Choose any oriented link type
and closed braid representatives
of
,
where
has minimal braid index among all closed braid representatives of
. The
main result of this paper is a ‘Markov theorem without stabilization’. It asserts
that there is a complexity function and a finite set of ‘templates’ such that
(possibly after initial complexity-reducing modifications in the choice of
and
which replace them
with closed braids
)
there is a sequence of closed braid representatives
such that each
passage
is
strictly complexity reducing and non-increasing on braid index. The templates which define
the passages
include 3 familiar ones, the destabilization, exchange move
and flype templates, and in addition, for each braid index
a finite set
of new ones. The number of
templates in
is a non-decreasing
function of
. We give
examples of members of
,
but not a complete listing. There are applications to contact geometry, which will be
given in a separate paper.
