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.