In a
–manifold
, let
be a knot and
be an annulus
which meets
transversely. We define the notion of the pair
being caught by
a surface
in the
exterior of the link
.
For a caught pair
,
we consider the knot
gotten by twisting
times
along
and give a lower bound on the bridge number of
with respect to
Heegaard splittings of
;
as a function of
,
the genus of the splitting, and the catching surface
. As a result, the bridge
number of tends to infinity
with
. In application, we
look at a family of knots
found by Teragaito that live in a small Seifert fiber space
and where each
admits a Dehn surgery
giving
. We show that the
bridge number of
with
respect to any genus-
Heegaard splitting of
tends to infinity with
.
This contrasts with other work of the authors as well as with the conjectured
picture for knots in lens spaces that admit Dehn surgeries giving
.
Keywords
Dehn surgery, bridge number, 3–manifolds, knot theory