Embedding knots and links in an open book III. On the braid index of satellite links

Abstract

Let V⊂S³ be a solid, knotted torus. Through the work of Birman and Menasco [2], the observation has been made that a satellite link L=C[midast ]P⊂V (with companion C, pattern P and essential torus T=∂V) falls into one of two broad categories: reverse string and non-reverse string. These categories are borne of the three embedding types identified in [2] and are distinguished by the existence of, or lack of, a meridional disc D⊂V with an orientation, whose point-intersections with the oriented satellite C[midast ]P are all similarly oriented. If no such D exists, then L is said to be a reverse string satellite.

If C[midast ]P is a non-reverse string satellite, it is known that the braid index b(C[midast ]P) is dependent only on b(C) and certain specified properties of the pattern and not on the choice f∈Z of framing parameter. In [2] it is conjectured that, if C[midast ]_fP is a reverse string satellite (in which the framing parameter is f), the braid index b(C[midast ]_fP) depends on the arc index α(C), properties of the pattern and also the framing. We study this dependence further via established upper and lower bounds for braid index. The upper bound comes from explicit closed braid diagrams of L, for which constructions are shown. The lower bound comes from the Homfly polynomial, via the Morton–Franks–Williams inequality [5, 8]. We will prove the following theorem.