula: see text]-span[Formula: see text] leads us to the simple braid index formula [Formula: see text]. In this paper, we derive explicit formulas for many alternating links based on minimum projections of these links by establishing the equality [Formula: see text]-span[Formula: see text]. Our methods depend on the structures of the link diagrams under consideration and our results lead to explicit braid index formulas that are applicable to a very large class of links, a proper subset of which contains all two bridge links, all alternating pretzel links, and more generally all alternating Montesinos links. The derived braid index formula for an alternating Montesinos link is a function whose inputs are the signs of the crossings in the rational tangles of the Montesinos link. Finally, by applying the now proven Jones Conjecture on the writhe of minimum braids of a link, our results also allow us to obtain explicit formulas of the writhe of minimum braids for the links discussed in this paper from the minimum projections of these links.